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AppliedMechanics



Chapter-1

Introduction

Structure of this unit

Mechanics and applied Mechanics

Learning Objectives

1. Concept of mechanics and applied mechanics

2. Explanation of Mechanics and applied Mechanics, its importance and necessity

3. giving suitable examples on bodies at rest and in motion

4. explanation of branches of this subject5. Concept of rigid bodies

1.1 Concept of mechanics and applied mechanics

Mechanics is the branch of science concerned with the behavior of physical bodies when

subjected to forces or displacements, and the subsequent effects of the bodies on their

environment. The scientific discipline has its origins in Ancient Greece with the writings of

Aristotle and Archimedes. During the early modern period, scientists such as Galileo, Kepler,

and especially Newton, laid the foundation for what is now known as classical mechanics. It is a branch of classical physics that deals with the particles that are moving either with less velocity

or that are at rest. It can also be defined as a branch of science which deals with the motion and

force of the particular object.

Applied mechanics is a branch of the physical sciences and the practical application of

mechanics. Applied mechanics examines the response of bodies (solids and fluids) or systems of

bodies to external forces. Some examples of mechanical systems include the flow of a liquid

under pressure, the fracture of a solid from an applied force, or the vibration of an ear in response

to sound. A practitioner of the discipline is known as a mechanician.

Applied mechanics, as its name suggests, bridges the gap between physical theory and its

application to technology. As such, applied mechanics is used in many fields of engineering,

especially mechanical engineering. In this context, it is commonly referred to as engineering

mechanics. Much of modern engineering mechanics is based on Isaac Newton's laws of motion

while the modern practice of their application can be traced back to Stephen Timoshenko, who is

said to be the father of modern engineering mechanics.



Within the theoretical sciences, applied mechanics is useful in formulating new ideas and

theories, discovering and interpreting phenomena, and developing experimental and

computational tools. In the application of the natural sciences, mechanics was said to be

complemented by thermodynamics by physical chemists Gilbert N. Lewis and Merle Randall,

the study of heat and more generally energy, and electro mechanics, the study of electricity and

magnetism.

1.2 Explanation of Mechanics and applied Mechanics, its importance and necessity

1.2.1 Types of mechanical bodies

Thus the often-used term body needs to stand for a wide assortment of objects, including

particles, projectiles, spacecraft, stars, parts of machinery, parts of solids, parts of fluids (gases

and liquids), etc.

Other distinctions between the various sub-disciplines of mechanics, concern the nature of the

bodies being described. Particles are bodies with little (known) internal structure, treated as

mathematical points in classical mechanics. Rigid bodies have size and shape, but retain a

simplicity close to that of the particle, adding just a few so-called degrees of freedom, such as

orientation in space.

Otherwise, bodies may be semi-rigid, i.e. elastic, or non-rigid, i.e. fluid. These subjects have both

classical and quantum divisions of study.

For instance, the motion of a spacecraft, regarding its orbit and attitude (rotation), is described by

the relativistic theory of classical mechanics, while the analogous movements of an atomic

nucleus are described by quantum mechanics.

1.2.2 Sub-disciplines in mechanics

The following are two lists of various subjects that are studied in mechanics.

Note that there is also the "theory of fields" which constitutes a separate discipline in physics,

formally treated as distinct from mechanics, whether classical fields or quantum fields. But in

actual practice, subjects belonging to mechanics and fields are closely interwoven. Thus, for

instance, forces that act on particles are frequently derived from fields (electromagnetic or

gravitational), and particles generate fields by acting as sources. In fact, in quantum mechanics,

particles themselves are fields, as described theoretically by the wave function.

1.2.2.1 Classical mechanics



Prof. Walter Lewin explains Newton's law of gravitation in MIT course 8.01

The following are described as forming Classical mechanics:

• Newtonian mechanics, the original theory of motion (kinematics) and forces (dynamics)

• Hamiltonian mechanics, a theoretical formalism, based on the principle of conservation

of energy

• Lagrangian mechanics, another theoretical formalism, based on the principle of the least

action

• Celestial mechanics, the motion of bodies in space: planets, comets, stars, galaxies, etc.

• Astrodynamics, spacecraft navigation, etc.

• Solid mechanics, elasticity, the properties of deformable bodies.

• Fracture mechanics

• Acoustics, sound ( = density variation propagation) in solids, fluids and gases.

• Statics, semi-rigid bodies in mechanical equilibrium

• Fluid mechanics, the motion of fluids

• Soil mechanics, mechanical behavior of soils

• Continuum mechanics, mechanics of continua (both solid and fluid)

• Hydraulics, mechanical properties of liquids

• Fluid statics, liquids in equilibrium

• Applied mechanics, or Engineering mechanics

• Biomechanics, solids, fluids, etc. in biology

• Biophysics, physical processes in living organisms

• Statistical mechanics, assemblies of particles too large to be described in a deterministicway

• Relativistic or Einsteinian mechanics, universal gravitation

1.2.2.2 Quantum mechanics



The following are categorized as being part of Quantum mechanics:

• Particle physics, the motion, structure, and reactions of particles

• Nuclear physics, the motion, structure, and reactions of nuclei

• Condensed matter physics, quantum gases, solids, liquids, etc.

• Quantum statistical mechanics, large assemblies of particles

1.2.2.3 Applied mechanics in practice

The advances and research in Applied Mechanics has wide application in many departments.

Some of the departments that put the subject into practice are Civil Engineering, Mechanical

Engineering, Construction Engineering, Materials Science and Engineering, Aerospace

Engineering, Chemical Engineering, Electrical Engineering, Nuclear Engineering, Structural

engineering and Bioengineering.

1.2.2.4 Applied mechanics in engineering

Typically, engineering mechanics is used to analyze and predict the acceleration and deformation

(both elastic and plastic) of objects under known forces (also called loads) or stresses.

When treated as an area of study within a larger engineering curriculum, engineering mechanics

can be subdivided into

• Statics, the study of non-moving bodies under known loads

• Dynamics (or kinetics), the study of how forces affect moving bodies

• Mechanics of materials or strength of materials, the study of how different materials

deform under various types of stress

• Deformation mechanics, the study of deformations typically in the elastic range

• Fluid mechanics, the study of how fluids react to forces. Note that fluid mechanics can

be further split into fluid statics and fluid dynamics, and is itself a subdiscipline of

continuum mechanics. The application of fluid mechanics in engineering is called

hydraulics.

• Continuum mechanics is a method of applying mechanics that assumes that all objects

are continuous. It is contrasted by discrete mechanics.

Major topics of applied mechanics

• Acoustics



• Analytical mechanics

• Computational mechanics

• Contact mechanics

• Continuum mechanics

• Dynamics (mechanics)

• Elasticity (physics)

• Experimental mechanics

• Fatigue (material)

• Finite element method

• Fluid mechanics


• Mechanics of materials

• Mechanics of structures

• Rotordynamics

• Solid mechanics• Soil mechanics

• Stress waves

• Viscoelasticity

Examples of applications

• Earthquake engineering

1.3 giving suitable examples on bodies at rest and in motion

A guy/girl driving a car is a good example of this question because when guy/girl is driving

his/her body is in rest but also in motion because of the movement of car. The tendency of a

body at rest to preserve its state of rest is called inertia of rest.

Example:

a) A passenger standing or sitting in a bus or a train falls back, when it starts suddenly. This is

because the lower part of the body moves forward with the bus while the upper part of the body

continues to be at rest due to inertia.

b) A coin is placed over a card on a tumbler. If we flip the card quickly away with a finger the

coin falls into the tumbler as the card moves away quickly.

Inertia of Motion:

The tendency of a moving body to preserve its motion in a straight line with uniform velocity is



known as inertia of motion.

Example:

a) A person in a moving vehicle falls forward when it suddenly stops. This is because the lower

part of the person comes to rest with the vehicle while the upper part of the body moves forward

due to inertia of motion.

b) Before taking a long jump an athlete runs some distance so that the inertia of motion might

help him to jump longer distance.

1.4 explanation of branches of this subject

Domains of major fields of physics

1.4.1 Classical mechanics

Classical mechanics is a model of the physics of forces acting upon bodies. It is often referred to

as "Newtonian mechanics" after Isaac Newton and his laws of motion. It is introduced by M.A

Hassan Younis.

classical mechanics came first, while quantum mechanics is a comparatively recent invention.

Classical mechanics originated with Isaac Newton's laws of motion in Principia Mathematica;Quantum Mechanics was discovered in 1925. Both are commonly held to constitute the most

certain knowledge that exists about physical nature. Classical mechanics has especially often

been viewed as a model for other so-called exact sciences. Essential in this respect is the

relentless use of mathematics in theories, as well as the decisive role played by experiment in

generating and testing them.



The following are described as forming Classical mechanics:

• Newtonian mechanics, the original theory of motion (kinematics) and forces (dynamics)

• Hamiltonian mechanics, a theoretical formalism, based on the principle of conservation

of energy

• Lagrangian mechanics, another theoretical formalism, based on the principle of the least

action

• Celestial mechanics, the motion of bodies in space: planets, comets, stars, galaxies, etc.

• Astrodynamics, spacecraft navigation, etc.

• Solid mechanics, elasticity, the properties of deformable bodies.


• Acoustics, sound ( = density variation propagation) in solids, fluids and gases.

• Statics, semi-rigid bodies in mechanical equilibrium

• Fluid mechanics, the motion of fluids

• Soil mechanics, mechanical behavior of soils

• Continuum mechanics, mechanics of continua (both solid and fluid)

• Hydraulics, mechanical properties of liquids

• Fluid statics, liquids in equilibrium

• Applied mechanics, or Engineering mechanics

• Biomechanics, solids, fluids, etc. in biology

• Biophysics, physical processes in living organisms

• Statistical mechanics, assemblies of particles too large to be described in a deterministic

way

• Relativistic or Einsteinian mechanics, universal gravitation

1.4.2 Thermodynamics and statistical mechanics

The first chapter of The Feynman Lectures on Physics is about the existence of atoms, which

Feynman considered to be the most compact statement of physics, from which science could

easily result even if all other knowledge was lost.[1]

By modeling matter as collections of hard

spheres, it is possible to describe the kinetic theory of gases, upon which classical

thermodynamics is based.

Thermodynamics studies the effects of changes in temperature, pressure, and volume on physical

systems on the macroscopic scale, and the transfer of energy as heat.[2][3]

Historically,

thermodynamics developed out of the desire to increase the efficiency of early steam engines.[4]

The starting point for most thermodynamic considerations is the laws of thermodynamics, which

postulate that energy can be exchanged between physical systems as heat or work.[5]

They also



postulate the existence of a quantity named entropy, which can be defined for any system.[6]

In

thermodynamics, interactions between large ensembles of objects are studied and categorized.

Central to this are the concepts of system and surroundings. A system is composed of particles,

whose average motions define its properties, which in turn are related to one another through

equations of state. Properties can be combined to express internal energy and thermodynamic

potentials, which are useful for determining conditions for equilibrium and spontaneous

processes.

1.4.3 Electromagnetism

1.4.4 Relativity

The special theory of relativity enjoys a relationship with electromagnetism and mechanics; that

is, the principle of relativity and the principle of stationary action in mechanics can be used to

derive Maxwell's equations, and vice versa.

The theory of special relativity was proposed in 1905 by Albert Einstein in his article "On the

Electrodynamics of Moving Bodies". The title of the article refers to the fact that special

relativity resolves an inconsistency between Maxwell's equations and classical mechanics. The

theory is based on two postulates:

(1) that the mathematical forms of the laws of physics are invariant in all inertial systems; and(2) that the speed of light in a vacuum is constant and independent of the source or observer.

Reconciling the two postulates requires a unification of space and time into the frame-dependent

concept of spacetime.

General relativity is the geometrical theory of gravitation published by Albert Einstein in

1915/16. It unifies special relativity, Newton's law of universal gravitation, and the insight that

Maxwell's equations of electromagnetism



gravitation can be described by the curvature of space and time. In general relativity, the

curvature of spacetime is produced by the energy of matter and radiation.

1.4.5 Quantum mechanics

Quantum mechanics is of a wider scope, as it encompasses classical mechanics as a sub-discipline which applies under certain restricted circumstances. According to the correspondence

principle, there is no contradiction or conflict between the two subjects, each simply pertains to

specific situations. The correspondence principle states that the behavior of systems described by

quantum theories reproduces classical physics in the limit of large quantum numbers. Quantum

mechanics has superseded classical mechanics at the foundational level and is indispensable for

the explanation and prediction of processes at molecular and (sub)atomic level. However, for

macroscopic processes classical mechanics is able to solve problems which are unmanageably

difficult in quantum mechanics and hence remains useful and well used. Modern descriptions of

such behavior begin with a careful definition of such quantities as displacement (distance

moved), time, velocity, acceleration, mass, and force. Until about 400 years ago, however,

motion was explained from a very different point of view. For example, following the ideas of

Greek philosopher and scientist Aristotle, scientists reasoned that a cannonball falls down

because its natural position is in the Earth; the sun, the moon, and the stars travel in circles

around the earth because it is the nature of heavenly objects to travel in perfect circles.

The Italian physicist and astronomer Galileo brought together the ideas of other great thinkers of

his time and began to analyze motion in terms of distance traveled from some starting position

and the time that it took. He showed that the speed of falling objects increases steadily during the

time of their fall. This acceleration is the same for heavy objects as for light ones, provided airfriction (air resistance) is discounted. The English mathematician and physicist Isaac Newton

improved this analysis by defining force and mass and relating these to acceleration. For objects

traveling at speeds close to the speed of light, Newton’s laws were superseded by Albert

Einstein’s theory of relativity. For atomic and subatomic particles, Newton’s laws were

superseded by quantum theory. For everyday phenomena, however, Newton’s three laws of

motion remain the cornerstone of dynamics, which is the study of what causes motion



The first few hydrogen atom electron orbitals shown as cross-sections with color-coded

probability density

1.4.5.1 Schrödinger equation of quantum mechanics

Quantum mechanics is the branch of physics treating atomic and subatomic systems and their

interaction with radiation. It is based on the observation that all forms of energy are released in

discrete units or bundles called "quanta". Remarkably, quantum theory typically permits only

probable or statistical calculation of the observed features of subatomic particles, understood in

terms of wave functions. The Schrödinger equation plays the role in quantum mechanics that

Newton's laws and conservation of energy serve in classical mechanics—i.e., it predicts the

future behavior of a dynamic system—and is a wave equation that is used to solve for wave

functions.

For example, the light, or electromagnetic radiation emitted or absorbed by an atom has only

certain frequencies (or wavelengths), as can be seen from the line spectrum associated with the

chemical element represented by that atom. The quantum theory shows that those frequencies

correspond to definite energies of the light quanta, or photons, and result from the fact that the

electrons of the atom can have only certain allowed energy values, or levels; when an electron

changes from one allowed level to another, a quantum of energy is emitted or absorbed whose

frequency is directly proportional to the energy difference between the two levels. The

photoelectric effect further confirmed the quantization of light.

In 1924, Louis de Broglie proposed that not only do light waves sometimes exhibit particle-like

properties, but particles may also exhibit wave-like properties. Two different formulations of

quantum mechanics were presented following de Broglie's suggestion. The wave mechanics of

Erwin Schrödinger (1926) involves the use of a mathematical entity, the wave function, which is



related to the probability of finding a particle at a given point in space. The matrix mechanics of

Werner Heisenberg (1925) makes no mention of wave functions or similar concepts but was

shown to be mathematically equivalent to Schrödinger's theory. A particularly important

discovery of the quantum theory is the uncertainty principle, enunciated by Heisenberg in 1927,

which places an absolute theoretical limit on the accuracy of certain measurements; as a result,

the assumption by earlier scientists that the physical state of a system could be measured exactly

and used to predict future states had to be abandoned. Quantum mechanics was combined with

the theory of relativity in the formulation of Paul Dirac. Other developments include quantum

statistics, quantum electrodynamics, concerned with interactions between charged particles and

electromagnetic fields; and its generalization, quantum field theory.

The following are categorized as being part of Quantum mechanics:

• Particle physics, the motion, structure, and reactions of particles

• Nuclear physics, the motion, structure, and reactions of nuclei

• Condensed matter physics, quantum gases, solids, liquids, etc.

• Quantum statistical mechanics, large assemblies of particles

Branches of mechanics



1.5 Concept of rigid bodies

In physics, a rigid body is an idealization of a solid body in which deformation is neglected. In

other words, the distance between any two given points of a rigid body remains constant in time

regardless of external forces exerted on it. Even though such an object cannot physically exist

due to relativity, objects can normally be assumed to be perfectly rigid if they are not moving

near the speed of light.

In classical mechanics a rigid body is usually considered as a continuous mass distribution, while

in quantum mechanics a rigid body is usually thought of as a collection of point masses. For

instance, in quantum mechanics molecules (consisting of the point masses: electrons and nuclei)

are often seen as rigid bodies

1.5.1 Kinematics

1.5.1.1 Linear and angular position

The position of a rigid body is the position of all the particles of which it is composed. To

simplify the description of this position, we exploit the property that the body is rigid, namely

that all its particles maintain the same distance relative to each other. If the body is rigid, it is

sufficient to describe the position of at least three non-collinear particles. This makes it possible

to reconstruct the position of all the other particles, provided that their time-invariant position

relative to the three selected particles is known. However, typically a different, mathematically

more convenient, but equivalent approach is used. The position of the whole body is represented

by:

1. the linear position or position of the body, namely the position of one of the particles of

the body, specifically chosen as a reference point (typically coinciding with the center of

mass or centroid of the body), together with

2. the angular position (also known as orientation, or attitude) of the body.

Thus, the position of a rigid body has two components: linear and angular, respectively. The

same is true for other kinematic and kinetic quantities describing the motion of a rigid body, such

as linear and angular velocity, acceleration, momentum, impulse, and kinetic energy.

The linear position can be represented by a vector with its tail at an arbitrary reference point in

space (the origin of a chosen coordinate system) and its tip at an arbitrary point of interest on the

rigid body, typically coinciding with its center of mass or centroid. This reference point may

define the origin of a coordinate system fixed to the body.



There are several ways to numerically describe the orientation of a rigid body, including a set of

three Euler angles, a quaternion, or a direction cosine matrix (also referred to as a rotation

matrix). All these methods actually define the orientation of a basis set (or coordinate system)

which has a fixed orientation relative to the body (i.e. rotates together with the body), relative to

another basis set (or coordinate system), from which the motion of the rigid body is observed.

For instance, a basis set with fixed orientation relative to an airplane can be defined as a set of

three orthogonal unit vectors b1, b2, b3, such that b1 is parallel to the chord line of the wing and

directed forward, b2 is normal to the plane of symmetry and directed rightward, and b 3 is given

by the cross product .

In general, when a rigid body moves, both its position and orientation vary with time. In the

kinematic sense, these changes are referred to as translation and rotation, respectively. Indeed,

the position of a rigid body can be viewed as a hypothetic translation and rotation (roto-

translation) of the body starting from a hypothetic reference position (not necessarily coinciding

with a position actually taken by the body during its motion).

1.5.1.2 Linear and angular velocity

Velocity (also called linear velocity) and angular velocity are measured with respect to a frame

of reference.

The linear velocity of a rigid body is a vector quantity, equal to the time rate of change of its

linear position. Thus, it is the velocity of a reference point fixed to the body. During purely

translational motion (motion with no rotation), all points on a rigid body move with the same

velocity. However, when motion involves rotation, the instantaneous velocity of any two pointson the body will generally not be the same. Two points of a rotating body will have the same

instantaneous velocity only if they happen to lie on an axis parallel to the instantaneous axis of

rotation.

Angular velocity is a vector quantity that describes the angular speed at which the orientation of

the rigid body is changing and the instantaneous axis about which it is rotating (the existence of

this instantaneous axis is guaranteed by the Euler's rotation theorem). All points on a rigid body

experience the same angular velocity at all times. During purely rotational motion, all points on

the body change position except for those lying on the instantaneous axis of rotation. The

relationship between orientation and angular velocity is not directly analogous to the relationship between position and velocity. Angular velocity is not the time rate of change of orientation,

because there is no such concept as an orientation vector that can be differentiated to obtain the

angular velocity.



1.5.2 Kinematical equations

1.5.2.1 Addition theorem for angular velocity

The angular velocity of a rigid body B in a reference frame N is equal to the sum of the angular

velocity of a rigid body D in N and the angular velocity of B with respect to D:

.

In this case, rigid bodies and reference frames are indistinguishable and completely

interchangeable.

1.5.2.2 Addition theorem for position

For any set of three points P, Q, and R, the position vector from P to R is the sum of the position

vector from P to Q and the position vector from Q to R:

.

1.5.2.3 Mathematical definition of velocity

The velocity of point P in reference frame N is defined using the time derivative in N of the

position vector from O to P:

where O is any arbitrary point fixed in reference frame N, and the N to the left of the d/dt

operator indicates that the derivative is taken in reference frame N. The result is independent of

the selection of O so long as O is fixed in N.

1.5.2.4 Mathematical definition of acceleration

The acceleration of point P in reference frame N is defined using the time derivative in N of its

velocity:



1.5.2.5 Velocity of two points fixed on a rigid body

For two points P and Q that are fixed on a rigid body B, where B has an angular velocity in

the reference frame N, the velocity of Q in N can be expressed as a function of the velocity of Pin N:

.

1.5.2.6 Acceleration of two points fixed on a rigid body

By differentiating the equation for the Velocity of two points fixed on a rigid body in N with

respect to time, the acceleration in reference frame N of a point Q fixed on a rigid body B can be

expressed as

where is the angular acceleration of B in the reference frame N.

1.5.2.7 Velocity of one point moving on a rigid body

If the point R is moving in rigid body B while B moves in reference frame N, then the velocity of

R in N is

.

where Q is the point fixed in B that is instantaneously coincident with R at the instant of interest.

This relation is often combined with the relation for the Velocity of two points fixed on a rigid

body.

1.5.2.8 Acceleration of one point moving on a rigid body

The acceleration in reference frame N of the point R moving in body B while B is moving in

frame N is given by



where Q is the point fixed in B that instantaneously coincident with R at the instant of interest.

This equation is often combined with Acceleration of two points fixed on a rigid body.

1.5.2.9 Other quantities

If C is the origin of a local coordinate system L, attached to the body,

• the spatial or twist acceleration of a rigid body is defined as the spatial acceleration of C

(as opposed to material acceleration above);

where

• represents the position of the point/particle with respect to the reference point of the

body in terms of the local coordinate system L (the rigidity of the body means that this

does not depend on time)

• is the orientation matrix, an orthogonal matrix with determinant 1, representing the

orientation (angular position) of the local coordinate system L, with respect to the

arbitrary reference orientation of another coordinate system G. Think of this matrix as

three orthogonal unit vectors, one in each column, which define the orientation of theaxes of L with respect to G.

• represents the angular velocity of the rigid body

• represents the total velocity of the point/particle

• represents the total acceleration of the point/particle

• represents the angular acceleration of the rigid body

• represents the spatial acceleration of the point/particle

• represents the spatial acceleration of the rigid body (i.e. the spatial acceleration of

the origin of L)

In 2D the angular velocity is a scalar, and matrix A(t) simply represents a rotation in the xy-plane

by an angle which is the integral of the angular velocity over time.

Vehicles, walking people, etc. usually rotate according to changes in the direction of the

velocity: they move forward with respect to their own orientation. Then, if the body follows a



closed orbit in a plane, the angular velocity integrated over a time interval in which the orbit is

completed once, is an integer times 360°. This integer is the winding number with respect to the

origin of the velocity. Compare the amount of rotation associated with the vertices of a polygon.

1.5.3 Kinetics

Any point that is rigidly connected to the body can be used as reference point (origin of

coordinate system L) to describe the linear motion of the body (the linear position, velocity and

acceleration vectors depend on the choice).

However, depending on the application, a convenient choice may be:

• the center of mass of the whole system, which generally has the simplest motion for a

body moving freely in space;

• a point such that the translational motion is zero or simplified, e.g. on an axle or hinge, at

the center of a ball and socket joint, etc.

When the center of mass is used as reference point:

• The (linear) momentum is independent of the rotational motion. At any time it is equal to

the total mass of the rigid body times the translational velocity.

• The angular momentum with respect to the center of mass is the same as without

translation: at any time it is equal to the inertia tensor times the angular velocity. When

the angular velocity is expressed with respect to a coordinate system coinciding with the

principal axes of the body, each component of the angular momentum is a product of a

moment of inertia (a principal value of the inertia tensor) times the corresponding

component of the angular velocity; the torque is the inertia tensor times the angular

acceleration.

• Possible motions in the absence of external forces are translation with constant velocity,

steady rotation about a fixed principal axis, and also torque-free precession.

• The net external force on the rigid body is always equal to the total mass times the

translational acceleration (i.e., Newton's second law holds for the translational motion,

even when the net external torque is nonzero, and/or the body rotates).

• The total kinetic energy is simply the sum of translational and rotational energy.

1.5.3.1 Geometry

Two rigid bodies are said to be different (not copies) if there is no proper rotation from one to the

other. A rigid body is called chiral if its mirror image is different in that sense, i.e., if it has either

no symmetry or its symmetry group contains only proper rotations. In the opposite case an object

is called achiral: the mirror image is a copy, not a different object. Such an object may have a



symmetry plane, but not necessarily: there may also be a plane of reflection with respect to

which the image of the object is a rotated version. The latter applies for S2n, of which the case n

= 1 is inversion symmetry.

For a (rigid) rectangular transparent sheet, inversion symmetry corresponds to having on one side

an image without rotational symmetry and on the other side an image such that what shines

through is the image at the top side, upside down. We can distinguish two cases:

• the sheet surface with the image is not symmetric - in this case the two sides are different,

but the mirror image of the object is the same, after a rotation by 180° about the axis

perpendicular to the mirror plane.

• the sheet surface with the image has a symmetry axis - in this case the two sides are the

same, and the mirror image of the object is also the same, again after a rotation by 180°

about the axis perpendicular to the mirror plane.

A sheet with a through and through image is achiral. We can distinguish again two cases:

• the sheet surface with the image has no symmetry axis - the two sides are different

• the sheet surface with the image has a symmetry axis - the two sides are the same

Review Questions

1. Describe the Concept of mechanics and applied mechanics.

2. Explain the Explanation of Mechanics and applied Mechanics.

3. Define Mechanics and applied Mechanics importance and necessity.

4. Give suitable examples on bodies at rest and in motion

5. Describe explanation of branches of this subject.

6. What do you mean by Concept of rigid bodies.



Chapter 2

Laws of Forces


Forces

Learning Objectives

1. Force and its effects, units and measurement of force

2. characteristics of force vector representation

3. Bow’s notation

4. Types of forces, action and reaction, tension, thrust and shear force.

5. Force systems : Coplaner and space force systems. Coplaner concurrent and

nonconcurrent forces.

6. Free body diagrams

7. Resultant and components concept of equlibirium

8. Parallelogram law of forces.

9. Equilibirium of two forces

10. superposition and transmissibility of forces

11. Newton’s third law12. triangle of forces

13. different cases of concurrent coplanar

14. two force systems

15. extension of parallelogram law and triangle law to many forces acting at one point

polygon law of forces

16. method of resolution into orthogonal components for finding the resultant

17. graphical methods

18. special case of three concurrent, coplanar forces

19. Lami’s theorem

2.1 Force and its effects



A force is any influence that causes an object to undergo a certain change, either concerning its

movement, direction, or geometrical construction. In other words, a force can cause an object

with mass to change its velocity (which includes to begin moving from a state of rest), i.e., to

accelerate, or a flexible object to deform, or both. Force can also be described by intuitive

concepts such as a push or a pull. A force has both magnitude and direction, making it a vector

quantity. It is measured in the SI unit of newtons and represented by the symbol F.

The original form of Newton's second law states that the net force acting upon an object is equal

to the rate at which its momentum changes with time. If the mass of the object is constant, this

law implies that the acceleration of an object is directly proportional to the net force acting on

the object, is in the direction of the net force, and is inversely proportional to the mass of the

object. As a formula, this is expressed as:

where the arrows imply a vector quantity possessing both magnitude and direction.

Related concepts to force include: thrust, which increases the velocity of an object; drag, which

decreases the velocity of an object; and torque which produces changes in rotational speed of an

object. In an extended body, each part usually applies forces on the adjacent parts; the

distribution of such forces through the body is the so-called mechanical stress. Pressure is a

simple type of stress. Stress usually causes deformation of solid materials, or flow in fluids.

2.1.1 units and measurement of force

The SI unit used to measure force is the newton (symbol N), which is equivalent to kg·m·s−2.

The earlier CGS unit is the dyne. The relationship F=m·a can be used with either of these. In

Imperial engineering units, if F is measured in "pounds force" or "lbf", and a in feet per second

squared, then m must be measured in slugs. Similarly, if mass is measured in pounds mass, and a

in feet per second squared, the force must be measured in poundals. The units of slugs and

poundals are specifically designed to avoid a constant of proportionality in this equation.

A more general form F=k·m·a is needed if consistent units are not used. Here, the constant k is a

conversion factor dependent upon the units being used.

When the standard 'g' (an acceleration of 9.80665 m/s²) is used to define pounds force, the mass

in pounds is numerically equal to the weight in pounds force. However, even at sea level on

Earth, the actual acceleration of free fall is quite variable, over 0.53% more at the poles than at

the equator. Thus, a mass of 1.0000 lb at sea level at the equator exerts a force due to gravity of



0.9973 lbf, whereas a mass of 1.000 lb at sea level at the poles exerts a force due to gravity of

1.0026 lbf. The normal average sea level acceleration on Earth (World Gravity Formula 1980) is

9.79764 m/s², so on average at sea level on Earth, 1.0000 lb will exerts a force of 0.9991 lbf.

Force is a quantity capable of changing the size, shape, or motion of an object. It is a vector

quantity and, as such, it has both direction and magnitude. In the SI system, the magnitude of a

force is measured in units called newtons, and in pounds in the British/American system. If a

body is in motion, the energy of that motion can be quantified as the momentum of the object,

the product of its mass and its velocity. If a body is free to move, the action of a force willchange the velocity of the body.

There are four basic forces in nature: gravitational, magnetic, strong nuclear, and weak nuclear

forces. The weakest of the four is the gravitational force. It is also the easiest to observe, because

it acts on all matter and it is always attractive, while having an infinite range. Its attraction

decreases with distance, but is always measurable. Therefore, positional "equilibrium" of a body

can only be achieved when gravitational pull is balanced by another force, such as the upward

force exerted on our feet by the earth's surface.



Figure Atmospheric Reference Gauge

Pressure is the ratio between a force acting on a surface and the area of that surface. Pressure is

measured in units of force divided by area: pounds per square inch (psi) or, in the SI system,

newtons per square meter, or pascals. When an external stress (pressure) is applied to an object

with the intent to cause a reduction in its volume, this process is called compression. Most

liquids and solids are practically incompressible, while gases are not.

The First Gas Law, called Boyle's law, states that the pressure and volume of a gas are inversely proportional to one another: PV = k, where P is pressure, V is volume and k is a constant o

proportionality. The Second Gas Law, Charles' Law, states that the volume of an enclosed gas is

directly proportional to its temperature: V = kT, where T is its absolute temperature. And,

according to the Third Gas Law, the pressure of a gas is directly proportional to its absolute

temperature: P = kT.

Figure Flexible Load-Cell Connections

Combining these three relationships yields the ideal gas law: PV = kT. This approximate

relationship holds true for many gases at relatively low pressures (not too close to the point

where liquification occurs) and high temperatures (not too close to the point where condensation

is imminent).

2.1.2 characteristics of force vector representation

Forces and vectors share three major characteristics:

1. Magnitude

2. Direction

3. Location



2.2 Bow’s notation

In the previous illustrations, the forces have been identified as F1, F

2, R, etc. Another system of

identifying forces, called Bow's notation, is helpful in solving force problems. In the space

diagram, a boldface capital letter, A, B, C, etc., is placed in the space between two forces and theforce is referred to by the two boldface capital letters in the adjoining spaces. The force AB in

the space diagram is represented by the vector ab in the force diagram, the letters a and b being

placed at the beginning and end, respectively, of the vector. The letters in the space diagram are

usually given in alphabetical order and in a clockwise direction.

2.3 Types of forces

2.3.1 Fundamental models

All of the forces in the universe are based on four fundamental interactions. The strong and weak

forces are nuclear forces that act only at very short distances, and are responsible for the

interactions between subatomic particles, including nucleons and compound nuclei. The

electromagnetic force acts between electric charges, and the gravitational force acts between

masses. All other forces in nature derive from these four fundamental interactions. For example,

friction is a manifestation of the electromagnetic force acting between the atoms of two surfaces,

and the Pauli Exclusion Principle, which does not permit atoms to pass through each other.

Similarly, the forces in springs, modeled by Hooke's law, are the result of electromagnetic forces

and the Exclusion Principle acting together to return an object to its equilibrium position.



Centrifugal forces are acceleration forces which arise simply from the acceleration of rotating

frames of reference.

The development of fundamental theories for forces proceeded along the lines of unification of

disparate ideas. For example, Isaac Newton unified the force responsible for objects falling at the

surface of the Earth with the force responsible for the orbits of celestial mechanics in his

universal theory of gravitation. Michael Faraday and James Clerk Maxwell demonstrated that

electric and magnetic forces were unified through one consistent theory of electromagnetism. In

the 20th century, the development of quantum mechanics led to a modern understanding that the

first three fundamental forces (all except gravity) are manifestations of matter (fermions)

interacting by exchanging virtual particles called gauge bosons. This standard model of particle

physics posits a similarity between the forces and led scientists to predict the unification of the

weak and electromagnetic forces in electroweak theory subsequently confirmed by observation.

The complete formulation of the standard model predicts an as yet unobserved Higgs

mechanism, but observations such as neutrino oscillations indicate that the standard model isincomplete. A grand unified theory allowing for the combination of the electroweak interaction

with the strong force is held out as a possibility with candidate theories such as supersymmetry

proposed to accommodate some of the outstanding unsolved problems in physics. Physicists are

still attempting to develop self-consistent unification models that would combine all four

fundamental interactions into a theory of everything. Einstein tried and failed at this endeavor,

but currently the most popular approach to answering this question is string theory.

2.3.1.1 Gravity

An initially stationary object which is allowed to fall freely under gravity drops a distance which

is proportional to the square of the elapsed time. An image was taken 20 flashes per second.

During the first 1/20th of a second the ball drops one unit of distance (here, a unit is about 12

mm); by 2/20ths it has dropped a total of 4 units; by 3/20ths, 9 units and so on.

What we now call gravity was not identified as a universal force until the work of Isaac Newton.

Before Newton, the tendency for objects to fall towards the Earth was not understood to be

related to the motions of celestial objects. Galileo was instrumental in describing the

characteristics of falling objects by determining that the acceleration of every object in free-fallwas constant and independent of the mass of the object. Today, this acceleration due to gravity

towards the surface of the Earth is usually designated as and has a magnitude of about 9.81

meters per second squared (this measurement is taken from sea level and may vary depending on

location), and points toward the center of the Earth. This observation means that the force of

gravity on an object at the Earth's surface is directly proportional to the object's mass. Thus an

object that has a mass of will experience a force:





In free-fall, this force is unopposed and therefore the net force on the object is its weight. For

objects not in free-fall, the force of gravity is opposed by the reactions of their supports. For

example, a person standing on the ground experiences zero net force, since his weight is

balanced by a normal force exerted by the ground.

Newton's contribution to gravitational theory was to unify the motions of heavenly bodies, which

Aristotle had assumed were in a natural state of constant motion, with falling motion observed on

the Earth. He proposed a law of gravity that could account for the celestial motions that had been

described earlier using Kepler's Laws of Planetary Motion.

Newton came to realize that the effects of gravity might be observed in different ways at larger

distances. In particular, Newton determined that the acceleration of the Moon around the Earth

could be ascribed to the same force of gravity if the acceleration due to gravity decreased as an

inverse square law. Further, Newton realized that the acceleration due to gravity is proportional

to the mass of the attracting body. Combining these ideas gives a formula that relates the mass

( ) and the radius ( ) of the Earth to the gravitational acceleration:

where the vector direction is given by , the unit vector directed outward from the center of the

Earth.

In this equation, a dimensional constant is used to describe the relative strength of gravity.

This constant has come to be known as Newton's Universal Gravitation Constant, though itsvalue was unknown in Newton's lifetime. Not until 1798 was Henry Cavendish able to make the

first measurement of using a torsion balance; this was widely reported in the press as a

measurement of the mass of the Earth since knowing could allow one to solve for the Earth's

mass given the above equation. Newton, however, realized that since all celestial bodies

followed the same laws of motion, his law of gravity had to be universal. Succinctly stated,

Newton's Law of Gravitation states that the force on a spherical object of mass due to the

gravitational pull of mass is

where is the distance between the two objects' centers of mass and is the unit vector pointed in

the direction away from the center of the first object toward the center of the second object.

This formula was powerful enough to stand as the basis for all subsequent descriptions of motion

within the solar system until the 20th century. During that time, sophisticated methods of



perturbation analysis were invented to calculate the deviations of orbits due to the influence of

multiple bodies on a planet, moon, comet, or asteroid. The formalism was exact enough to allow

mathematicians to predict the existence of the planet Neptune before it was observed.

It was only the orbit of the planet Mercury that Newton's Law of Gravitation seemed not to fully

explain. Some astrophysicists predicted the existence of another planet (Vulcan) that would

explain the discrepancies; however, despite some early indications, no such planet could be

found. When Albert Einstein finally formulated his theory of general relativity (GR) he turned

his attention to the problem of Mercury's orbit and found that his theory added a correction

which could account for the discrepancy. This was the first time that Newton's Theory of Gravity

had been shown to be less correct than an alternative.

Since then, and so far, general relativity has been acknowledged as the theory which best

explains gravity. In GR, gravitation is not viewed as a force, but rather, objects moving freely in

gravitational fields travel under their own inertia in straight lines through curved space-time –

defined as the shortest space-time path between two space-time events. From the perspective of

the object, all motion occurs as if there were no gravitation whatsoever. It is only when

observing the motion in a global sense that the curvature of space-time can be observed and the

force is inferred from the object's curved path. Thus, the straight line path in space-time is seen

as a curved line in space, and it is called the ballistic trajectory of the object. For example, a

basketball thrown from the ground moves in a parabola, as it is in a uniform gravitational field.

Its space-time trajectory (when the extra ct dimension is added) is almost a straight line, slightly

curved (with the radius of curvature of the order of few light-years). The time derivative of the

changing momentum of the object is what we label as "gravitational force".

2.3.1.2 Electromagnetic forces

The electrostatic force was first described in 1784 by Coulomb as a force which existed

intrinsically between two charges. The properties of the electrostatic force were that it varied as

an inverse square law directed in the radial direction, was both attractive and repulsive (there was

intrinsic polarity), was independent of the mass of the charged objects, and followed the

superposition principle. Coulomb's Law unifies all these observations into one succinct

statement.

Subsequent mathematicians and physicists found the construct of the electric field to be usefulfor determining the electrostatic force on an electric charge at any point in space. The electric

field was based on using a hypothetical "test charge" anywhere in space and then using

Coulomb's Law to determine the electrostatic force.[37]

Thus the electric field anywhere in space

is defined as



where is the magnitude of the hypothetical test charge.

Meanwhile, the Lorentz force of magnetism was discovered to exist between two electric

currents. It has the same mathematical character as Coulomb's Law with the proviso that like

currents attract and unlike currents repel. Similar to the electric field, the magnetic field can be

used to determine the magnetic force on an electric current at any point in space. In this case, the

magnitude of the magnetic field was determined to be

where is the magnitude of the hypothetical test current and is the length of hypothetical wirethrough which the test current flows. The magnetic field exerts a force on all magnets including,

for example, those used in compasses. The fact that the Earth's magnetic field is aligned closely

with the orientation of the Earth's axis causes compass magnets to become oriented because of

the magnetic force pulling on the needle.

Through combining the definition of electric current as the time rate of change of electric charge,

a rule of vector multiplication called Lorentz's Law describes the force on a charge moving in a

magnetic field. The connection between electricity and magnetism allows for the description of a

unified electromagnetic force that acts on a charge. This force can be written as a sum of the

electrostatic force (due to the electric field) and the magnetic force (due to the magnetic field).

Fully stated, this is the law:

where is the electromagnetic force, is the magnitude of the charge of the particle, is the

electric field, is the velocity of the particle which is crossed with the magnetic field ( ).

The origin of electric and magnetic fields would not be fully explained until 1864 when James

Clerk Maxwell unified a number of earlier theories into a set of 20 scalar equations, which were

later reformulated into 4 vector equations by Oliver Heaviside and Josiah Willard Gibbs. These

"Maxwell Equations" fully described the sources of the fields as being stationary and moving

charges, and the interactions of the fields themselves. This led Maxwell to discover that electric

and magnetic fields could be "self-generating" through a wave that traveled at a speed which he

calculated to be the speed of light. This insight united the nascent fields of electromagnetic

theory with optics and led directly to a complete description of the electromagnetic spectrum.



However, attempting to reconcile electromagnetic theory with two observations, the

photoelectric effect, and the nonexistence of the ultraviolet catastrophe, proved troublesome.

Through the work of leading theoretical physicists, a new theory of electromagnetism was

developed using quantum mechanics. This final modification to electromagnetic theory

ultimately led to quantum electrodynamics (or QED), which fully describes all electromagnetic

phenomena as being mediated by wave-particles known as photons. In QED, photons are the

fundamental exchange particle which described all interactions relating to electromagnetism

including the electromagnetic force.

It is a common misconception to ascribe the stiffness and rigidity of solid matter to the repulsion

of like charges under the influence of the electromagnetic force. However, these characteristics

actually result from the Pauli Exclusion Principle. Since electrons are fermions, they cannot

occupy the same quantum mechanical state as other electrons. When the electrons in a material

are densely packed together, there are not enough lower energy quantum mechanical states for

them all, so some of them must be in higher energy states. This means that it takes energy to pack them together. While this effect is manifested macroscopically as a structural force, it is

technically only the result of the existence of a finite set of electron states.

2.3.1.3 Nuclear forces

There are two "nuclear forces" which today are usually described as interactions that take place

in quantum theories of particle physics. The strong nuclear force is the force responsible for the

structural integrity of atomic nuclei while the weak nuclear force is responsible for the decay ofcertain nucleons into leptons and other types of hadrons.

The strong force is today understood to represent the interactions between quarks and gluons as

detailed by the theory of quantum chromodynamics (QCD). The strong force is the fundamental

force mediated by gluons, acting upon quarks, antiquarks, and the gluons themselves. The (aptly

named) strong interaction is the "strongest" of the four fundamental forces.

The strong force only acts directly upon elementary particles. However, a residual of the force is

observed between hadrons (the best known example being the force that acts between nucleons

in atomic nuclei) as the nuclear force. Here the strong force acts indirectly, transmitted as gluonswhich form part of the virtual pi and rho mesons which classically transmit the nuclear force (see

this topic for more). The failure of many searches for free quarks has shown that the elementary

particles affected are not directly observable. This phenomenon is called color confinement.

The weak force is due to the exchange of the heavy W and Z bosons. Its most familiar effect is

beta decay (of neutrons in atomic nuclei) and the associated radioactivity. The word "weak"



derives from the fact that the field strength is some 1013

times less than that of the strong force.

Still, it is stronger than gravity over short distances. A consistent electroweak theory has also

been developed which shows that electromagnetic forces and the weak force are

indistinguishable at a temperatures in excess of approximately 1015

kelvins. Such temperatures

have been probed in modern particle accelerators and show the conditions of the universe in the

early moments of the Big Bang.

2.3.2 Non-fundamental forces

Some forces are consequences of the fundamental ones. In such situations, idealized models can

be utilized to gain physical insight.

2.3.2.1 Normal force

F N represents the normal force exerted on the object.

The normal force is due to repulsive forces of interaction between atoms at close contact. When

their electron clouds overlap, Pauli repulsion (due to fermionic nature of electrons) follows

resulting in the force which acts in a direction normal to the surface interface between two

objects. The normal force, for example, is responsible for the structural integrity of tables and

floors as well as being the force that responds whenever an external force pushes on a solid

object. An example of the normal force in action is the impact force on an object crashing into an

immobile surface.

2.3.2.2 Friction

Friction is a surface force that opposes relative motion. The frictional force is directly related to

the normal force which acts to keep two solid objects separated at the point of contact. There are

two broad classifications of frictional forces: static friction and kinetic friction.

The static friction force ( ) will exactly oppose forces applied to an object parallel to a surface

contact up to the limit specified by the coefficient of static friction ( ) multiplied by the

normal force ( ). In other words the magnitude of the static friction force satisfies the

inequality:



.

The kinetic friction force ( ) is independent of both the forces applied and the movement of

the object. Thus, the magnitude of the force equals:

,

where is the coefficient of kinetic friction. For most surface interfaces, the coefficient of

kinetic friction is less than the coefficient of static friction.[3]

2.3.2.3 Tension

Tension forces can be modeled using ideal strings which are massless, frictionless, unbreakable,

and unstretchable. They can be combined with ideal pulleys which allow ideal strings to switch

physical direction. Ideal strings transmit tension forces instantaneously in action-reaction pairs sothat if two objects are connected by an ideal string, any force directed along the string by the first

object is accompanied by a force directed along the string in the opposite direction by the second

object. By connecting the same string multiple times to the same object through the use of a set-

up that uses movable pulleys, the tension force on a load can be multiplied. For every string that

acts on a load, another factor of the tension force in the string acts on the load. However, even

though such machines allow for an increase in force, there is a corresponding increase in the

length of string that must be displaced in order to move the load. These tandem effects result

ultimately in the conservation of mechanical energy since the work done on the load is the same

no matter how complicated the machine.

2.3.2.4 Elastic force

An elastic force acts to return a spring to its natural length. An ideal spring is taken to be

massless, frictionless, unbreakable, and infinitely stretchable. Such springs exert forces that push

when contracted, or pull when extended, in proportion to the displacement of the spring from its

equilibrium position. This linear relationship was described by Robert Hooke in 1676, for whom

Hooke's law is named. If is the displacement, the force exerted by an ideal spring equals:

where is the spring constant (or force constant), which is particular to the spring. The minus

sign accounts for the tendency of the force to act in opposition to the applied load.



Fk is the force that responds to the load on the spring

2.3.2.5 Continuum mechanics

When the drag force ( ) associated with air resistance becomes equal in magnitude to the force

of gravity on a falling object ( ), the object reaches a state of dynamic equilibrium at terminalvelocity.

Newton's laws and Newtonian mechanics in general were first developed to describe how forces

affect idealized point particles rather than three-dimensional objects. However, in real life,

matter has extended structure and forces that act on one part of an object might affect other parts

of an object. For situations where lattice holding together the atoms in an object is able to flow,



contract, expand, or otherwise change shape, the theories of continuum mechanics describe the

way forces affect the material. For example, in extended fluids, differences in pressure result in

forces being directed along the pressure gradients as follows:

where is the volume of the object in the fluid and is the scalar function that describes the

pressure at all locations in space. Pressure gradients and differentials result in the buoyant force

for fluids suspended in gravitational fields, winds in atmospheric science, and the lift associated

with aerodynamics and flight.

A specific instance of such a force that is associated with dynamic pressure is fluid resistance: a

body force that resists the motion of an object through a fluid due to viscosity. For so-called

"Stokes' drag" the force is approximately proportional to the velocity, but opposite in direction:

where:

is a constant that depends on the properties of the fluid and the dimensions of the object

(usually the cross-sectional area), and

is the velocity of the object.

More formally, forces in continuum mechanics are fully described by a stress-tensor with termsthat are roughly defined as

where is the relevant cross-sectional area for the volume for which the stress-tensor is being

calculated. This formalism includes pressure terms associated with forces that act normal to the

cross-sectional area (the matrix diagonals of the tensor) as well as shear terms associated with

forces that act parallel to the cross-sectional area (the off-diagonal elements). The stress tensor

accounts for forces that cause all deformations including also tensile stresses and compressions.

2.3.2.6 Fictitious forces

There are forces which are frame dependent, meaning that they appear due to the adoption of

non-Newtonian (that is, non-inertial) reference frames. Such forces include the centrifugal force



and the Coriolis force. These forces are considered fictitious because they do not exist in frames

of reference that are not accelerating.

In general relativity, gravity becomes a fictitious force that arises in situations where spacetime

deviates from a flat geometry. As an extension, Kaluza-Klein theory and string theory ascribe

electromagnetism and the other fundamental forces respectively to the curvature of differently

scaled dimensions, which would ultimately imply that all forces are fictitious.

2.4 action and reaction, tension, thrust and shear force

2.4.1 Reaction

The third of Newton's laws of motion of classical mechanics states that forces always occur in

pairs. This is related to the fact that a force results from the interaction of two objects. Everyforce ('action') on one object is accompanied by a 'reaction' on another, of equal magnitude but

opposite direction. The attribution of which of the two forces is action or reaction is arbitrary.

Each of the two forces can be considered the action, the other force is its associated reaction.

2.4.1.1 Examples

Interaction with ground

When something is exerting force on the ground, the ground will push back with equal force in

the opposite direction. In certain fields of applied physics, such as biomechanics, this force by

the ground is called 'ground reaction force'; the force by the object on the ground is viewed as the

'action'.

When someone wants to jump, he or she exerts additional downward force on the ground

('action'). Simultaneously, the ground exerts upward force on the person ('reaction'). If this

upward force is greater than the person's weight, this will result in upward acceleration. Because

these forces are perpendicular to the ground, they are also called normal force.

Likewise, the spinning wheels of a vehicle attempt to slide backward across the ground. If the

ground is not too slippery, this results in a pair of friction forces: the 'action' by the wheel on the

ground in backward direction, and the 'reaction' by the ground on the wheel in forward direction.

This forward force propels the vehicle.

Gravitational forces



The Earth orbits around the Sun because the gravitational force exerted by the Sun on the Earth,

the action, serves as the centripetal force that maintains the planet in the neighborhood of the

Sun. Simultaneously, the Earth exerts a gravitational attraction on the Sun, the reaction, which

has the same amplitude as the action and an opposite direction (in this case, pulling the Sun

towards the Earth). Since the Sun's mass is very much larger than the Earth's, it does not appear

to be reacting to the pull of the Earth, but in fact it does. A correct way of describing the

combined motion of both objects (ignoring all other celestial bodies for the moment) is to say

that they both orbit around the center of mass of the combined system.

Supported mass

Any mass on earth is pulled down by the gravitational force of the earth; this force is also called

its weight. The corresponding 'reaction' is the gravitational force that mass exerts on the planet.

If the object is supported so that it remains at rest, for instance by a cable from which it is

hanging, or by a surface underneath, or by a liquid on which it is floating, there is also a support

force in upward direction (tension force, normal force, buoyant force, respectively). This support

force is an 'equal and opposite' force; we know this not because of Newton's Third Law, but

because the object remains at rest, so that the forces must be balanced.

To this support force there is also a 'reaction': the object pulls down on the supporting cable, or

pushes down on the supporting surface or liquid. In this case, there are therefore four forces of

equal magnitude:

• F1. gravitational force by earth on object (downward)

• F2. gravitational force by object on earth (upward)

• F3. force by support on object (upward)

• F4. force by object on support (downward)

Forces F1 and F2 are equal because of Newton's Third Law; the same is true for forces F3 and

F4. Forces F1 and F3 are only equal if the object is in equilibrium, and no other forces are

applied. This has nothing to do with Newton's Third Law.

Mass on a spring

If a mass is hanging from a spring, the same considerations apply as before. However, if this

system is then perturbed (e.g., the mass is given a slight kick upwards or downwards, say), the

mass starts to oscillate up and down. Because of these accelerations (and subsequent

decelerations), we conclude from Newton's second law that a net force is responsible for the

observed change in velocity. The gravitational force pulling down on the mass is no longer equal



to the upward elastic force of the spring. In the terminology of the previous section, F1 and F3

are no longer equal.

However, it is still true that F1 = F2 and F3 = F4, as this is required by Newton's Third Law.

Causal misinterpretation

The terms 'action' and 'reaction' have the unfortunate suggestion of causality, as if the 'action' is

the cause and 'reaction' is the effect. It is therefore easy to think of the second force as being

there because of the first, and even happening some time after the first. This is incorrect; the

forces are perfectly simultaneous, and are there for the same reason.

When the forces are caused by a person's volition (e.g. a soccer player kicks a ball), this

volitional cause often leads to an asymmetric interpretation, where the force by the player on the

ball is considered the 'action' and the force by the ball on the player, the 'reaction'. But

physically, the situation is symmetric. The forces on ball and player are both explained by their

nearness, which results in a pair of contact forces (ultimately due to electric repulsion). That this

nearness is caused by a decision of the player has no bearing on the physical analysis. As far as

the physics is concerned, the labels 'action' and 'reaction' can be flipped.

'Equal and opposite'

One problem frequently observed by physics educators is that students tend to apply Newton's

Third Law to pairs of 'equal and opposite' forces acting on the same object. This is incorrect; the

Third Law refers to forces on two different objects. For example, a book lying on a table is

subject to a downward gravitational force (exerted by the earth) and to an upward normal force

by the table. Since the book is not accelerating, these forces must be exactly balanced, according

to Newton's First or Second law. They are therefore 'equal and opposite'. However, these forces

are not always equally strong; they will be different if the book is pushed down by a third force,

or if the table is slanted, or if the table-and-book system is in an accelerating elevator. The case

of three or more forces is covered by considering sum of all forces.

A possible cause of this problem is that the Third Law is often stated in an abbreviated form: For

every action there is an equal and opposite reaction, without the details, namely that these forces

act on two different objects. Moreover, there is a causal connection between the weight ofsomething and the normal force: if an object had no weight, it would not experience support

force from the table, and the weight dictates how strong the support force will be. This causal

relationship is not due to the Third Law but to other physical relations in the system.

Centripetal and centrifugal force



Another common mistake is to state that The centrifugal force that an object experiences is the

reaction to the centripetal force on that object.

If an object were simultaneously subject to both a centripetal force and an equal and opposite

centrifugal force, the resultant force would vanish and the object could not experience a circular

motion. The centrifugal force is sometimes called a fictitious force or pseudo force, to

underscore the fact that such a force only appears when calculations or measurements are

conducted in non-inertial reference frames.

2.4.2 Tension

The forces involved in supporting a ball by a rope. Tension is the force of the rope on thescaffold, the force of the rope on the ball, and the balanced forces acting on and produced by

segments of the rope.

In physics, tension is the pulling force exerted by a string, cable, chain, or similar solid object on

another object. It results from the net electrostatic attraction between the particles in a solid when

it is deformed so that the particles are further apart from each other than when at equilibrium,



where this force is balanced by repulsion due to electron shells; as such, it is the pull exerted by a

solid trying to restore its original, more compressed shape. Tension is the opposite of

compression. Slackening is the reduction of tension.

As tension is the magnitude of a force, it is measured in newtons (or sometimes pounds-force)

and is always measured parallel to the string on which it applies. There are two basic possibilities

for systems of objects held by strings: Either acceleration is zero and the system is therefore in

equilibrium, or there is acceleration and therefore a net force is present. Note that a string is

assumed to have negligible mass.

2.4.3 Thrust

Thrust is a reaction force described quantitatively by Newton's second and third laws. When a

system expels or accelerates mass in one direction, the accelerated mass will cause a force ofequal magnitude but opposite direction on that system. The force applied on a surface in a

direction perpendicular or normal to the surface is called thrust.

In mechanical engineering, force orthogonal to the main load (such as in parallel helical gears) is

referred to as thrust.

2.4.3.1 Examples

Forces on an aerofoil cross section

A fixed-wing aircraft generates forward thrust when air is pushed in the direction opposite to

flight. This can be done in several ways including by the spinning blades of a propeller, or a

rotating fan pushing air out from the back of a jet engine, or by ejecting hot gases from a rocket

engine. The forward thrust is proportional to the mass of the airstream multiplied by thedifference in velocity of the airstream. Reverse thrust can be generated to aid braking after

landing by reversing the pitch of variable pitch propeller blades, or using a thrust reverser on a

jet engine. Rotary wing aircraft and thrust vectoring V/STOL aircraft use engine thrust to support

the weight of the aircraft, and vector sum of this thrust fore and aft to control forward speed.

Birds normally achieve thrust during flight by flapping their wings.



A motorboat generates thrust (or reverse thrust) when the propellers are turned to accelerate

water backwards (or forwards). The resulting thrust pushes the boat in the opposite direction to

the sum of the momentum change in the water flowing through the propeller.

A rocket is propelled forward by a thrust force equal in magnitude, but opposite in direction, to

the time-rate of momentum change of the exhaust gas accelerated from the combustion chamber

through the rocket engine nozzle. This is the exhaust velocity with respect to the rocket, times

the time-rate at which the mass is expelled, or in mathematical terms:

where:

• T is the thrust generated (force)

• is the rate of change of mass with respect to time (mass flow rate of exhaust);

• v is the speed of the exhaust gases measured relative to the rocket.

For vertical launch of a rocket the initial thrust must be more than the weight.

Each of the three Space Shuttle Main Engines could produce a thrust of 1.8 MN, and each of the

Space Shuttle's two Solid Rocket Boosters 14.7 MN, together 29.4 MN. Compare with the mass

at lift-off of 2,040,000 kg, hence a weight of 20 MN.

By contrast, the simplified Aid for EVA Rescue (SAFER) has 24 thrusters of 3.56 N each.

In the air-breathing category, the AMT-USA AT-180 jet engine developed for radio-controlled

aircraft produce 90 N (20 lbf) of thrust. The GE90-115B engine fitted on the Boeing 777-300ER,

recognized by the Guinness Book of World Records as the "World's Most Powerful Commercial

Jet Engine," has a thrust of 569 kN (127,900 lbf).

2.4.3.2 Thrust to power

The power needed to generate thrust and the force of the thrust can be related in a non-linear

way. In general, . The proportionality constant varies, and can be solved for a uniform

flow:



Note that these calculations are only valid for when the incoming air is accelerated from a

standstill - for example when hovering.

The inverse of the proportionality constant, the "efficiency" of an otherwise-perfect thruster, is

proportional to the area of the cross section of the propelled volume of fluid ( ) and the density

of the fluid ( ). This helps to explain why moving through water is easier and why aircraft have

much larger propellers than watercraft do.

2.4.3.3 Thrust to propulsive power

A very common question is how to contrast the thrust rating of a jet engine with the power rating

of a piston engine. Such comparison is difficult, as these quantities are not equivalent. A piston

engine does not move the aircraft by itself (the propeller does that), so piston engines are usually

rated by how much power they deliver to the propeller. Except for changes in temperature and

air pressure, this quantity depends basically on the throttle setting.

A jet engine has no propeller, so the propulsive power of a jet engine is determined from its

thrust as follows. Power is the force (F) it takes to move something over some distance (d)

divided by the time (t) it takes to move that distance:

In case of a rocket or a jet aircraft, the force is exactly the thrust produced by the engine. If the

rocket or aircraft is moving at about a constant speed, then distance divided by time is just speed,

so power is thrust times speed:

This formula looks very surprising, but it is correct: the propulsive power (or power available) of

a jet engine increases with its speed. If the speed is zero, then the propulsive power is zero. If a

jet aircraft is at full throttle but is tied to a very strong tree with a very strong chain, then the jet

engine produces no propulsive power. It certainly transfers a lot of power around, but all that is



wasted. Compare that to a piston engine. The combination piston engine–propeller also has a

propulsive power with exactly the same formula, and it will also be zero at zero speed –- but that

is for the engine–propeller set. The engine alone will continue to produce its rated power at a

constant rate, whether the aircraft is moving or not.

Two aircraft tied to a tree

Now, imagine the strong chain is broken, and the jet and the piston aircraft start to move. At low

speeds:

The piston engine will have constant 100% power, and the propeller's thrust will vary with speed

The jet engine will have constant 100% thrust, and the engine's power will vary with speed

This shows why one cannot compare the rated power of a piston engine with the propulsive

power of a jet engine – these are different quantities (even if the name "power" is the same).

There isn't any useful power measurement in a jet engine that compares directly to a piston

engine rated power. However, instead of comparing engine performance, the gross aircraft

performance as complete systems can be compared using first principle definitions of power,

force and work with the requisite considerations of constantly changing effects like drag and the

mass (of the fuel) in both systems. There is of course an implicit relationship between thrust and

their engines. Thrust specific fuel consumption is a useful measure for comparing engines.

2.4.4 Shear force



Shearing forces push in one direction at the top, and the opposite direction at the bottom, causing

shearing deformation.

A crack or tear may develop in a body from parallel shearing forces pushing in opposite

directions at different points of the body. If the forces were aligned and aimed straight into each

other, they would pinch or compress the body, rather than tear or crack it.

Shearing forces are unaligned forces pushing one part of a body in one direction, and another

part the body in the opposite direction. When the forces are aligned into each other, they are

called compression forces. An example is a deck of cards being pushed one way on the top, and

the other at the bottom, causing the cards to slide. Another example is when wind blows at theside of a peaked roof of a home - the side walls experience a force at their top pushing in the

direction of the wind, and their bottom in the opposite direction, from the ground or foundation.

William A. Nash defines shear force thus: "If a plane is passed through a body, a force acting

along this plane is called shear force or shearing force.

2.5 Force systems : Coplaner and space force systems. Coplaner concurrent and

nonconcurrent Forces

2.5.1 Concurrent Force System in Space

The same method used to solve coplanar concurrent force systems is used to solve noncoplanar

concurrent systems. The plane-table (an early surveying instrument) weighs 40 pounds and is

supported by a tripod, the legs of which are pushed into the ground. The force in each leg may be

considered to act along the leg.



Using the free body and the equations given, solve for the magnitude of the force in each leg.



2.6 Free body diagrams

Space diagram represents the sketch of the physical problem. The free body diagram

selects the significant particle or points and draws the force system on that particle or point.

Steps:

1. Imagine the particle to be isolated or cut free from its surroundings. Draw or sketch its

outlined shape.

2. Indicate on this sketch all the forces that act on the particle.

These include active forces - tend to set the particle in motion e.g. from cables and weights

and reactive forces caused by constraints or supports that prevent motion.

3. Label known forces with their magnitudes and directions. use letters to represent

magnitudes and directions of unknown forces.

Assume direction of force which may be corrected later.

The crate below has a weight of 50 kg. Draw a free body diagram of the crate, the cord BD

and the ring at B.



(a) Crate

FD ( force of cord acting on crate)

50 kg (wt. of crate)

(b) Cord BD

FB (force of ring acting on cord)

FD (force of crate acting on cord)

CRATE

B ring C

A

D

45o



(c) Ring

F A (Force of cord BA acting along ring)

FC (force of cord BC acting on ring)

FB (force of cord BD acting on ring)



F F

F BC

AC

o

o AC = =

sin

cos. .............( )

75

75373 1

∑ Fy = 0 i.e. FBC sin 75o - F AC cos 75o - 1962 = 0

F

F

F BC

AC

AC =

+

= +

1962 0 26

0966 20312 0 27 2.

. . . ......( )

From Equations (1) and (2), 3.73 F AC = 2031.2 + 0.27 F AC

F AC = 587 N

From (1), FBC = 3.73 x 587 = 2190 N

2.7 Resultant and components concept of equlibirium

2.7.1 Orthogonal components of forces

The determination of the resultant of three or more forces using strictly the Parallelogram Law in

the form of Equations is somewhat tedious and in the long run almost useless. We need better

tools !!!

Three forces , F1, F2, and F3 are shown acting on a particle A . Also shown is an orthogonal

coordinate system whose axes I labelled x and y. The location of its origin and the alignment of

its axes with the borders of the figure are arbitrary choices of mine. Our task is now to develop a

more efficient way to determine the magnitude and direction of the resultant of the three forces

shown.

The trick we will be employing is the following. We interpret each of the three forces as the

resultant of two forces, one aligned

Figure 3 forces on a particle



with the x- the other with the y-axis as shown in Figure 3.1b for the force F 1.

We call these two new forces the x- and y-component of the force F1.

Their values can be easily calculated if the magnitude (the absolute value) of F1 and its

orientation (the angle α) are known.

Note that the angle α is measured between the positive x-axis and the force in counterclockwise

direction.

Also, depending on the value of the angle α one or both of the components might have a negative

value, indicating that the component is pointing in the direction of the minus x-axis for example.

2.7.2 Determination of resultant of forces

we now replace the force F1 by its x- and y-component and repeat this step for the two other

forces involved. The result is that we have replaced the original three forces by six new forces, of

which three are aligned with the x-axis and three with the y-axis of our coordinate system.

The final step is then to add the three force components in the x-direction (no sweat here, that

would be just adding/subtracting numbers) to get the x-component of the resultant. The y-

component of the resultant is obtained in similar fashion.Formally we write this as :

Figure Force F1 and its components



Once we have these components we can determine the magnitude of the resultant and the angle β

between the resultant and the x-axis :

Equation 3.2d has always two solutions for the angle β. If for given R x and R your calculator

gives β=80° for example then β=110° is a solution as well. But which value is correct, 80 or 110°

?

The answer to this question can be found by looking at the signs (+/-) of the components R x and

R y which inform you in which quadrant of the unit circle your resultant R lies.

2.7.3 Resultant of forces, a sample case

When applying these equations it is extremely important to know about the sign (plus/minus)

conventions which go along with the cos() and sin() function used in the Equations. Of course in

Statics we don't make up our own rules but follow strictly the rules of trigonometry. Here is a

short sample case I would recommend you read carefully. To some of you it might seem silly to

harp on sign conventions. However, in practical engineering applications not observing the

correct sign amounts often to the difference between a well designed structure and a failing

structure with possible loss of human life and/or millions of dollars.

So, here is the problem as depicted in Figure 3.3a. Originally you know only the magnitudes (

400 N , 350 N, 600 N, and 100 N ) and the orientations (angles 50, 70, 30, and 15 degrees) of the

four forces. Our task is to find the resultant of these four forces, that is to find that single force

which has the same action on particle A as the four given forces.

If the labels ( F1, F2, F3, and F4) are not given, you must label them. I furthermore entered already

an x-y coordinate system. If it is not given, you must make a choice. I aligned the x-axis with the

line a-a.



Figure

Find resultant of four forces

After these preliminary steps the real work begins. Here are my calculations for the x-components of the four forces and then the x-component of the resultant :

Three points of interest :

1. The angle used as argument of the cosine function is always determined by going on an

arc from the positive x-axis in counter-clockwise direction towards the force of which

you want to determine the x-component.

2. Two of the forces have negative x-components ( cos(110) is negative as is cos(210) ). A

negative value of the x-component means that the x- component is pointing in the

direction of the minus x-axis.

3. The plus/minus sign of the obtained x-components has to be entered when calculating the

x-component of the resultant. Here R x comes out to be negative itself, meaning that the

combined action of the forces is to pull to the left in minus x-direction (forgetting at the

moment about what happens in the y-direction).



Please, do determine the value of the y-component of the resultant yourself. My results are

displayed in Figure.

Figure

Resultant of four forces

The action of the four original forces becomes now clear. They will pull the body A to the left

and upwards. Also, please check out whether I got the magnitude and angle for the resultant

right.

2.8 Parallelogram law of forces

The parallelogram of forces is a method for solving (or visualizing) the results of applying two

forces to an object.

Figure: Parallelogram construction for adding vectors

When more than two forces are involved, the geometry is no longer parallelogrammatic, but the

same principles apply. Forces, being vectors are observed to obey the laws of vector addition,

and so the overall (resultant) force due to the application of a number of forces can be found

geometrically by drawing vector arrows for each force. For example, see Figure 1. This



construction has the same result as moving F2 so its tail coincides with the head of F1, and taking

the net force as the vector joining the tail of F1 to the head of F2. This procedure can be repeated

to add F3 to the resultant F1 + F2, and so forth

Proof

Figure: Parallelogram of velocity

the parallelogram of velocity

Suppose a particle moves at a uniform rate along a line from A to B (Figure 2) in a given time

(say, one second), while in the same time, the line AB moves uniformly from its position at AB

to a position at DC, remaining parallel to its original orientation throughout. Accounting for both

motions, the particle traces the line AC. Because a displacement in a given time is a measure of

velocity, the length of AB is a measure of the particle's velocity along AB, the length of AD is a

measure of the line's velocity along AD, and the length of AC is a measure of the particle's

velocity along AC. The particle's motion is the same as if it had moved with a single velocity

along AC.[1]

Newton's proof of the parallelogram of force

Suppose two forces act on a particle at the origin (the "tails" of the vectors) of Figure 1. Let the

lengths of the vectors F1 and F2 represent the velocities the two forces could produce in the

particle by acting for a given time, and let the direction of each represent the direction in which

they act. Each force acts independently and will produce its particular velocity whether the other

force acts or not. At the end of the given time, the particle has both velocities. By the above

proof, they are equivalent to a single velocity, Fnet. By Newton's second law, this vector is also a

measure of the force which would produce that velocity, thus the two forces are equivalent to a

single force.

2.9 Equilibirium of two forces



Equilibrium occurs when the resultant force acting on a point particle is zero (that is, the vector

sum of all forces is zero). When dealing with an extended body, it is also necessary that the net

torque in it is 0.

There are two kinds of equilibrium:

1. static equilibrium

2. dynamic equilibrium.

Static equilibrium

Static equilibrium was understood well before the invention of classical mechanics. Objects

which are at rest have zero net force acting on them.

The simplest case of static equilibrium occurs when two forces are equal in magnitude but

opposite in direction. For example, an object on a level surface is pulled (attracted) downwardtoward the center of the Earth by the force of gravity. At the same time, surface forces resist the

downward force with equal upward force (called the normal force). The situation is one of zero

net force and no acceleration.

Pushing against an object on a frictional surface can result in a situation where the object does

not move because the applied force is opposed by static friction, generated between the object

and the table surface. For a situation with no movement, the static friction force exactly balances

the applied force resulting in no acceleration. The static friction increases or decreases in

response to the applied force up to an upper limit determined by the characteristics of the contact

between the surface and the object.

A static equilibrium between two forces is the most usual way of measuring forces, using simple

devices such as weighing scales and spring balances. For example, an object suspended on a

vertical spring scale experiences the force of gravity acting on the object balanced by a force

applied by the "spring reaction force" which equals the object's weight. Using such tools, some

quantitative force laws were discovered: that the force of gravity is proportional to volume for

objects of constant density (widely exploited for millennia to define standard weights);

Archimedes' principle for buoyancy; Archimedes' analysis of the lever; Boyle's law for gas

pressure; and Hooke's law for springs. These were all formulated and experimentally verified before Isaac Newton expounded his Three Laws of Motion.

Dynamic equilibrium

Galileo Galilei was the first to point out the inherent contradictions contained in Aristotle's

description of forces.



Dynamic equilibrium was first described by Galileo who noticed that certain assumptions of

Aristotelian physics were contradicted by observations and logic. Galileo realized that simple

velocity addition demands that the concept of an "absolute rest frame" did not exist. Galileo

concluded that motion in a constant velocity was completely equivalent to rest. This was

contrary to Aristotle's notion of a "natural state" of rest that objects with mass naturally

approached. Simple experiments showed that Galileo's understanding of the equivalence of

constant velocity and rest were correct. For example, if a mariner dropped a cannonball from the

crow's nest of a ship moving at a constant velocity, Aristotelian physics would have the

cannonball fall straight down while the ship moved beneath it. Thus, in an Aristotelian universe,

the falling cannonball would land behind the foot of the mast of a moving ship. However, when

this experiment is actually conducted, the cannonball always falls at the foot of the mast, as if the

cannonball knows to travel with the ship despite being separated from it. Since there is no

forward horizontal force being applied on the cannonball as it falls, the only conclusion left is

that the cannonball continues to move with the same velocity as the boat as it falls. Thus, no

force is required to keep the cannonball moving at the constant forward velocity.

Moreover, any object traveling at a constant velocity must be subject to zero net force (resultant

force). This is the definition of dynamic equilibrium: when all the forces on an object balance but

it still moves at a constant velocity.

A simple case of dynamic equilibrium occurs in constant velocity motion across a surface with

kinetic friction. In such a situation, a force is applied in the direction of motion while the kinetic

friction force exactly opposes the applied force. This results in zero net force, but since the object

started with a non-zero velocity, it continues to move with a non-zero velocity. Aristotle

misinterpreted this motion as being caused by the applied force. However, when kinetic frictionis taken into consideration it is clear that there is no net force causing constant velocity motion.

2.10 superposition and transmissibility of forces

2.10.1 Principle of superposition of forces

Net effect of forces applied in any sequence on a body is given by the algebraic sum of effect of

individual forces on the body.



2.10.2 Principle of transmissibility of forces



The point of application of a force on a rigid body can be changed along the same line of action

maintaining the same magnitude and direction without affecting the effect of the force on the

body.

Limitation of principle of transmissibility: Principle of transmissibility can be used only for

rigid bodies and cannot be used for deformable bodies.

2.11 Newton’s third law

Newton's Third Law is a result of applying symmetry to situations where forces can be attributed

to the presence of different objects. The third law means that all forces are interactions between

different bodies,[17][18]

and thus that there is no such thing as a unidirectional force or a force that

acts on only one body. Whenever a first body exerts a force F on a second body, the second body

exerts a force −F on the first body. F and −F are equal in magnitude and opposite in direction.This law is sometimes referred to as the action-reaction law, with F called the "action" and −F

the "reaction". The action and the reaction are simultaneous:

If object 1 and object 2 are considered to be in the same system, then the net force on the system

due to the interactions between objects 1 and 2 is zero since

This means that in a closed system of particles, there are no internal forces that are unbalanced.

That is, the action-reaction force shared between any two objects in a closed system will not

cause the center of mass of the system to accelerate. The constituent objects only accelerate with

respect to each other, the system itself remains unaccelerated. Alternatively, if an external force

acts on the system, then the center of mass will experience an acceleration proportional to the

magnitude of the external force divided by the mass of the system.

Combining Newton's Second and Third Laws, it is possible to show that the linear momentum of

a system is conserved. Using



and integrating with respect to time, the equation:

is obtained. For a system which includes objects 1 and 2,

which is the conservation of linear momentum. Using the similar arguments, it is possible to

generalize this to a system of an arbitrary number of particles. This shows that exchanging

momentum between constituent objects will not affect the net momentum of a system. In

general, as long as all forces are due to the interaction of objects with mass, it is possible to

define a system such that net momentum is never lost nor gained.

2.12 triangle of forces

When there are three forces acting on a body and they are in equilibrium, we use the triangle

law to solve such problems:

If three forces acting at a point are in equilibrium, they can be represented in magnitude

and direction by the sides of a triangle taken in order.

When the triangle law is applied to three forces in equilibrium, the resulting triangle will be a

closed figure, ie all the vectors will be head-to-tail. Such a vector diagram implies that the

resultant force is zero.

Example 1

A mass of 40 kg is suspended from the ceiling by a length of rope. The mass is pulled sideways

by a horizontal force of 231 N until the rope makes an angle of 30° with the vertical. The mass is

now stationary. Determine the magnitude of the force in the rope.



First draw a diagram showing all the forces acting on the mass. This is called a force diagram or

a free body diagram. Now we represent the three forces by means of a triangle. The mass is

stationary, so the forces are in equilibrium and the sides of the triangle must all be head-to-tail.



Example 2

A lamp of mass 2 kg hangs from the ceiling. In order to make it illuminate the required area, a

string (F) is attached to its cord and then attached to the ceiling. The angles made by the cord and

the string are as indicated in the diagram.



(i) the magnitude of the tension in the cord is 13,2 N

(ii) the magnitude of the tension in the string is 17,6 N

The sequence in which the triangle is drawn is:

1 Draw the vertical line 50 mm long.

2 Draw the 60° angle at the top of the vertical line.

3 Draw the 40° angle at the bottom of the vertical line.

4 Where the lines meet is the third corner of the triangle.

5 Make sure the arrows are all head-to-tail.



2.13 two force systems

A two-force member is a rigid body with no force couples, acted upon by a system o

forces composed of, or reducible to, two forces at different locations.

The most common example of the a two force member is a structural brace where each

end is pinned to other members as shown at the left. In the diagram, notice that member

BD is pinned at only two locations and thus only two forces will be acting on the

member (not considering components, just the total force at the pinned joint).

Two-force members are special since the two forces must be co-linear and equal. This

can be proven by taking a two force member with forces at arbitrary angles as shown at

the left. If moments are summed at point B then force FD cannot not have any horizontal

component. This requires FD to be vertical. Then the forces are summed in both

directions, it shows FB must also be vertical. Furthermore, the two forces must be equal.

There are three criteria for a two-force member:

1. The forces are directed along a line that intersects their points of application.

2. The forces are equal in magnitude.

3. The forces are opposite in direction.



2.14 extension of parallelogram law and triangle law to many forces acting at one point

polygon law of forces

If two forces acting at a point are represented, in magnitude and direction, by the sides of a

parallelogram drawn from the point, their resultant force is represented, both in magnitude and in

direction, by the diagonal of the parallelogram drawn through that point.

let the two forces F1 and F2, acting at the point O be represented, in magnitude and direction,

by the directed line OA and OB inclined at an angle θ with each other. Then if the parallelogram

OACB be completed, the resultant force, R , will be represented by the diagonal OC.



2.15 method of resolution into orthogonal components for finding the resultant

The method to find a resultant, as used in leaflet 1.5 (Force as a Vector), is generally slow and

can be complicated. Taking components of forces can be used to find the resultant force more

quickly. In two dimensions, a force can be resolved into two mutually perpendicular components

whose vector sum is equal to the given force. The components are often taken to be parallel to

the x- and y-axes. In two dimensions we use the perpendicular unit vectors i and j (and in three

dimensions they are i, j and k). Let F be a force, of magnitude F with components X and Y in the

directions of the x- and y-axes, respectively.



2.16 graphical methods

The graphical method of solving mechanical problems involving forces is often used because it

is quick and accurate. The force is shown graphically. To describe completely the force, the

following particulars must be given:

1. Its magnitude

2. Its point of application

3. Its direction

4. Its sense, i.e., whether it is pushing or pulling

A line is drawn to a given length to represent the magnitude of the force. The direction of this

line is parallel to the direction of the force. The sense of the force is indicated by an arrow on the

line indicating whether it is acting toward or away from the point of application. The graphical

representation of the force is called a vector.

Thus a pull of 6 tons (T) acting at a point A at 45° to the horizontal would be represented by thevector AB. Using the scale .25 in. = I T, the length of the vector would be 1.50 in.



A body is said to be in equilibrium if the forces acting at a point balance one another. If two

equal and opposite forces act at a point in a straight line, the body is in equilibrium. Examples

are tie bars, which are bars under pull or tension, and struts or columns, which are bars under

push or compression

2.16.1 TWO FORCES ACTING AT A POINT

Two or more forces acting at a point may be replaced by one force that will produce the same

effect. This force is called the resultant of the forces. If two opposite forces of 8 and 5 T act at a

point 0 in a straight line, a resultant force of 3 T acting in the same direction as the 8 T force

could replace the two original forces.



If two opposite forces F1

and F2

act at point O at angles of 120° to each other, the resultant force

R may be found by drawing the two forces to scale and completing the parallelogram. The

diagonal OC would be the resultant, and the magnitude of the force could be measured. This

method is called the parallelogram of forces. Accuracy of direction and distance is important in

laying out forces.

Another way of finding the resultant is the triangle of force method. The known force

vectors are laid end to end with the forces traveling in the same direction. The resultant R is

found by joining the beginning of the first vector to the end of the last vec-tor, as shown in Fig.

28-1-5A, and the direction of the resultant force is in the combined direction of the other two

forces.

If a force equal to the resultant of forces F1

and F2, but acting in the opposite direction, was to act

at O, as shown in Fig. 28-1-5B, the object would be in equilibrium, since the forces acting at



point O tend to balance one another. This force balancing the other forces is known as the

equilibrant.

The equilibrant is found in a similar manner to the resultant, by using the triangle of force

method. Note that the arrows representing the direction of the forces are pointing the same way

around the triangle.

2.16.2 MORE THAN TWO FORCES ACTING AT A POINT

Resultants or equilibrants may be found for any number of forces acting at a point and in one plane. Let A, B, C, and D represent forces acting at a point 0.

Using the parallelogram of forces method, we find the resultant R 1

for forces A and B and

resultant R 2

for forces C and D. Using resultants R 1

and R 2

instead of the forces A, B, C, and D,

we find the resultant R of the four forces. The equilibrant or force required to keep the forces A,

B, C, and D in equilibrium would be equal to R but would act in the opposite direction.



2.17 Lami’s theorem

In statics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent

and non-collinear forces, which keeps an object in static equilibrium, with the angles directly

opposite to the corresponding forces. A,B,C

where A, B and C are the magnitudes of three coplanar, concurrent and non-collinear forces,

which keep the object in static equilibrium, and α, β and γ are the angles directly opposite to the

forces A, B and C respectively.

Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is

named after Bernard Lamy.

Proof of Lami's Theorem

Suppose there are three coplanar, concurrent and non-collinear forces, which keeps the object in

static equilibrium. By the triangle law, we can re-construct the diagram as follow:



By the law of sines,

Review Questions

1. What do you mean by Force and its effects?

2. Describe units and measurement of force.

3. What are characteristics of force vector representation?

4. Describe Bow’s notation.

5. What are different types of Types of forces?

6. Explain action and reaction, tension, thrust and shear force.

7. Define Force systems : Coplaner and space force systems. Coplaner concurrent and

nonconcurrent forces.

8. Explain Free body diagrams.

9. Define Resultant and components concept of equlibirium.



10. Explain Parallelogram law of forces.

11. Explain Equilibirium of two forces.

12. Describe superposition and transmissibility of forces

13. Describe Newton’s third law.

14. What is triangle of forces?

15. Describe two force systems.

16. Define extension of parallelogram law and triangle law to many forces acting at one point

polygon law of forces.

17. Explain method of resolution into orthogonal components for finding the resultant.

18. What do you mean by graphical methods.

19. Describe Lami’s theorem.



Chapter-3

Moments


Moment and forces

Learning Objectives

1. Concept of moment

2. Varignon’s theorem – statement only

3. Principle of moments –application of moments to simple mechanism, parallel forces,

calculation of their resultant

4. concept of couple properties and effect

5. moving a force parallel to its line of action

6. general cases of coplanar force system

7. general conditions of equilibrium of bodies under coplanar forces

3.1 Concept of moment

In physics, moment relates to the perpendicular distance from a point to a line or a surface, and

is derived from the mathematical concept of moments. It is frequently used in combination with

other physical quantities as in moment of inertia, moment of force, moment of momentum,magnetic moment and so on.

Moment is also used colloquially for different physical quantities that depend upon distance. For

example, in engineering and kinesiology the term moment is often used instead of the more

complete term moment of force. A moment of force being the product of the distance of a force

from an axis times the magnitude of the force, i.e., F × d, where F is the magnitude of the force

and d is the moment of the force. See torque for a more complete description of moments of

force or couple for the related concept free moment of force also known as a force couple.

It may also be used when the distance is squared, as in moment of inertia. The moment of inertiais the "second moment" of mass of a physical object. This is the object's resistance or inertia to

changes in its angular motion. It is roughly the sum of the squared distances (i.e., moments) of

the object's mass particles about a particular axis .



3.2 Varignon’s theorem – statement only

The Varignon Theorem is a theorem by French mathematician Pierre Varignon (1654-1722),

published in 1687 in his book Projet d' unè nouvelle mèchanique. The theorem states that the

moment of a force about any point is equal to the sum of the moments of its components about

the same point.

Proof

For a set n of vectors that concurs at point O in space. The resultant is:

The moment of each vector is:

In terms of summary, taking out the common factor , and considering the expression

of R , results:

3.3 Principle of moments –application of moments to simple mechanism



The Principle of Moments, also known as Varignon's Theorem , states that the moment of any

force is equal to the algebraic sum of the moments of the components of that force. It is a very

important principle that is often used in conjunction with the Principle of Transmissibility in

order to solve systems of forces that are acting upon and/or within a structure. This concept will

be illustrated by calculating the moment around the bolt caused by the 100 pound force at points

A, B, C, D, and E in the illustration.

First consider the 100 pound force

Since the line of action of the force is not perpendicular to the wrench at A, the force is broken

down into its orthagonal components by inspection. The line of action of the the 100 pound force

can be inspected to determine if there are any convenient geometries to aid in the decomposition

of the 100 pound force.

The 4 inch horizontal and the 5 inch diagonal

measurement near point A should be

recognized as belonging to a 3-4-5 triangle.

Therefore, Fx = -4/5(100 pounds) or -80 pounds

and Fy = -3/5(100 pounds) or -60 pounds.



Consider Point A

The line of action of Fx at A passes through the handle of the wrench to the bolt (which is also

the center of moments). This means that the magnitude of the moment arm is zero and therefore

the moment due to FAx is zero. FAy at A has a moment arm of twenty inches and will tend to

cause a positive moment.

FAy d = (60 pounds)(20in) = 1200 pound-inches or 100 pound-feet

The total moment caused by the 100 pound force F at point A is 1200 pound-inches.

Consider Point B

At this point the 100 pound force is perpendicular to the wrench. Thus, the total moment due to

the force can easily be found without breaking it into components.

FB d = (100 pounds)(12in) = 1200 pound-inches

The total moment caused by the 100 pound force F at point B is again 1200 pound-inches.



Consider Point C

The force must once again be decomposed into components. This time the vertical component

passes through the center of moments. The horizontal component FCx causes the entire moment.

FCx d = (80 pounds)(15inches) = 1200 pound-inches

Consider Point D

The force must once again be decomposed into components. Both components will contribute to

the total moment.

FDx d = (80 pounds)(21inches) = 1680 pound-inches

FDy d = (60 pounds)(8in) = -480 pound-inches

Note that the y component in this case would create a counter clockwise or negative rotation.

The total moment at D due to the 100 pound force is determined by adding the two component

moments. Not surprisingly, this yields 1200 pound-inches.

Consider Point E



Varignon's Theorem applies even though point E is removed from the physical object. Following

the same procedure as at point D;

FEx d = (80 pounds)(3in) = -240 pound-inches

FEy d = (60 pounds)(24in) = 1440 pound-inches

However, this time Fx tends to cause a negative moment. Once again the total moment is 1200

pound-inches.

Summary for the example At each point, A, B, C, D and E the total moment around the bolt

caused by the 100 pound force equalled 1200 pound-inches. In fact, the total moment would

equal 1200 pound-inches at ANY point along the line of action of the force. This is Varignon's

Theorem.

3.4 parallel forces

Statics refers to the bodies in equilibrium. Equilibrium deals with the absence of a net force.

When the net equals zero, the forces are in equilibrium provided they are concurrent (they

intersect). If they are non-concurrent, the body may rotate even if the vector sum of the forces

equals zero. Hence, there must be another condition to set forces in equilibrium – that under the

influence of forces, the body must have no tendency toward translational or rotary motion.

An example of non-concurrent forces where the vector sum may be equal to zero but it still

causes the body to move is parallel forces. They act in the same or opposite directions. Theirlines of action are parallel.

Forces acting in the same or opposite directions are parallel.

3.4.1 TORQUE (MOMENT OF FORCE)

Torque or moment of force refers to the turning effect of the force upon a body about a point

(fulcrum). It is the product of the magnitude of the force and perpendicular distance from the line

of action of the force to the fulcrum. This perpendicular distance is called moment arm or leverarm.

• The greater the distance from the axis to the point where we apply the force, the greater

the torque.

• Maximum torque occurs when the direction of the applied force is perpendicular to a line

drawn between the axis and the point where the force is applied.





Graphical placing of the resultant force

3.6 concepts of couple properties and effect

In mechanics, a couple is a system of forces with a resultant (a.k.a. net or sum) moment but no

resultant force. A better term is force couple or pure moment. Its effect is to create rotation

without translation, or more generally without any acceleration of the centre of mass. In rigid

body mechanics, force couples are free vectors, meaning their effects on a body are independentof the point of application.

The resultant moment of a couple is called a torque. This is not to be confused with the term

torque as it is used in physics, where it is merely a synonym of moment. Instead, torque is a

special case of moment. Torque has special properties that moment does not have, in particular

the property of being independent of reference point, as described below.

3.6.1 Simple couple

Definition- A couple is a pair of forces, equal in magnitude, oppositely directed, and displaced by perpendicular distance or moment.

The simplest kind of couple consists of two equal and opposite forces whose lines of action do

not coincide. This is called a "simple couple". The forces have a turning effect or moment called

a torque about an axis which is normal to the plane of the forces. The SI unit for the torque of the

couple is newton metre.



If the two forces are F and −F, then the magnitude of the torque is given by the following

formula:

where

is the torque

F is the magnitude of one of the forces

d is the perpendicular distance between the forces, sometimes called the arm of the

couple

The magnitude of the torque is always equal to F d, with the direction of the torque given by the

unit vector , which is perpendicular to the plane containing the two forces. When d is taken as a

vector between the points of action of the forces, then the couple is the cross product of d and F.

I.e.,

3.6.2 Independence of reference point

The moment of a force is only defined with respect to a certain point P (it is said to be the

"moment about P"), and in general when P is changed, the moment changes. However, the

moment (torque) of a couple is independent of the reference point P: Any point will give the

same moment.[1]

In other words, a torque vector, unlike any other moment vector, is a "free

vector".

(This fact is called Varignon's Second Moment Theorem.)

The proof of this claim is as follows: Suppose there are a set of force vectors F1, F2, etc. that

form a couple, with position vectors (about some origin P) r1, r2, etc., respectively. The moment

about P is

Now we pick a new reference point P' that differs from P by the vector r. The new moment is

Now the distributive property of the cross product implies



However, the definition of a force couple means that

Therefore,

This proves that the moment is independent of reference point, which is proof that a couple is a

free vector.

3.6.3 Forces and couples

A force F applied to a rigid body at a distance d from the center of mass has the same effect as

the same force applied directly to the center of mass and a couple Cℓ = Fd. The couple produces

an angular acceleration of the rigid body at right angles to the plane of the couple.[4]

The force at

the center of mass accelerates the body in the direction of the force without change in

orientation. The general theorems are:[4]



A single force acting at any point O′ of a rigid body can be replaced by an equal and parallel

force F acting at any given point O and a couple with forces parallel to F whose moment is M =

Fd, d being the separation of O and O′. Conversely, a couple and a force in the plane of the

couple can be replaced by a single force, appropriately located.

Any couple can be replaced by another in the same plane of the same direction and moment,having any desired force or any desired arm.

3.6.4 Applications

Couples are very important in mechanical engineering and the physical sciences. A few

examples are:

• The forces exerted by one's hand on a screw-driver• The forces exerted by the tip of a screw-driver on the head of a screw

• Drag forces acting on a spinning propeller

• Forces on an electric dipole in a uniform electric field.

• The reaction control system on a spacecraft.

In a liquid crystal it is the rotation of an optic axis called the director that produces the

functionality of these compounds. As Jerald Ericksen explained

At first glance, it may seem that it is optics or electronics which is involved, rather than

mechanics. Actually, the changes in optical behavior, etc. are associated with changes inorientation. In turn, these are produced by couples. Very roughly, it is similar to bending a wire,

by applying couples.

3.7 moving a force parallel to its line of action

In physics, net force is the overall force acting on an object. In order to perform this calculation

the body is isolated and interactions with the environment or constraints are introduced as forces

and torques forming a free-body diagram.

The net force does not have the same effect on the movement of the object as the original system

forces, unless the point of application of the net force and an associated torque are determined so

that they form the resultant force and torque. It is always possible to determine the torque

associated with a point of application of a net force so that it maintains the movement of the

object under the original system of forces.



With its associated torque, the net force becomes the resultant force and has the same effect on

the rotational motion of the object as all actual forces taken together. It is possible for a system

of forces to define a torque-free resultant force. In this case, the net force when applied at the

proper line of action has the same effect on the body as all of the forces at their points of

application. It is not always possible to find a torque-free resultant force.

3.7.1 Total force

The sum of forces acting on a particle is called the total force or the net force. The net force is a

single force that replaces the effect of the original forces on the particle's motion. It gives the

particle the same acceleration as all those actual forces together as described by the Newton's

second law of motion.

Force is a vector quantity, which means that it has a magnitude and a direction, and it is usually

denoted using boldface such as F or by using an arrow over the symbol, such as .

Graphically a force is represented as line segment from its point of application A to a point B

which defines its direction and magnitude. The length of the segment AB represents the

magnitude of the force.

Another method for diagramming addition of forces

Vector calculus was developed in the late 1800s and early 1900s, however, the parallelogramrule for addition of forces is said to date from the ancient times, and it is explicitly noted by

Galileo and Newton.

The diagram shows the addition of the forces and . The sum of the two forces is drawn as

the diagonal of a parallelogram defined by the two forces.



Forces applied to an extended body can have different points of application. Forces are bound

vectors and can be added only if they are applied at the same point. The net force obtained from

all the forces acting on a body will not preserve its motion unless they are applied at the same

point and the appropriate torque associated with the new point of application is determined. The

net force on a body applied at a single point with the appropriate torque is known as the resultant

force and torque.

3.7.2 Parallelogram rule for the addition of forces

A force is known as a bound vector which means it has a direction and magnitude and a point of

application. A convenient way to define a force is by a line segment from a point A to a point B.

If we denote the coordinates of these points as A=(Ax, Ay, Az) and B=(Bx, By, Bz), then the force

vector applied at A is given by

The length of the vector B-A defines the magnitude of F, and is given by

The sum of two forces F1 and F2 applied at A can be computed from the sum of the segments

that define them. Let F1=B-A and F2=D-A, then the sum of these two vectors is

which can be written as



where E is the midpoint of the segment BD that joins the points B and D.

Thus, the sum of the forces F1 and F2 is twice the segment joining A to the midpoint E of the

segment joining the endpoints B and D of the two forces. The doubling of this length is easily

achieved by defining a segments BC and DC parallel to AD and AB, respectively, to complete

the parallelogram ABCD. The diagonal AC of this parallelogram is the sum of the two force

vectors. This is known as the parallelogram rule for the addition of forces.

3.7.3 Translation and rotation due to a force

3.7.3.1 Point forces

When a force acts on a particle, it is applied to a single point (the particle volume is negligible):

this is a point force and the particle is its application point. But an external force on an extended

body (object) can be applied to a number of its constituent particles, i.e. can be "spread" over

some volume or surface of the body. However, in order to determine its rotational effect on the

body, it is necessary to specify its point of application (actually, the line of application, as

explained below). The problem is usually resolved in the following ways:

• Often the volume or surface on which the force acts is relatively small compared to the

size of the body, so that it can be approximated by a point. It is usually not difficult to

determine whether the error caused by such approximation is acceptable.

• If it is not acceptable (obviously e.g. in the case of gravitational force), such

"volume/surface" force should be described as a system of forces (components), each

acting on a single particle, and then the calculation should be done for each of them

separately. Such a calculation is typically simplified by the use of differential elements of

the body volume/surface, and the integral calculus. In a number of cases, though, it can

be shown that such a system of forces may be replaced by a single point force without the

actual calculation (as in the case of uniform gravitational force).

In any case, the analysis of the rigid body motion begins with the point force model. And when a

force acting on a body is shown graphically, the oriented line segment representing the force is

usually drawn so as to "begin" (or "end") at the application point.

3.7.4 Rigid bodies



How a force accelerates a body

In the example shown on the diagram, a single force acts at the application point H on a free

rigid body. The body has the mass and its center of mass is the point C. In the constant mass

approximation, the force causes changes in the body motion described by the following

expressions:

is the center of mass acceleration; and

is the angular acceleration of the body.

In the second expression, is the torque or moment of force, whereas is the moment of inertia of

the body. A torque caused by a force is a vector quantity defined with respect to some

reference point:

is the torque vector, and

is the amount of torque.

The vector is the position vector of the force application point, and in this example it is drawn

from the center of mass as the reference point (see diagram). The straight line segment is thelever arm of the force with respect to the center of mass. As the illustration suggests, the torque

does not change (the same lever arm) if the application point is moved along the line of the

application of the force (dotted black line). More formally, this follows from the properties of the

vector product, and shows that rotational effect of the force depends only on the position of its

line of application, and not on the particular choice of the point of application along that line.



The torque vector is perpendicular to the plane defined by the force and the vector , and in this

example it is directed towards the observer; the angular acceleration vector has the same

direction. The right hand rule relates this direction to the clockwise or counter-clockwise rotation

in the plane of the drawing.

The moment of inertia is calculated with respect to the axis through the center of mass that is

parallel with the torque. If the body shown in the illustration is a homogenous disc, this moment

of inertia is . If the disc has the mass 0,5 kg and the radius 0,8 m, the moment of inertia

is 0,16 kgm2. If the amount of force is 2 N, and the lever arm 0,6 m, the amount of torque is 1,2

Nm. At the instant shown, the force gives to the disc the angular acceleration α = τ/I = 7,5 rad/s2,

and to its center of mass it gives the linear acceleration a = F/m = 4 m/s2.

3.7.5 Resultant force

Graphical placing of the resultant force

Resultant force and torque replaces the effects of a system of forces acting on the movement of a

rigid body. An interesting special case is a torque-free resultant which can be found as follows:

1. First, vector addition is used to find the net force;

2. Then use the equation to determine the point of application with zero torque:

where is the net force, locates its application point, and individual forces are with

application points. It may be that there is no point of application that yields a torque-free

resultant.



The diagram illustrates simple graphical methods for finding the line of application of theresultant force of simple planar systems.

1. Lines of application of the actual forces and on the leftmost illustration intersect.

After vector addition is performed "at the location of ", the net force obtained is

translated so that its line of application passes through the common intersection point.

With respect to that point all torques are zero, so the torque of the resultant force is

equal to the sum of the torques of the actual forces.

2. Illustration in the middle of the diagram shows two parallel actual forces. After vector

addition "at the location of ", the net force is translated to the appropriate line of

application, where it becomes the resultant force . The procedure is based on

decomposition of all forces into components for which the lines of application (pale

dotted lines) intersect at one point (the so-called pole, arbitrarily set at the right side of

the illustration). Then the arguments from the previous case are applied to the forces and

their components to demonstrate the torque relationships.

3. The rightmost illustration shows a couple, two equal but opposite forces for which the

amount of the net force is zero, but they produce the net torque where is the

distance between their lines of application. This is "pure" torque, since there is no

resultant force.

3.7.6 Usage



Vector diagram for addition of non-parallel forces

Generally, a system of forces acting on a rigid body can always be replaced by one force plus

one "pure" torque. The force is the net force, but in order to calculate the additional torque, the

net force must be assigned the line of action. The line of action can be selected arbitrarily, but the

additional "pure" torque will depend on this choice. In a special case it is possible to find such

line of action that this additional torque is zero.

The resultant force and torque can be determined for any configuration of forces. However, an

interesting special case is a torque-free resultant which it is useful both conceptually and

practically, because the body moves without rotating as if it was a particle.

Some authors do not distinguish the resultant force from the net force and use the terms as

synonyms.

3.8 general cases of coplanar force system

3.8.1 Nonconcurrent Force Systems

You already have some understanding of the conditions which determine whether a body subject

to nonconcurrent forces is in equilibrium. Look at the following cases and tell in which ones

1. Sum of Forces = 0

2. The system is likely to be in equilibrium

3.9 general conditions of equilibrium of bodies under coplanar forces

3.9.1 Equilibrium of Concurrent Force System

In static, a body is said to be in equilibrium when the force system acting upon it has a zero

resultant.



Conditions of Static Equilibrium of Concurrent Forces

The sum of all forces in the x-direction or horizontal is zero.

or

The sum of all forces in the y-direction or vertical is zero.

or

Important Points for Equilibrium Forces

• Two forces are in equilibrium if they are equal and oppositely directed.

• Three coplanar forces in equilibrium are concurrent.

• Three or more concurrent forces in equilibrium form a close polygon when connected in

head-to-tail manner.

Review Questions

1. Describe Concept of moment.

2. Explain Varignon’s theorem – statement only.

3. What is Principle of moments?

4. Describe application of moments to simple mechanism.

5. What do you mean by parallel forces and calculation of their resultant?

6. Describe concept of couple properties and effect.

7. Explain moving a force parallel to its line of action.

8. Define general cases of coplanar force system.

9. Describe general conditions of equilibirium of bodies under coplanar forces.



Chapter 4

Friction


Friction

Learning Objectives

1. Concept of friction

2. laws of friction

3. limiting friction and coefficient of friction

4. sliding friction

4.1 Concept of friction

Friction is the force resisting the relative motion of solid surfaces, fluid layers, and material

elements sliding against each other. There are several types of friction:

• Dry friction resists relative lateral motion of two solid surfaces in contact. Dry friction is

subdivided into static friction ("stiction") between non-moving surfaces, and kinetic

friction between moving surfaces.

• Fluid friction describes the friction between layers within a viscous fluid that are moving

relative to each other.

• Lubricated friction is a case of fluid friction where a fluid separates two solid surfaces.

• Skin friction is a component of drag, the force resisting the motion of a fluid across the

surface of a body.

• Internal friction is the force resisting motion between the elements making up a solid

material while it undergoes deformation.

When surfaces in contact move relative to each other, the friction between the two surfacesconverts kinetic energy into heat. This property can have dramatic consequences, as illustrated

by the use of friction created by rubbing pieces of wood together to start a fire. Kinetic energy is

converted to heat whenever motion with friction occurs, for example when a viscous fluid is

stirred. Another important consequence of many types of friction can be wear, which may lead to

performance degradation and/or damage to components. Friction is a component of the science

of tribology.



Friction is not itself a fundamental force but arises from fundamental electromagnetic forces

between the charged particles constituting the two contacting surfaces. The complexity of these

interactions makes the calculation of friction from first principles impractical and necessitates the

use of empirical methods for analysis and the development of theory.

4.1.1 Energy of friction

According to the law of conservation of energy, no energy is destroyed due to friction, though it

may be lost to the system of concern. Energy is transformed from other forms into heat. A

sliding hockey puck comes to rest because friction converts its kinetic energy into heat. Since

heat quickly dissipates, many early philosophers, including Aristotle, wrongly concluded that

moving objects lose energy without a driving force.

When an object is pushed along a surface, the energy converted to heat is given by:

where

is the normal force,

is the coefficient of kinetic friction,

is the coordinate along which the object transverses.

Energy lost to a system as a result of friction is a classic example of thermodynamic

irreversibility.

4.1.2 Work of friction

In the reference frame of the interface between two surfaces, static friction does no work,

because there is never displacement between the surfaces. In the same reference frame, kinetic

friction is always in the direction opposite the motion, and does negative work.[41]

However,

friction can do positive work in certain frames of reference. One can see this by placing a heavy

box on a rug, then pulling on the rug quickly. In this case, the box slides backwards relative tothe rug, but moves forward relative to the frame of reference in which the floor is stationary.

Thus, the kinetic friction between the box and rug accelerates the box in the same direction that

the box moves, doing positive work.

The work done by friction can translate into deformation, wear, and heat that can affect the

contact surface properties (even the coefficient of friction between the surfaces). This can be



beneficial as in polishing. The work of friction is used to mix and join materials such as in the

process of friction welding. Excessive erosion or wear of mating sliding surfaces occurs when

work due frictional forces rise to unacceptable levels. Harder corrosion particles caught between

mating surfaces in relative motion (fretting) exacerbates wear of frictional forces. Bearing

seizure or failure may result from excessive wear due to work of friction. As surfaces are worn

by work due to friction, fit and surface finish of an object may degrade until it no longer

functions properly.

4.1.3 Applications

Friction is an important factor in many engineering disciplines.

Transportation

• Automobile brakes inherently rely on friction, slowing a vehicle by converting its kinetic

energy into heat. Incidentally, dispersing this large amount of heat safely is one technical

challenge in designing brake systems.

• Rail adhesion refers to the grip wheels of a train have on the rails, see Frictional contact

mechanics.

• Road slipperiness is an important design and safety factor for automobiles

o Split friction is a particularly dangerous condition arising due to varying friction

on either side of a car.

o Road texture affects the interaction of tires and the driving surface.

Measurement

• A tribometer is an instrument that measures friction on a surface.

• A profilograph is a device used to measure pavement surface roughness.

Household usage

• Friction is used to heat and ignite matchsticks (friction between the head of a matchstick

and the rubbing surface of the match box).

4.2 laws of friction

The Three Laws of Friction

• The frictional force being independent of the area of contact

• The frictional force being proportional to the load

• The frictional force being independent of the speed of movement



Friction

Smooth surfaces are defined by the properties that when they are in contact, the surfaces are

always perpendicular to their common tangent plane. It can, however, be verified experimentally

that no surface is perfectly smooth and that whenever there is a tendency for two bodies in

contact to move relative to each other, a force known as the force of friction tends to prevent the

relative motion. The mathematical discussion of the force of friction depends on certain

assumptions which are embodied in the so called laws of friction and are found to be in close

agreement with experiments.

Law 1

When two bodies are in contact the direction of the forces of Friction on one of them at it's

point of contact, is opposite to the the direction in which the point of contact tends to move

relative to the other.

Law 2

If the bodies are in equilibrium, the force of Friction is just sufficient to prevent motion and

may therefore be determined by applying the conditions of equilibrium of all the forces

acting on the body.

The amount of Friction that can be exerted between two surfaces is limited and if the forces

acting on the body are made sufficiently great, motion will occur. Hence, we define

limiting friction as the friction which is exerted when equilibrium is on the point of being broken by one body sliding on another. The magnitude of limiting friction is given by the

following three laws.

Law 3

The ratio of the limiting friction to the Normal reaction between two surfaces depends on

the substances of which the surfaces are composed, and not on the magnitude of the Normal

reaction.

This ratio is usually denoted by . Thus if the Normal reaction is R, the limiting friction isFor given materials polished to the same standard is found to be constant and

independent of R.

is called The Coefficient of friction

Law 4



The amount of limiting friction is independent of the area of contact between the two

surfaces and the shape of the surfaces, provided that the Normal reaction is unaltered.

Law 5

When motion takes place, the direction of friction is opposite to the direction of relative

motion and is independent of velocity. The magnitude of the force of friction is in a

constant ratio to the Normal reaction, but this ratio may be slightly less than when the body

is just on the point of moving.

It should be stressed that the above laws are experimental and are accepted as the basis for the

mathematical treatment of friction. Modern theory suggests that the force of friction is in fact due

to the non-rigidity of bodies. When one body rests on another, there is always an area of contact,

which is much smaller than the apparent area and also depends on the the normal pressure between the bodies. Friction is considered to be due to the fusion of materials (of which the

bodies are composed) over the area of contact. Therefore friction would be proportional to the

area of contact, and therefore proportional to the normal pressure, as assumed in the above laws.

4.3 limiting friction and coefficient of friction

The maximum value of static friction, when motion is impending, is sometimes referred to as

limiting friction, although this term is not used universally.

we find that the maximum value of static friction and the force of kinetic friction are each

proportional to the normal force; that is,

f s,max = s n

and

f k = k n

These 's are called the coefficients of friction. s is the coefficient of static friction and k is the

coefficient of kinetic friction. Since f s,max > f k , this means s > k . If it is clear from context, it is

common to say simply the "coefficient of friction" and to label it merely as .

Now let us return to earlier examples:



Example

Once again, we have a man pulling a crate along a concrete floor. This time, let's be specific. The

crate has a mass of 100 kg and the man pulls with a force of 1 250 N. The coefficient of friction

between the crate and the floor is 0.2. What is the acceleration of the crate? For this example,

take g = 10 m/s2 for arithmetic convenience.

The free-body diagram looks about as it did earlier -- except there is an additonal force now, the

force of kinetic friction, f k .



Applying F = m a to the y-component forces, we find

n = w = m g = (500 kg) (10 m/s2)

n = 5 000 N

f k = n

f k = (0.2) (5 000 N)

f k = 1 000 N

Now we know the values of all the forces involved and we can proceed

Fnet = F - f k

Fnet = 1 250 N - 1 000 N

Fnet = 250 N

Fnet = 250 N = m a

250 N = (500 kg) a

a = 500 kg / 250 N

a = 2 [ kg / N ] [ N / (kg m/s2)]

a = 2 m/s2

Example

Find the acceleration of an inclined Atwoods machine with a hanging mass of m1 = 1 kg and a

mass of m2 = 5 kg sitting on an inclined plane which is inclined at 30o from the horizontal. The

coefficient of kinetic friction between this mass and the plane is 0.25.



The forces on the hanging mass, m1, are just as they were before:

But the forces on the other mass, m2, which sits on the plane now have a friction force to be

included:



Now we apply Newton's Second Law to these forces acting on mass m2.

Fy,net = 0

Fy,ne = 0 because there is no motion -- and, certainly, no acceleration -- in the y-direction.

Fy,net = n - m2 g cos 30o = 0

n = m2 g cos 30o

n = (5 kg) (10 m/s2) (0.866)

n = 43.3 N

Notice that the normal force is not equal to the weight! This is important. Now that we know

the normal force, we can immediately calculate the kinetic friction force,

f k = n

f k = (0.25) (43.3 N)

f k = 10.8 N



Now we can apply F = m a to the x-component forces to find

Fx,net = m 2 g sin 30o - T - 10.8 N = m 2 a

(5 kg) (10 m/s2) (0.5) - T - 10.8 N = (5 kg) a

25 N - T - 10.8 N = (5 kg) a

14.2 N - T = (5 kg) a

We still have one equation with two unknowns. But from the forces on the hanging mass, m1,

we know

T - m1 g = m 1 a

T = m1 g + m 1 a

T = (1 kg) (10 m/s2) + ( 1 kg) a

T = 10 N + (1 kg) a

Now we substitute that to find

14.2 N - [10 N + (1 kg) a] = (5 kg) a

14.2 N - 10 N - (1 kg) a = (5 kg) a

4.2 N - (1 kg) a = (5 kg) a + (1 kg) a = (6 kg) a

a = 6 kg / 4.2 N

a = 1.43 m/s2

4.4 sliding friction

Sliding friction is the kind of friction that is caused by two surfaces that slide against each other.

This kind of friction is alternatively known as kinetic. Sliding friction is intended to stop an

object from moving.

Understanding Sliding Friction



The amount of sliding friction created by objects is expressed as a coefficient which takes into

consideration the various factors that can affect the level of friction. These various factors that

can impact sliding friction include the following:

• The surface deformation of objects

• The roughness/smoothness of the surface of the objeects

• The original speed of either object

• The size of object

• The amount of pressure on either object

• The adhesion of the surface

Everyday Examples of Sliding Friction

Specific examples of sliding friction include:

• Rubbing both hands together to create heat

• A sled sliding across snow or ice

• Skis sliding against smow

• A person sliding down a slide is an example of sliding friction

• A coaster sliding against a table

• A washing machine pushed along a floor

• An iron being pushed across material

• The frame and the edge of door sliding against one another

• The bottom of a trashcan sliding against the concrete

• A block being slid across the floor

• Two cement blocks being slid into place next to each other

• Two cards in a deck sliding against each other

• The bottom of a glass being pushed across a table

• A couch sliding against the steps when being moved

• A dresser's legs on the carpet when being slid to another part of the room

• The rope and the pulley on a set of blinds or curtains

• The friction between two books when sliding one into place on a bookshelf

• The friction between the bottom of a book and the shelf when being slid into place

• A vegetable drawer sliding against the holder in the fridge

• A check being slid across the counter at the bank• A paper sliding against the paper holder once emitted from a copy machine

• A paper on the roller as it slides through a fax machine

• The bottom of a chair leg and the floor when a chair is moved out

• The bottom of the coffee pot when slid out from the maker

• The sliding of the brew basket of the coffee maker against the internal parts when it is

removed



• The tube on a lotion bottle and the opening to the lotion when it is pushed down to let out

lotion

• A rag and the counter it is being used to clean

• A ski board on the snow on a mountain

• Jeans on your legs when putting them on

• A card and an envelope when the card is being slid into the envelope

• A sliding glass door against both the track in which it is moving, and the other door

As these examples show, there are many different situations where sliding friction exists and

where sliding friction creates resistance as objects rub against each other. This type of friction is

different than rolling friction where one item can roll and the friction typically slows the rate of

movement.

There is a good chance you have encountered sliding friction examples in the real world. Now

you will be able to recognize these examples when you come upon them since you have a better

understanding of what sliding friction means.

Review Questions

1. Describe Concept of friction.

2. Define laws of friction.3. Explain limiting friction and coefficient of friction.

4. Describe sliding friction.



Chapter-5

Centre of Gravity


Gravity, centroid and center of gravity

Learning Objectives

1. Concept of gravity

2. gravitational force

3. centroid and center of gravity

4. centroid for regular lamina and center of gravity for regular solids

5. Position of center of gravity of compound bodies and centroid of composition area

6. CG of bodies with portions removed

5.1 Concept of gravity

Gravity is a physical phenomenon, specifically the mutual attraction between all objects in the

universe. In a gaming setting, gravity determines the relationship between the player and the

"ground," preventing the player or game objects from flying off into space, and hopefully acting

in a predictable/realistic manner.

Gravity is the weakest of the four fundamental forces, yet it is the dominant force in the universefor shaping the large scale structure of galaxies, stars, etc. The gravitational force between two

masses m1 and m 2 is given by the relationship:

This is often called the "universal law of gravitation" and G the universal gravitation constant. It

is an example of an inverse square law force. The force is always attractive and acts along theline joining the centers of mass of the two masses. The forces on the two masses are equal in size

but opposite in direction, obeying Newton's third law. Viewed as an exchange force, the massless

exchange particle is called the graviton.

The gravity force has the same form as Coulomb's law for the forces between electric charges,

i.e., it is an inverse square law force which depends upon the product of the two interacting



sources. This led Einstein to start with the electromagnetic force and gravity as the first attempt

to demonstrate the unification of the fundamental forces. It turns out that this was the wrong

place to start, and that gravity will be the last of the forces to unify with the other three forces.

Electroweak unification (unification of the electromagnetic and weak forces) was demonstrated

in 1983, a result which could not be anticipated in the time of Einstein's search. It now appears

that the common form of the gravity and electromagnetic forces arises from the fact that each of

them involves an exchange particle of zero mass, not because of an inherent symmetry which

would make them easy to unify.

5.2 gravitational force

Newton's law of universal gravitation states that every point mass in the universe attracts every

other point mass with a force that is directly proportional to the product of their masses and

inversely proportional to the square of the distance between them. (Separately it was shown that

large spherically symmetrical masses attract and are attracted as if all their mass were

concentrated at their centers.) This is a general physical law derived from empirical observations

by what Newton called induction. It is a part of classical mechanics and was formulated in

Newton's work Philosophiæ Naturalis Principia Mathematica ("the Principia"), first published on

5 July 1687. (When Newton's book was presented in 1686 to the Royal Society, Robert Hooke

made a claim that Newton had obtained the inverse square law from him – see History section

below.) In modern language, the law states the following:

Every point mass attracts every single other point mass by a force pointing along the line

intersecting both points. The force is proportional to the product of the two masses andinversely proportional to the square of the distance between them:

where:

• F is the force between the masses,

• G is the gravitational constant,

• m1 is the first mass,

• m2 is the second mass, and

• r is the distance between the centers of the masses.



Assuming SI units, F is measured in newtons (N), m1 and m 2 in kilograms (kg), r in meters (m),

and the constant G is approximately equal to 6.674×10−11

N m2 kg

−2.[4]

The value of the constant

G was first accurately determined from the results of the Cavendish experiment conducted by the

British scientist Henry Cavendish in 1798, although Cavendish did not himself calculate a

numerical value for G.[5]

This experiment was also the first test of Newton's theory of gravitation

between masses in the laboratory. It took place 111 years after the publication of Newton's

Principia and 71 years after Newton's death, so none of Newton's calculations could use the

value of G; instead he could only calculate a force relative to another force.

Newton's law of gravitation resembles Coulomb's law of electrical forces, which is used to

calculate the magnitude of electrical force between two charged bodies. Both are inverse-square

laws, in which force is inversely proportional to the square of the distance between the bodies.

Coulomb's law has the product of two charges in place of the product of the masses, and the

electrostatic constant in place of the gravitational constant.

Newton's law has since been superseded by Einstein's theory of general relativity, but it

continues to be used as an excellent approximation of the effects of gravity. Relativity is required

only when there is a need for extreme precision, or when dealing with gravitation for extremely

massive and dense objects.

5.3 centroid and center of gravity

In general when a rigid body lies in a field of force acts on each particle of the body. We

equivalently represent the system of forces by single force acting at a specific point. This point is

known as centre of gravity. We can extend this concept in many ways and get the various

equivalent parameters of a body, which could help us in dealing the situation directly on a rigid body rather than considering each individual particle of the rigid body. Various such parameters

include centre of gravity, moment of inertia, centroid , first and second moment of inertias of a

line or a rigid body. These parameters simplify the analysis of structures such as beams. Further

we will also study the surface area or volume of revolution of a line or area respectively.



5.3.1 CENTRE OF GRAVITY

Consider the following lamina. Let’s assume that it has been exposed to gravitational field.

Obviously every single element will experience a gravitational force towards the centre of earth.

Further let’s assume the body has practical dimensions, then we can easily conclude that all

elementary forces will be unidirectional and parallel.

Consider G to be the centroid of the irregular lamina. As shown in first figure we can easily

represent the net force passing through the single point G. We can also divide the entire region

into let’s say n small elements. Let’s say the coordinates to be (x1,y1), (x2,y2), (x3,y3)……….

(xn,yn) as shown in figure . Let ΔW1, ΔW2, ΔW3,……., ΔWn be the elementary forces acting on

the elementary elements

Clearly,

W = ΔW1+ ΔW2+ ΔW3 +…………..+ ΔWn

When n tends to infinity ΔW becomes infinitesimally small and can be replaced as dW.

Centre of gravity :

xc= / ,

yc= /

zc= /

in case of a three dimensional body)

where x,y are the coordinate of the small element and dw(or ΔW) the elemental force.

And we have seen that W.



For some type of surfaces of bodies there lies a probability that the centre of gravity may lie

outside the body. Secondly centre of gravity represents the entire lamina, therefore we can

replace the entire body by the single point with a force acting on it when needed. There is a

major difference between centre of mass and centre of gravity of a body. For centre of gravity we

integrate with respect to dW whereas for centre of mass we integrate with respect to dm. Mass is

a scalar quantity and force a vector quantity. For general practical size objects both of them turn

out to be the same as both of them are proportional and the force is unidirected (dW = dm*g)

.But when we consider large size objects such as a continent, results would turn out to be

different because here the vector nature of dW comes into play.

5.3.2 CENROIDS OF AREAS AND LINES

We have seen one method to find out the centre of gravity, there are other ways too. Let’s

consider plate of uniform thickness and a homogenous density. Now weight of small element is

directly proportional to its thickness, area and density as:

ΔW = t dA.

Where is the density per unit volume, t is the thickness , dA is the area of the small element.

Let’s consider plate of uniform thickness and a homogenous density. Now weight of small

element is directly proportional to its thickness, area and density as:

ΔW = t dA.

Where is the density per unit volume, t is the thickness, dA is the area of the small element.

So we can replace ΔW with this relationship in the expression we obtained in the prior topic.

Therefore we get:



Centroid of area :

xc= / ,

yc= /

zc= /

in case of a three dimensional body)

Where x,y are the coordinate of the small element and da(or ΔA) the elemental force.

Also A (total area of the plate).

(xc ,y c,zc) is called the centroid of area of the lamina. If the surface is homogenous we conclude

that it is the same as centre of gravity.

There can also arise a case where in cross-sectional area is constant and length is variable as in

the case of a rope or slender rod.

In such cases the situation modifies to:

ΔW = a dl.

Where is the weight per unit length, per unit cross-sectional area, A is the area of cross –

section, and dl the variable length.

So the above results reduce to:





x

Because of symmetrical nature will always turn out to be zero. Hence Qy = 0. So we can

conclude that the first moment about the axis will be zero about the axis of symmetry(y axis in

the above example). Further centroid also lies on the axis of symmetry (figure out why?). If a

body has more than one axis of symmetry then centroid will lie on the point of intersection of the

axes.

5.4 centroid for regular lamina and center of gravity for regular solids

5.4.1 Locating the centroid

5.4.1.1 Plumb line method

The centroid of a uniform planar lamina, such as

(a) below, may be determined, experimentally, by using a plumbline and a pin to find the center

of mass of a thin body of uniform density having the same shape. The body is held by the pin

inserted at a point near the body's perimeter, in such a way that it can freely rotate around the

pin; and the plumb line is dropped from the pin.

(b). The position of the plumbline is traced on the body. The experiment is repeated with the pin

inserted at a different point of the object. The intersection of the two lines is the centroid of the

figure (c).



(a) (b) (c)

This method can be extended (in theory) to concave shapes where the centroid lies outside the

shape, and to solids (of uniform density), but the positions of the plumb lines need to be recorded

by means other than drawing.

5.4.1.2 Balancing method

For convex two-dimensional shapes, the centroid can be found by balancing the shape on a

smaller shape, such as the top of a narrow cylinder. The centroid occurs somewhere within the

range of contact between the two shapes. In principle, progressively narrower cylinders can be

used to find the centroid to arbitrary accuracy. In practice air currents make this unfeasible.

However, by marking the overlap range from multiple balances, one can achieve a considerable

level of accuracy.

Of a finite set of points

The centroid of a finite set of points in is

This point minimizes the sum of squared Euclidean distances between itself and each point in the

set.

By geometric decomposition

The centroid of a plane figure can be computed by dividing it into a finite number of simpler

figures , computing the centroid and area of each part, and then

computing



Holes in the figure , overlaps between the parts, or parts that extend outside the figure can all

be handled using negative areas . Namely, the measures should be taken with positive andnegative signs in such a way that the sum of the signs of for all parts that enclose a given

point is 1 if belongs to , and 0 otherwise.

For example, the figure below (a) is easily divided into a square and a triangle, both with positive

area; and a circular hole, with negative area (b).

(a) 2D Object

(b) Object described using simpler elements

(c) Centroids of elements of the object

The centroid of each part can be found in any list of centroids of simple shapes (c). Then the

centroid of the figure is the weighted average of the three points. The horizontal position of the

centroid, from the left edge of the figure is



The vertical position of the centroid is found in the same way.

The same formula holds for any three-dimensional objects, except that each should be the

volume of , rather than its area. It also holds for any subset of , for any dimension , with

the areas replaced by the -dimensional measures of the parts.

By integral formula

The centroid of a subset X of can also be computed by the integral

where the integrals are taken over the whole space , and g is the characteristic function of the

subset, which is 1 inside X and 0 outside it. Note that the denominator is simply the measure of

the set X. This formula cannot be applied if the set X has zero measure, or if either integral

diverges.

Another formula for the centroid is

where Ck is the kth coordinate of C, and S k (z) is the measure of the intersection of X with the

hyperplane defined by the equation xk = z. Again, the denominator is simply the measure of X.

For a plane figure, in particular, the barycenter coordinates are

where A is the area of the figure X; Sy(x) is the length of the intersection of X with the vertical

line at abscissa x; and Sx(y) is the analogous quantity for the swapped axes.





This is a method of determining the centroid of an L-shaped object.

1. Divide the shape into two rectangles, as shown in fig 2. Find the centroids of these two

rectangles by drawing the diagonals. Draw a line joining the centroids. The centroid of

the shape must lie on this line AB.

2. Divide the shape into two other rectangles, as shown in fig 3. Find the centroids of these

two rectangles by drawing the diagonals. Draw a line joining the centroids. The centroid

of the L-shape must lie on this line CD.

3. As the centroid of the shape must lie along AB and also along CD, it is obvious that it is

at the intersection of these two lines, at O. The point O might not lie inside the L-shaped

object.

Of triangle and tetrahedron

The centroid of a triangle is the point of intersection of its medians (the lines joining each vertex

with the midpoint of the opposite side). The centroid divides each of the medians in the ratio 2:1,

which is to say it is located ⅓ of the perpendicular distance between each side and the opposing

point (see figures at right). Its Cartesian coordinates are the means of the coordinates of the three

vertices. That is, if the three vertices are , , and , thenthe centroid is



The centroid is therefore at in barycentric coordinates.

The centroid is also the physical center of mass if the triangle is made from a uniform sheet of

material; or if all the mass is concentrated at the three vertices, and evenly divided among them.

On the other hand, if the mass is distributed along the triangle's perimeter, with uniform linear

density, then the center of mass lies at the Spieker center (the incenter of the medial triangle),

which does not (in general) coincide with the geometric centroid of the full triangle.

The area of the triangle is 1.5 times the length of any side times the perpendicular distance from

the side to the centroid.

A triangle's centroid lies on its Euler line between its orthocenter and its circumcenter, exactly

twice as close to the latter as to the former.

Similar results hold for a tetrahedron: its centroid is the intersection of all line segments that

connect each vertex to the centroid of the opposite face. These line segments are divided by the

centroid in the ratio 3:1. The result generalizes to any n-dimensional simplex in the obvious way.

If the set of vertices of a simplex is , then considering the vertices as vectors, the

centroid is

The geometric centroid coincides with the center of mass if the mass is uniformly distributedover the whole simplex, or concentrated at the vertices as n equal masses.

The isogonal conjugate of a triangle's centroid is its symmedian point.

Centroid of polygon

The centroid of a non-self-intersecting closed polygon defined by n vertices (x0,y0), (x1,y1), ...,

(xn−1,yn−1) is the point (Cx, Cy), where



and where A is the polygon's signed area,

In these formulas, the vertices are assumed to be numbered in order of their occurrence along the

polygon's perimeter, and the vertex ( xn, yn ) is assumed to be the same as ( x 0, y0 ). Note that if

the points are numbered in clockwise order the area A, computed as above, will have a negative

sign; but the centroid coordinates will be correct even in this case.

Centroid of cone or pyramid

The centroid of a cone or pyramid is located on the line segment that connects the apex to the

centroid of the base. For a solid cone or pyramid, the centroid is 1/4 the distance from the base to

the apex. For a cone or pyramid that is just a shell (hollow) with no base, the centroid is 1/3 thedistance from the base plane to the apex.

Finding the Centers of Circles, Rectangles and Parallelograms From a pasteboard we draw and

cut out a circle 7 or 8 cm in diameter. If the circle is drawn with compasses, the center should be

marked with a pen, and marked with an X.

If the circle is drawn with a glass turned upside down, the center can be found by the intersection

of two diameters. The diameters can be drawn with a ruler. But it is difficult to be sure if the

ruler passes exactly through the center when we do not know exactly where the center is located.

An alternative procedure to find the diameter and center of the circle involves the paper. Laterwe will perform experiments with the pasteboards, so it is better not to fold them. For this reason

the folding we discuss here should be done with similar figures made from sheets of paper. For

example, we place the paste board circle on a sheet of paper and cut out a similar circle of paper.

We then fold the paper circle in two equal halves. We fold it once more so that it is divided into

four equal parts. We can then use a pen to draw the diameters in the paper circle. The center of

the circle is the intersection of the diameters. A hole should be made at the center. By placing the

paper circle on the pasteboard circle, we can mark the center of the circle on the pasteboard.



We cut out a pasteboard in the shape of a rectangle with sides of 6 cm and 12 cm. There are two

ways to find the center. The simplest one is to connect the opposite vertices. The center of the

rectangle is the intersection of these diagonals, marked with the X.

The other way is to find (with a ruler or by folding) the central point of each side. We then

connect the middle points of opposite sides. The center is the intersection of these straight lines.

The parallelogram is a plane quadrilateral in which the opposite sides are parallel to one another.

A parallelogram is cut out from a pasteboard with sides

of 6 cm and 12 cm, with the smallest internal angle being 30o (or 45o). The center of this

parallelogram can be found by the two methods we used for the rectangle.

The Triangle Centers

There are three types of triangle: equilateral (three equal sides), isosceles (only two sides of the

same length), and scalene (with three different sides). Every triangle has four special centers:

circumcenter (C), barycenter or triangle centroid (B), orthocenter (O), and incenter (I). We will

find these four special points in the case of an isosceles triangle with a base of 6 cm and height of



12 cm. With these dimensions each one of the equal sides has a length of 12.37 cm. We draw and

cut out a triangle of this size from a pasteboard. We also cut out four equal triangles from a sheet

of paper. Each one of these four paper triangles will be used to draw the straight lines and locate

one of the

special points. When necessary, also the folding should be done with these paper triangles.

The circumcenter, C, is the intersection of the perpendicular bisectors. A perpendicular bisector

of a straight line AB is a straight line perpendicular to AB and passing through its midpoint M.

To find the midpoint of each side we can use a ruler. With a T-square or using the pasteboard

rectangle we draw a straight line perpendicular to each side through its midpoint. The

intersection of these lines is the circumcenter (C).



5.5 Position of center of gravity of compound bodies and centroid of composition area

5.5.1 Of an L-shaped object

This is a method of determining the center of mass of an L-shaped object.

1. Divide the shape into two rectangles. Find the center of masses of these two rectangles bydrawing the diagonals. Draw a line joining the centers of mass. The center of mass of the

shape must lie on this line AB.

2. Divide the shape into two other rectangles, as shown in fig 3. Find the centers of mass of

these two rectangles by drawing the diagonals. Draw a line joining the centers of mass.

The center of mass of the L-shape must lie on this line CD.



3. As the center of mass of the shape must lie along AB and also along CD, it is obvious

that it is at the intersection of these two lines, at O. (The point O may or may not lie

inside the L-shaped object.)

5.5.2 Of a composite shape

This method is useful when one wishes to find the location of the centroid or center of mass of

an object that is easily divided into elementary shapes, whose centers of mass are easy to find

(see List of centroids). Here the center of mass will only be found in the x direction. The same

procedure may be followed to locate the center of mass in the y direction.

The shape. It is easily divided into a square, triangle, and circle. Note that the circle will have

negative area. From the List of centroids, we note the coordinates of the individual centroids.

From equation 1 above:

units.

The center of mass of this figure is at a distance of 8.5 units from the left corner of the figure.



5.5.3 CENTROIDS OF COMPOSITE AREAS

We can end up in situations where the given plate can be broken up into various segments. In

such cases we can replace the separate sections by their centre of gravity. One centroid takes care

of the entire weight of the section.

Further overall centre of gravity can be found out using the same concept we studied before.

Xc (W 1 + W 2 + W 3+…..+Wn) = xc1W1 + x c2W2 + x c3W3+…….……..+xcnWn

Yc (W 1 + W 2 + W 3+…..+Wn) = yc1W1 + y c2W2 + y c3W3+…….……..+ycnWn

Once again if the plate is homogenous and of uniform thickness, centre of gravity turns out to be

equal to the centroid of the area. In a similar way we can also define centroid of this composite

area by:

Xc (A 1 + A 2 + A 3+…..+An) = xc1A1 + x c2A2 + x c3A3+…….……..+xcnAn

Yc (A 1 + A 2 + A 3+…..+An) = yc1A1 + y c2A2 + y c3A3+…….……..+ycnAn

We can also introduce the concept of negative area. It simply denotes the region where any area

is left vacant. We will see its usage in the coming problems.

The following diagrams depict a list of centroids. A centroid of an object in -dimensional

space is the intersection of all hyperplanes that divide into two parts of equal moment about

the hyperplane. Informally, it is the "average" of all points of . For an object of uniform

composition (mass, density, etc.) the centroid of a body is also its centre of mass.





5.6 CG of bodies with portions removed

Rigid body is composed of very large numbers of particles. Mass of rigid body is distributed

closely. Thus, the distribution of mass can be treated as continuous. The mathematical expression

for rigid body, therefore, is modified involving integration. The integral expressions of the

components of position of COM in three mutually perpendicular directions are :

Note that the term in the numerator of the expression is nothing but the product of the mass of

particle like small volumetric element and its distance from the origin along the axis. Evidently,

this terms when integrated is equal to sum of all such products of mass elements constituting the

rigid body.

Evaluation of above integrals is simplified, if the density of the rigid body is uniform. In that

case,

Substituting,

We must understand here that once we determine COM of a rigid body, the same can be treated

as a particle at COM with all the mass assigned to that particle. This concept helps to find COM

of a system of rigid bodies, comprising of many rigid bodies. Similarly, when a portion is

removed from a rigid body, the COM of the rigid body can be obtained by treating the "portion

removed" and the "remaining body" as particles. We shall see the working of this concept in the

example given in the next section.

Symmetry and COM of rigid body

Evaluation of the integrals for determining COM is very difficult for irregularly shaped bodies.

On the other hand, symmetry plays important role in determining COM of a regularly shaped

rigid body. There are certain simplifying facts about symmetry and COM :

1. If symmetry is about a point, then COM lies on that point. For example, COM of a

spherical ball of uniform density is its center.

2. If symmetry is about a line, then COM lies on that line. For example, COM of a cone of

uniform density lies on cone axis.

3. If symmetry is about a plane, then COM lies on that plane. For example, COM of a

cricket bat lies on the central plane.

The test of symmetry about a straight line or a plane is that the body on one side is replicated on

the other side.



Example

Problem : A small circular portion of radius r/4 is taken out from a circular disc of uniform

thickness having radius "r" and mass "m" as shown in the figure. Determine the COM of the

remaining portion of the uniform disc.

COM of remaining portion of circular disc

Figure : A small circular portion of radius r/4 is

taken out.

Solution : If we start from integral to determine COM of the remaining disc portion, then itwould be a really complex proposition. Here, we shall make use of the connection between

symmetry and COM. We note that the COM of the given disc is "O" and COM of the smaller

disc removed is "O'". How can we use these fact to find center of mass of the remaining portion ?

The main idea here is that we can treat regular bodies with known COM as particles, which are

separated by a known distance. Then, we shall employ the expression of COM for two particles

to determine the COM of the remaining portion. We must realize that when a portion is removed

from the bigger disc on the right side, the COM of the remaining portion shifts towards left side

(heavier side).

The test of symmetry about a straight line or a plane is that the body on one side is replicated on

the other side. We see here that the remaining portion of the disc is not symmetric about y-axis,

but is symmetric about x-axis. It means that the COM of the remaining portion lies on the x-axis

on the left side of the center of original disc. It also means that we need to employ the expression

of COM for one dimension only.



While employing expression of COM, we use the logic as explained here. The original disc (with

known COM) is equivalent to two particles system comprising of (i) remaining portion (with

unknown COM) and (ii) smaller disc (with known COM). Now, x-component of the COM of

original disc is :

COM of remaining portion of circular disc

Figure : A small circular portion of radius r/4 is

taken out.

But, x-component of the COM of original disc coincides with origin of the coordinate system.

Further, let us denote the remaining disc by subscript "r" and the smaller circular disk removed

by subscript "s".

As evident from figure, . We, now, need to find mass of smaller disc, , and mass of remaining

portion, , using the fact that the density is uniform.

Review Questions

1. Describe Concept of gravity.

2. Define gravitational force.

3. Explain centroid and center of gravity.

4. Explain centroid for regular lamina and center of gravity for regular solids.

5. Define Position of center of gravity of compound bodies and centroid of composition

area.

6. Define CG of bodies with portions removed.



Chapter-6

Laws of Motion


Momentum

Learning Objectives

1. Concept of momentum

2. Newton’s laws of motion, their application

3. derivation of force equation from second law of motion, numerical problems on second

law of motion

4. piles, lifts, bodies tied with string

5. Newton’s third law of motion and numerical problems based on it

6. conservation of momentum

7. impulsive force (definition only).

6.1 Concept of momentum

In classical mechanics, linear momentum or translational momentum (pl. momenta; SI

unit kg m/s, or equivalently, N s) is the product of the mass and velocity of an object. Forexample, a heavy truck moving fast has a large momentum—it takes a large and prolonged

force to get the truck up to this speed, and it takes a large and prolonged force to bring it to a

stop afterwards. If the truck were lighter, or moving more slowly, then it would have less

momentum.

Like velocity, linear momentum is a vector quantity, possessing a direction as well as a

magnitude:

Linear momentum is also a conserved quantity, meaning that if a closed system is not

affected by external forces, its total linear momentum cannot change. In classical mechanics,

conservation of linear momentum is implied by Newton's laws; but it also holds in special

relativity (with a modified formula) and, with appropriate definitions, a (generalized) linear

momentum conservation law holds in electrodynamics, quantum mechanics, quantum field

theory, and general relativity.



6.1.1 Newtonian mechanics

Momentum has a direction as well as magnitude. Quantities that have both a magnitude and a

direction are known as vector quantities. Because momentum has a direction, it can be used to

predict the resulting direction of objects after they collide, as well as their speeds. Below, the

basic properties of momentum are described in one dimension. The vector equations are almost

identical to the scalar equations (see multiple dimensions).

6.1.1.1 Single particle

The momentum of a particle is traditionally represented by the letter p. It is the product of two

quantities, the mass (represented by the letter m) and velocity (v):

The units of momentum are the product of the units of mass and velocity. In SI units, if the massis in kilograms and the velocity in meters per second, then the momentum is in kilograms

meters/second (kg m/s). Being a vector, momentum has magnitude and direction. For example, a

model airplane of 1 kg, traveling due north at 1 m/s in straight and level flight, has a momentum

of 1 kg m/s due north measured from the ground.

6.1.1.2 Many particles

The momentum of a system of particles is the sum of their momenta. If two particles have

masses m1 and m 2, and velocities v1 and v 2, the total momentum is

The momenta of more than two particles can be added in the same way.

A system of particles has a center of mass, a point determined by the weighted sum of their

positions:

If all the particles are moving, the center of mass will generally be moving as well. If the center

of mass is moving at velocity vcm, the momentum is:



This is known as Euler's first law.

6.1.2 Relation to force

If a force F is applied to a particle for a time interval Δt, the momentum of the particle changes

by an amount

In differential form, this gives Newton's second law: the rate of change of the momentum of a

particle is equal to the force F acting on it:

If the force depends on time, the change in momentum (or impulse) between times t1 and t 2 is

The second law only applies to a particle that does not exchange matter with its surroundings,

and so it is equivalent to write

so the force is equal to mass times acceleration.

Example: a model airplane of 1 kg accelerates from rest to a velocity of 6 m/s due north in 2 s.

The thrust required to produce this acceleration is 3 newton. The change in momentum is

6 kg m/s. The rate of change of momentum is 3 (kg m/s)/s = 3 N.

6.1.3 Conservation



A Newton's cradle demonstrates conservation of momentum.

In a closed system (one that does not exchange any matter with the outside and is not acted on by

outside forces) the total momentum is constant. This fact, known as the law of conservation of

momentum, is implied by Newton's laws of motion.[5]

Suppose, for example, that two particles

interact. Because of the third law, the forces between them are equal and opposite. If the particles

are numbered 1 and 2, the second law states that F1 = dp 1/dt and F2 = dp 2/dt. Therefore

or

If the velocities of the particles are u1 and u 2 before the interaction, and afterwards they are v 1

and v2, then

This law holds no matter how complicated the force is between particles. Similarly, if there are

several particles, the momentum exchanged between each pair of particles adds up to zero, so the

total change in momentum is zero. This conservation law applies to all interactions, including

collisions and separations caused by explosive forces. It can also be generalized to situations

where Newton's laws do not hold, for example in the theory of relativity and in electrodynamics.

6.1.4 Dependence on reference frame

Newton's apple in Einstein's elevator. In person A's frame of reference, the apple has non-zero

velocity and momentum. In the elevator's and person B's frames of reference, it has zero velocity

and momentum.



Momentum is a measurable quantity, and the measurement depends on the motion of the

observer. For example, if an apple is sitting in a glass elevator that is descending, an outside

observer looking into the elevator sees the apple moving, so to that observer the apple has a

nonzero momentum. To someone inside the elevator, the apple does not move, so it has zero

momentum. The two observers each have a frame of reference in which they observe motions,

and if the elevator is descending steadily they will see behavior that is consistent with the same

physical laws.

Suppose a particle has position x in a stationary frame of reference. From the point of view of

another frame of reference moving at a uniform speed u, the position (represented by a primed

coordinate) changes with time as

This is called a Galilean transformation. If the particle is moving at speed dx/dt = v in the first

frame of reference, in the second it is moving at speed

Since u does not change, the accelerations are the same:

Thus, momentum is conserved in both reference frames. Moreover, as long as the force has the

same form in both frames, Newton's second law is unchanged. Forces such as Newtonian

gravity, which depend only on the scalar distance between objects, satisfy this criterion. This

independence of reference frame is called Newtonian relativity or Galilean invariance.[7]

A change of reference frame can often simplify calculations of motion. For example, in a

collision of two particles a reference frame can be chosen where one particle begins at rest.

Another commonly used reference frame is the center of mass frame, one that is moving with the

center of mass. In this frame, the total momentum is zero.

6.1.5 Application to collisions

By itself, the law of conservation of momentum is not enough to determine the motion of

particles after a collision. Another property of the motion, kinetic energy, must be known. This is

not necessarily conserved. If it is conserved, the collision is called an elastic collision; if not, it is

an inelastic collision.



6.1.5.1 Elastic collisions

Elastic collision of equal masses

Elastic collision of unequal masses

An elastic collision is one in which no kinetic energy is lost. Perfectly elastic "collisions" can

occur when the objects do not touch each other, as for example in atomic or nuclear scattering

where electric repulsion keeps them apart. A slingshot maneuver of a satellite around a planetcan also be viewed as a perfectly elastic collision from a distance. A collision between two pool

balls is a good example of an almost totally elastic collision, due to their high rigidity; but when

bodies come in contact there is always some dissipation.[8]

A head-on elastic collision between two bodies can be represented by velocities in one

dimension, along a line passing through the bodies. If the velocities are u1 and u 2 before the

collision and v1 and v 2 after, the equations expressing conservation of momentum and kinetic

energy are:

A change of reference frame can often simplify the analysis of a collision. For example, suppose

there are two bodies of equal mass m, one stationary and one approaching the other at a speed v

(as in the figure). The center of mass is moving at speed v/2 and both bodies are moving towards

it at speed v/2. Because of the symmetry, after the collision both must be moving away from the

center of mass at the same speed. Adding the speed of the center of mass to both, we find that the

body that was moving is now stopped and the other is moving away at speed v. The bodies have

exchanged their velocities. Regardless of the velocities of the bodies, a switch to the center of

mass frame leads us to the same conclusion. Therefore, the final velocities are given by

In general, when the initial velocities are known, the final velocities are given by[9]



If one body has much greater mass than the other, its velocity will be little affected by a collision

while the other body will experience a large change.

6.1.5.2 Inelastic collisions

a perfectly inelastic collision between equal masses

In an inelastic collision, some of the kinetic energy of the colliding bodies is converted into other

forms of energy such as heat or sound. Examples include traffic collisions, in which the effect of

lost kinetic energy can be seen in the damage to the vehicles; electrons losing some of their

energy to atoms (as in the Franck–Hertz experiment); and particle accelerators in which the

kinetic energy is converted into mass in the form of new particles.

In a perfectly inelastic collision (such as a bug hitting a windshield), both bodies have the same

motion afterwards. If one body is motionless to begin with, the equation for conservation of

momentum is

so

In a frame of reference moving at the speed v), the objects are brought to rest by the collision and

100% of the kinetic energy is converted.

One measure of the inelasticity of the collision is the coefficient of restitution CR , defined as the

ratio of relative velocity of separation to relative velocity of approach. In applying this measure

to ball sports, this can be easily measured using the following formula:



The momentum and energy equations also apply to the motions of objects that begin together

and then move apart. For example, an explosion is the result of a chain reaction that transforms

potential energy stored in chemical, mechanical, or nuclear form into kinetic energy, acoustic

energy, and electromagnetic radiation. Rockets also make use of conservation of momentum:

propellant is thrust outward, gaining momentum, and an equal and opposite momentum is

imparted to the rocket.

Multiple dimensions

Two-dimensional elastic collision.

There is no motion perpendicular to the image, so only two components are needed to represent

the velocities and momenta. The two blue vectors represent velocities after the collision and add

vectorially to get the initial (red) velocity.

Real motion has both direction and magnitude and must be represented by a vector. In a

coordinate system with x, y, z axes, velocity has components vx in the x direction, v y in the y

direction, vz in the z direction. The vector is represented by a boldface symbol:

Similarly, the momentum is a vector quantity and is represented by a boldface symbol:

The equations in the previous sections work in vector form if the scalars p and v are replaced by

vectors p and v. Each vector equation represents three scalar equations. For example,

represents three equations:



The kinetic energy equations are exceptions to the above replacement rule. The equations arestill one-dimensional, but each scalar represents the magnitude of the vector, for example,

Each vector equation represents three scalar equations. Often coordinates can be chosen so that

only two components are needed, as in the figure. Each component can be obtained separately

and the results combined to produce a vector result.

A simple construction involving the center of mass frame can be used to show that if a stationary

elastic sphere is struck by a moving sphere, the two will head off at right angles after thecollision (as in the figure).

6.1.6 Objects of variable mass

The concept of momentum plays a fundamental role in explaining the behavior of variable-mass

objects such as a rocket ejecting fuel or a star accreting gas. In analyzing such an object, one

treats the object's mass as a function that varies with time: m(t). The momentum of the object at

time t is therefore p(t) = m(t)v(t). One might then try to invoke Newton's second law of motion

by saying that the external force F on the object is related to its momentum p(t) by F = dp/dt, but

this is incorrect, as is the related expression found by applying the product rule to d(mv)/dt:

This equation does not correctly describe the motion of variable-mass objects. The correct

equation is

where u is the velocity of the ejected/accreted mass as seen in the object's rest frame. This is

distinct from v, which is the velocity of the object itself as seen in an inertial frame.

This equation is derived by keeping track of both the momentum of the object as well as the

momentum of the ejected/accreted mass. When considered together, the object and the mass

constitute a closed system in which total momentum is conserved.



6.1.6.1 Generalized coordinates

Newton's laws can be difficult to apply to many kinds of motion because the motion is limited by

constraints. For example, a bead on an abacus is constrained to move along its wire and a

pendulum bob is constrained to swing at a fixed distance from the pivot. Many such constraints

can be incorporated by changing the normal Cartesian coordinates to a set of generalized

coordinates that may be fewer in number. Refined mathematical methods have been developed

for solving mechanics problems in generalized coordinates. They introduce a generalized

momentum, also known as the canonical or conjugate momentum, that extends the concepts of

both linear momentum and angular momentum. To distinguish it from generalized momentum,

the product of mass and velocity is also referred to as mechanical, kinetic or kinematic

momentum. The two main methods are described below.

6.1.7 Lagrangian mechanics

In Lagrangian mechanics, a Lagrangian is defined as the difference between the kinetic energy T

and the potential energy V:

If the generalized coordinates are represented as a vector q = (q 1, q2, ... , q N) and time

differentiation is represented by a dot over the variable, then the equations of motion (known as

the Lagrange or Euler–Lagrange equations) are a set of N equations:

If a coordinate qi is not a Cartesian coordinate, the associated generalized momentum component

pi does not necessarily have the dimensions of linear momentum. Even if q i is a Cartesian

coordinate, pi will not be the same as the mechanical momentum if the potential depends on

velocity.[6]

Some sources represent the kinematic momentum by the symbol Π.

In this mathematical framework, a generalized momentum is associated with the generalized

coordinates. Its components are defined as

Each component p j is said to be the conjugate momentum for the coordinate q j.



Now if a given coordinate qi does not appear in the Lagrangian (although its time derivative

might appear), then

This is the generalization of the conservation of momentum.

Even if the generalized coordinates are just the ordinary spatial coordinates, the conjugate

momenta are not necessarily the ordinary momentum coordinates. An example is found in the

section on electromagnetism.

6.1.8 Hamiltonian mechanics

In Hamiltonian mechanics, the Lagrangian (a function of generalized coordinates and their

derivatives) is replaced by a Hamiltonian that is a function of generalized coordinates and

momentum. The Hamiltonian is defined as

where the momentum is obtained by differentiating the Lagrangian as above. The Hamiltonian

equations of motion are

As in Lagrangian mechanics, if a generalized coordinate does not appear in the Hamiltonian, its

conjugate momentum component is conserved.

Symmetry and conservation

Conservation of momentum is a mathematical consequence of the homogeneity (shift symmetry)

of space (position in space is the canonical conjugate quantity to momentum). That is,

conservation of momentum is a consequence of the fact that the laws of physics do not depend

on position; this is a special case of Noether's theorem.

6.1.9 Relativistic mechanics



Lorentz invariance

Newtonian physics assumes that absolute time and space exist outside of any observer; this gives

rise to the Galilean invariance described earlier. It also results in a prediction that the speed of

light can vary from one reference frame to another. This is contrary to observation. In the special

theory of relativity, Einstein keeps the postulate that the equations of motion do not depend on

the reference frame, but assumes that the speed of light c is invariant. As a result, position and

time in two reference frames are related by the Lorentz transformation instead of the Galilean

transformation.

Consider, for example, a reference frame moving relative to another at velocity v in the x

direction. The Galilean transformation gives the coordinates of the moving frame as

while the Lorentz transformation gives

where γ is the Lorentz factor:

Newton's second law, with mass fixed, is not invariant under a Lorentz transformation. However,

it can be made invariant by making the inertial mass m of an object a function of velocity:

m0 is the object's invariant mass.

The modified momentum,

obeys Newton's second law:



Within the domain of classical mechanics, relativistic momentum closely approximates

Newtonian momentum: at low velocity, γm0v is approximately equal to m 0v, the Newtonian

expression for momentum.

Four-vector formulation

In the theory of relativity, physical quantities are expressed in terms of four-vectors that include

time as a fourth coordinate along with the three space coordinates. These vectors are generally

represented by capital letters, for example R for position. The expression for the four-momentum

depends on how the coordinates are expressed. Time may be given in its normal units or

multiplied by the speed of light so that all the components of the four-vector have dimensions of

length. If the latter scaling is used, an interval of proper time, τ, defined by

is invariant under Lorentz transformations (in this expression and in what follows the (+ − − −)

metric signature has been used, different authors use different conventions). Mathematically this

invariance can be ensured in one of two ways: by treating the four-vectors as Euclidean vectors

and multiplying time by the square root of -1; or by keeping time a real quantity and embedding

the vectors in a Minkowski space. In a Minkowski space, the scalar product of two four-vectors

U = (U 0,U1,U2,U3) and V = (V 0,V1,V2,V3) is defined as

In all the coordinate systems, the (contravariant) relativistic four-velocity is defined by

and the (contravariant) four-momentum is

where m0 is the invariant mass. If R = (ct,x,y,z) (in Minkowski space), then

Using Einstein's mass-energy equivalence, E = mc2, this can be rewritten as



Thus, conservation of four-momentum is Lorentz-invariant and implies conservation of both

mass and energy.

The magnitude of the momentum four-vector is equal to m0c:

and is invariant across all reference frames.

The relativistic energy–momentum relationship holds even for massless particles such as

photons; by setting m0 = 0 it follows that

In a game of relativistic "billiards", if a stationary particle is hit by a moving particle in an elastic

collision, the paths formed by the two afterwards will form an acute angle. This is unlike the

non-relativistic case where they travel at right angles.

6.1.10 Classical electromagnetism

In Newtonian mechanics, the law of conservation of momentum can be derived from the law of

action and reaction, which states that the forces between two particles are equal and opposite.Electromagnetic forces violate this law. Under some circumstances one moving charged particle

can exert a force on another without any return force.[31]

Moreover, Maxwell's equations, the

foundation of classical electrodynamics, are Lorentz-invariant. However, momentum is still

conserved.

Vacuum

In Maxwell's equations, the forces between particles are mediated by electric and magnetic

fields. The electromagnetic force (Lorentz force) on a particle with charge q due to a

combination of electric field E and magnetic field (as given by the "B-field" B) is

This force imparts a momentum to the particle, so by Newton's second law the particle must

impart a momentum to the electromagnetic fields.[32]



In a vacuum, the momentum per unit volume is

where μ0 is the vacuum permeability and c is the speed of light. The momentum density is

proportional to the Poynting vector S which gives the directional rate of energy transfer per unit

area:

If momentum is to be conserved in a volume V, changes in the momentum of matter through the

Lorentz force must be balanced by changes in the momentum of the electromagnetic field and

outflow of momentum. If Pmech

is the momentum of all the particles in a volume V, and the

particles are treated as a continuum, then Newton's second law gives

The electromagnetic momentum is

and the equation for conservation of each component i of the momentum is

The term on the right is an integral over the surface S representing momentum flow into and out

of the volume, and n j is a component of the surface normal of S. The quantity T i j is called the

Maxwell stress tensor, defined as

[32]

Media



The above results are for the microscopic Maxwell equations, applicable to electromagnetic

forces in a vacuum (or on a very small scale in media). It is more difficult to define momentum

density in media because the division into electromagnetic and mechanical is arbitrary. The

definition of electromagnetic momentum density is modified to

where the H-field H is related to the B-field and the magnetization M by

The electromagnetic stress tensor depends on the properties of the media.

Particle in field

If a charged particle q moves in an electromagnetic field, its kinematic momentum m v is not

conserved. However, it has a canonical momentum that is conserved.

Lagrangian and Hamiltonian formulation

The kinetic momentum p is different to the canonical momentum P (synonymous with the

generalized momentum) conjugate to the ordinary position coordinates r, because P includes a

contribution from the electric potential φ(r, t) and vector potential A(r, t):

Classical mechanics Relativistic mechanics

Lagrangian

Canonical

momentum

Kinetic

momentum



Hamiltonian

where = v is the velocity (see time derivative) and e is the electric charge of the particle. See

also Electromagnetism (momentum). If neither φ nor A depends on position, P is conserved.

The classical Hamiltonian for a particle in any field equals the total energy of the system - the

kinetic energy T = p2/2m (where p

2 = p·p, see dot product) plus the potential energy V. For a

particle in an electromagnetic field, the potential energy is V = eφ, and since the kinetic energy T

always corresponds to the kinetic momentum p, replacing the kinetic momentum by the above

equation (p = P − e A) leads to the Hamiltonian in the table.

These Lagrangian and Hamiltonian expressons can derive the Lorentz force.

Canonical commutation relations

The kinetic momentum (p above) satisfies the commutation relation:

where: j, k, ℓ are indices labelling vector components, B ℓ is a component of the magnetic field,

and εkjℓ is the Levi-Civita symbol, here in 3-dimensions.

6.1.11 Quantum mechanics

In quantum mechanics, momentum is defined as an operator on the wave function. The

Heisenberg uncertainty principle defines limits on how accurately the momentum and position of

a single observable system can be known at once. In quantum mechanics, position and

momentum are conjugate variables.

For a single particle described in the position basis the momentum operator can be written as



where is the gradient operator, ħ is the reduced Planck constant, and i is the imaginary unit.

This is a commonly encountered form of the momentum operator, though the momentum

operator in other bases can take other forms. For example, in momentum space the momentum

operator is represented as

where the operator p acting on a wave function ψ(p) yields that wave function multiplied by the

value p, in an analogous fashion to the way that the position operator acting on a wave function

ψ(x) yields that wave function multiplied by the value x.

For both massive and massless objects, relativistic momentum is related to the de Broglie

wavelength λ by

Electromagnetic radiation (including visible light, ultraviolet light, and radio waves) is carried by

photons. Even though photons (the particle aspect of light) have no mass, they still carry

momentum. This leads to applications such as the solar sail. The calculation of the momentum of

light within dielectric media is somewhat controversial.

6.2 Newton’s laws of motion

Newton's First Law of Motion:

I. Every object in a state of uniform motion tends to remain in

that state of motion unless an external force is applied to it.

This we recognize as essentially Galileo's concept of inertia, and this is often termed simply the

"Law of Inertia".

Newton's Second Law of Motion:



II. The relationship between an object's mass m, its acceleration

a, and the applied force F is F = ma. Acceleration and force are

vectors (as indicated by their symbols being displayed in slant

bold font); in this law the direction of the force vector is the sameas the direction of the acceleration vector.

This is the most powerful of Newton's three Laws, because it allows quantitative calculations of

dynamics: how do velocities change when forces are applied. Notice the fundamental difference

between Newton's 2nd Law and the dynamics of Aristotle: according to Newton, a force causes

only a change in velocity (an acceleration); it does not maintain the velocity as Aristotle held.

This is sometimes summarized by saying that under Newton, F = ma, but under Aristotle F =

mv, where v is the velocity. Thus, according to Aristotle there is only a velocity if there is a

force, but according to Newton an object with a certain velocity maintains that velocity unless a

force acts on it to cause an acceleration (that is, a change in the velocity). As we have noted

earlier in conjunction with the discussion of Galileo, Aristotle's view seems to be more in accord

with common sense, but that is because of a failure to appreciate the role played by frictional

forces. Once account is taken of all forces acting in a given situation it is the dynamics of Galileo

and Newton, not of Aristotle, that are found to be in accord with the observations.

Newton's Third Law of Motion:

III. For every action there is an equal and opposite reaction.

This law is exemplified by what happens if we step off a boat onto the bank of a lake: as we

move in the direction of the shore, the boat tends to move in the opposite direction (leaving us

facedown in the water, if we aren't careful!).

6.3 their application

Everyday Applications of Newton's First Law



There are many applications of Newton's first law of motion. Consider some of your experiences

in an automobile. Have you ever observed the behavior of coffee in a coffee cup filled to the rim

while starting a car from rest or while bringing a car to rest from a state of motion? Coffee

"keeps on doing what it is doing." When you accelerate a car from rest, the road provides an

unbalanced force on the spinning wheels to push the car forward; yet the coffee (that was at rest)

wants to stay at rest. While the car accelerates forward, the coffee remains in the same position;

subsequently, the car accelerates out from under the coffee and the coffee spills in your lap. On

the other hand, when braking from a state of motion the coffee continues forward with the same

speed and in the same direction, ultimately hitting the windshield or the dash. Coffee in motion

stays in motion.

There are many more applications of Newton's first law of motion. Several applications are listed

below. Perhaps you could think about the law of inertia and provide explanations for each

application.

• Blood rushes from your head to your feet while quickly stopping when riding

on a descending elevator.

• The head of a hammer can be tightened onto the wooden handle by banging

the bottom of the handle against a hard surface.

• A brick is painlessly broken over the hand of a physics teacher by slamming it

with a hammer. (CAUTION: do not attempt this at home!)

• To dislodge ketchup from the bottom of a ketchup bottle, it is often turned

upside down and thrusted downward at high speeds and then abruptly halted.

• Headrests are placed in cars to prevent whiplash injuries during rear-endcollisions.

• While riding a skateboard (or wagon or bicycle), you fly forward off the board

when hitting a curb or rock or other object that abruptly halts the motion of the

skateboard.

Applications of Newton's Second Law

An apple falling to the ground must be under the influence of a force, according to his second

law. That force is gravity, which causes the apple to accelerate toward Earth's center.

Applications of Newton's third Law

Newton reasoned that the moon might be under the influence of Earth's gravity, as well, but

he had to explain why the moon didn't fall into Earth. Unlike the falling apple, it moved

parallel to Earth's surface.



6.4 derivation of force equation from second law of motion

The second law states that the net force on an object is equal to the rate of change (that is, the

derivative) of its linear momentum p in an inertial reference frame:

The second law can also be stated in terms of an object's acceleration. Since the law is valid

only for constant-mass systems, the mass can be taken outside the differentiation operator by

the constant factor rule in differentiation. Thus,

where F is the net force applied, m is the mass of the body, and a is the body's acceleration.

Thus, the net force applied to a body produces a proportional acceleration. In other words, if

a body is accelerating, then there is a force on it.

Consistent with the first law, the time derivative of the momentum is non-zero when the

momentum changes direction, even if there is no change in its magnitude; such is the case

with uniform circular motion. The relationship also implies the conservation of momentum:

when the net force on the body is zero, the momentum of the body is constant. Any net forceis equal to the rate of change of the momentum.

Any mass that is gained or lost by the system will cause a change in momentum that is not

the result of an external force. A different equation is necessary for variable-mass systems.

Newton's second law requires modification if the effects of special relativity are to be taken

into account, because at high speeds the approximation that momentum is the product of rest

mass and velocity is not accurate.

6.4.1 Impulse

An impulse J occurs when a force F acts over an interval of time Δt, and it is given by



Since force is the time derivative of momentum, it follows that

This relation between impulse and momentum is closer to Newton's wording of the second

law.

Impulse is a concept frequently used in the analysis of collisions and impacts.

6.4.2 Variable-mass systems

Variable-mass systems, like a rocket burning fuel and ejecting spent gases, are not closed and

cannot be directly treated by making mass a function of time in the second law; that is, the

following formula is wrong:

The falsehood of this formula can be seen by noting that it does not respect Galilean

invariance: a variable-mass object with F = 0 in one frame will be seen to have F ≠ 0 in

another frame.

The correct equation of motion for a body whose mass m varies with time by either ejecting

or accreting mass is obtained by applying the second law to the entire, constant-mass system

consisting of the body and its ejected/accreted mass; the result is

where u is the relative velocity of the escaping or incoming mass as seen by the body. From

this equation one can derive the Tsiolkovsky rocket equation.

Under some conventions, the quantity u dm/dt on the left-hand side, known as the thrust, is

defined as a force (the force exerted on the body by the changing mass, such as rocket

exhaust) and is included in the quantity F. Then, by substituting the definition of

acceleration, the equation becomes F = m a.

6.4.3 numerical problems on second law of motion



Type – 1 :-

Question – 1 – Calculate the force needed to speed up a car with a rate of 5ms-2

, if the mass

of the car is 1000 kg.

Solution:

According to questions:

Acceleration (a) = 5m/s2 and Mass (m) = 1000 kg, therefore, Force (F) =?

We know that, F = m x a

= 1000 kg x 5m/s2

= 5000 kg m/s2

Therefore, required Force = 5000 m/s2 or 5000 N

Question – 2- If the mass of a moving object is 50 kg, what force will be required to speed up

the object at a rate of 2ms-2

?

Solution:-

According to the question,

Acceleration (a) = 2ms-2

and Mass (m) = 50 kg, therefore, Force (F) =?


= 50 kg x 2m/s2

= 100 kg m/s2


Question – 3 – To accelerate a vehicle to 3m/s2 what force will be needed if the mass of the

vehicle is equal to 100 kg?

Solution:




Acceleration (a) = 3m/s2 and Mass (m) = 100 kg, therefore, Force (F) =?


= 100 kg x 3m/s2

= 300 kg m/s2


Type -II

Question -1 – To accelerate an object to a rate of 2m/s2, 10 N force is required. Find the mass

of object.

Solution:

According to the question:

Acceleration (a) = 2m/s2, Force (F) = 10N, therefore, Mass (m) = ?


Thus, the mass of the object = 5 kg

Question – 2 – If 1000 N force is required to accelerate an object to the rate of 5m/s2, what

will be the weight of the object?

Solution:


Acceleration (a) = 2m/s

2

, Force (F) = 1000N, therefore, Mass (m) = ?




Thus, the mass of the object = 200 kg

Question – 3 – A vehicle accelerate at the rate of 10m/s2 after the applying of force equal to

50000 N. Find the mass of the vehicle.

Solution:


Acceleration (a) = 10 m/s2, Force (F) = 50000N, therefore, Mass (m) = ?


Thus, the mass of the vehicle = 5000 kg

Type - III

Question – 1 - What the acceleration a vehicle having 1000 kg of mass will get after applying

a force of 5000N?

Solution:

According to question:

Mass (m) = 1000 kg, Force (F) = 5000N, Acceleration (a) =?

We know that, Force = Mass x Acceleration or F = m x a

Therefore,

Thus acceleration of the vehicle = 5 ms-2

Question – 2 – After applying a force of 1000 N an object of mass 2000 kg will achieve what

acceleration?

Solution:






Therefore,

Thus acceleration of the vehicle = 0.5 ms-2

Question – 3 – An object requires the force of 100N to achieve the acceleration ‘a’. If the

mass of the object is 500 kg what will be the value of ‘a’?

Solution:




Therefore,

Thus acceleration of the vehicle = 0.2 ms-2

6.5 piles, lifts, bodies tied with string

6.5.1 Piles

The response of a laterally loaded pile within a group of closely spaced piles is often

substantially different than a single isolated pile. This difference is attributed to the following

three items:

1. The rotational restraint at the pile cap connection. The greater the rotational restraint, the

smaller the deflection caused by a given lateral load.



2. The additional lateral resistance provided by the pile cap. verifying and quantifying the cap

resistance is the primary focus of this research.

3. The interference that occurs between adjacent piles through the supporting soil. Interference

between zones of influence causes a pile within a group to deflect more than a single isolated

pile, as a result of pile-soil-pile interaction.

A comprehensive literature review was conducted as part of this research to examine the current

state of knowledge regarding pile cap resistance and pile group behavior. Over 350 journal

articles and other publications pertaining to lateral resistance, testing, and analysis of pile caps,

piles, and pile groups were collected and reviewed.

6.5.2 lifts

6.5.2.1 LIFTING FUNCTIONS

Attachments:Chains

Cables

Ropes

Webbing

BASIC GUIDELINES:

Locations of attachment should be:

Directly over/in alignment with the load's center of gravity (CG).

Above the load's CG.

Rigid objects should be supported by at least two attachments along with balancing supportattachments.

Weight on the carrying attachments is more important than the total weight of the load.

Angles of attachments influence hauling system effectiveness.

When center of gravity of load lines up over the fulcrum or pivot point, balance point has

been reached, and load is at static equilibrium.

When making a vertical lift, attachments should be above center of gravity when possible.

This will keep load from rotating, and under control.

6.5.2.2 CRITICAL ANGLE CONSIDERATIONS

The angle of a rigging strap/ cable attachment in relation to the lifting point greatly effects the

vertical and horizontal forces placed on the anchor attachments as well as the forces in the

strap/cable.

These forces are easily calculated, based on the properties of the triangle that is created.



A circle can be divided into three 120 degree sections.

If the included angle of the rope system is equal to 120 degrees, the force in the rope and it’s

attachment is equal to the supported load.

If the angle becomes greater by pulling the load line tighter, a greater force is placed on the

rope and the anchors.

If the included angle is less, the force in the rope is less.

In lifting systems the angle should be as small as Possible Applying this concept to rigging can

be done by inverting the triangle.

The higher the point of attachment is over the objects CG the lesser the forces on the sling and

it’s attachments.

The flatter the angle, the greater the forces.

Keep this in mind when you begin any lifting operation.

- In some cases lifting a fairly light object with a flat lifting angle will create forces ubstantial

enough to break the sling and/or blow-out the anchor points.

6.5.3 bodies tied with string

block of mass 2 kg sits on a frictionless ramp and is tied to the wall with a string as shown. The

string is horizontal and tied to the center of the block. If the ramp is inclined at 20 degrees, what

is the magnitude of the force from the block on the ramp?

(1) Comprehend the Problem

We have a block sitting on a ramp without any friction. A horizontal string is tied to the ramp.

The tension in the string, coupled with the normal force from the ramp on the block, keeps the

block from sliding down the ramp. We’re asked to find the magnitude (strength) of the force

with which the block is pushing down on the ramp. The block’s free body diagram has three





backward, while the water simultaneously pushes the person forward—both the person and

the water push against each other. The reaction forces account for the motion in these

examples. These forces depend on friction; a person or car on ice, for example, may be

unable to exert the action force to produce the needed reaction force.

An illustration of Newton's third law in which two skaters push against each other. The first

skater on the left exerts a normal force N12 on the second skater directed towards the right,

and the second skater exerts a normal force N21 on the first skater directed towards the left.

The magnitude of both forces are equal, but they have opposite directions, as dictated by

Newton's third law.

Example –

(a) Walking of a person - A person is able to walk because of the Newton’s Third Law of

Motion. During walking, a person pushes the ground in backward direction and in the reaction

the ground also pushes the person with equal magnitude of force but in opposite direction. This

enables him to move in forward direction against the push.

(b) Recoil of gun - When bullet is fired from a gun, the bullet also pushes the gun in opposite

direction, with equal magnitude of force. This results in gunman feeling a backward push from

the butt of gun.

(c) Propulsion of a boat in forward direction – Sailor pushes water with oar in backwarddirection; resulting water pushing the oar in forward direction. Consequently, the boat is pushed

in forward direction. Force applied by oar and water are of equal magnitude but in opposite

directions.





Therefore, rate of change of momentum of A during collision,

Similarly, the rate of change of momentum of B during collision,

Since, according to the Newton’s Third Law of Motion, action of the object A (force exerted by

A) will be equal to reaction of the object B (force exerted by B). But the force exerted in the

course of action and reaction is in opposite direction.

Therefore,

Above equation says that total momentum of object A and B before collision is equal to the total

momentum of object A and B after collision. This means there is no loss of momentum, i.e.momentum is conserved. This situation is considered assuming there is no external force acting

upon the object.

This is the Law of Conservation of Momentum, which states that in a closed system the total

momentum is constant.

In the condition of collision, the velocity of the object which is moving faster is decreased and

the velocity of the object which is moving slower is increased after collision. The magnitude of

loss of momentum of faster object is equal to the magnitude of gain of momentum by slower

object after collision.

6.7.1 Conservation of Momentum – Practical Application

• Bullet and Gun – When bullet is fired from a gun, gun recoils in the opposite direction of

bullet. The momentum of bullet is equal to momentum of gun. Since, the bullet is has

very small mass compared to the gun, hence velocity of bullet is very high compared to



the recoil of gun. In the case of firing of bullet, law of conservation of momentum is

applied as usual.

• In the collision of atoms, the conservation of momentum is applied.

• In the game of snooker, when a ball is hit by stick, the conservation of momentum isapplied.

• When the mouth of an inflated balloon is let open, it starts flying, because of

conservation of momentum.

• When a cricket ball is hit by bat, the Law of Conservation of Momentum is applied.

• When the coins of carom board are hit by striker, the Law of Conservation of Momentum

is applied.

• Newton’s cradle is one of the best examples of conservation of momentum.

6.8 impulsive force (definition only)

The force that two colliding bodies exert on one another acts only for a short time, giving a brie

but strong push. This force is called an impulsive force. During the collision, the impulsive

force is much stronger than any other forces that may be present; consequently, the impulsive

force produces a large change in the motion while the other forces produce only small and

insignificant changes. For example, during the automobile collision shown in Figure, the only

important force is the push of the wall on the front end of the automobile; the effects produced

by gravity and by the friction force of the road during the collision are insignificant.

Suppose that a collision lasts a time t, say, from t = 0 to t = t, and that during this time an

impulsive force F acts on one of the colliding bodies. The force is zero before t = 0 and it is zero

after t = t, but is large between these times. For example, Figure 11.2 shows a plot of the force

experienced by an automobile in a collision with a solid wall lasting 0.120 s. The force is zero

before t = 0 and after t = 0.120 s, and varies in a complicated way between these times. The

impulse delivered by such a force F to the body is defined as the integral of the force over time,



According to this equation, the x component of the impulse for the force shown in Figure 11.2 is

the area between the curve Fx(t) and the t axis, and similarly for the y component and the z

component.

The units of impulse are N · s or kg · m/s in the metric system and lbf · s in the British system;

these units are the same as those of momentum.

The definition (1) of the impulse is not restricted to forces of short duration -- it is equally valid

if the duration t of the impulsive force is long. However, in most of our applications of the

concept of impulse in this chapter, the force will be of short duration.

By means of the equation of motion

where p is the momentum before the collision (at time t = 0) and p is the momentum after the

collision (at time t = t). Thus, the impulse of a force is simply equal to the momentum change

produced by this force. However, since the force acting during a collision is usually not known indetail, Eq. (3) is not very helpful for calculating momentum changes. It is often best to apply Eq.

(3) in reverse for calculating the average force from the known momentum change. The time-

average force is defined by



In a plot of force as a function of time, such as shown in Figure 11.2, the time-average force

simply represents the mean height of the function above the t axis; this mean height is shown by

the colored horizontal line in Figure 11.2. By means of Eq. (3) we can write the time-average

force as

This relation gives a quick estimate of the average magnitude of the impulsive force if the

duration of the collision and the momentum change are known.

EXAMPLE

The collision between the automobile and wall shown in Figure 11.1 lasts 0.120 s. The mass of

the automobile is 1700 kg and the initial and final velocities are v = 13.6 m/s and v = -1.3 m/s,

respectively. Evaluate the impulse and the time-average force from these data.

SOLUTION: With the x axis along the direction of the initial motion, the change in momentum

is

Review Questions

1. Describe Concept of momentum.

2. Define Newton’s laws of motion and their application.

3. Explain derivation of force equation from second law of motion and numerical problems

on second law of motion.

4. Describe piles, lifts, bodies tied with string.



5. Explain Newton’s third law of motion and numerical problems based on it.

6. Explain conservation of momentum.

7. Describe impulsive force (definition only).



Chaper-7

Simple Machines


Machine

Learning Objectives

1. Concept of machine

2. mechanical advantage

3. velocity ratio and efficiency of a machine their relationship

4. law of machine

5. simple machines (lever, wheel and axle, pulleys, jacks winch crabs only).

7.1 Concept of machine

A machine is a tool that consists of one or more parts, and uses energy to meet a particular goal.

Machines are usually powered by mechanical, chemical, thermal, or electrical means, and are

often motorized. Historically, a power tool also required moving parts to classify as a machine.

However, the advent of electronics technology has led to the development of power tools without

moving parts that are considered machines.

A simple machine is a device that simply transforms the direction or magnitude of a force, but a

large number of more complex machines exist. Examples include vehicles, electronic systems,

molecular machines, computers, television, and radio.

7.1.1 Types

Types of machines and related components

Classification Machine(s)

Simple machines Inclined plane, Wheel and axle, Lever, Pulley, Wedge, Screw

Mechanical componentsAxle, Bearings, Belts, Bucket, Fastener, Gear, Key, Link chains, Rack

and pinion, Roller chains, Rope, Seals, Spring, Wheel



Clock Atomic clock, Watch, Pendulum clock, Quartz clock

Compressors and PumpsArchimedes' screw, Eductor-jet pump, Hydraulic ram, Pump, Trompe,

Vacuum pump

Heat

engines

External

combustion enginesSteam engine, Stirling engine

Internal combustion

enginesReciprocating engine, Gas turbine

Heat pumpsAbsorption refrigerator, Thermoelectric refrigerator, Regenerative

cooling

Linkages Pantograph, Cam, Peaucellier-Lipkin

TurbineGas turbine, Jet engine, Steam turbine, Water turbine, Wind generator,

Windmill

Aerofoil Sail, Wing, Rudder, Flap, Propeller

Electronic devicesVacuum tube, Transistor, Diode, Resistor, Capacitor, Inductor,

Memristor, Semiconductor, Computer

Robots Actuator, Servo, Servomechanism, Stepper motor, Computer

MiscellaneousVending machine, Wind tunnel, Check weighing machines, Riveting

machines

Mechanical

The word mechanical refers to the work that has been produced by machines or the machinery. It

mostly relates to the machinery tools and the mechanical applications of science. Some of its

synonyms are automatic and mechanic.



7.1.1.1 Simple machines

Table of simple mechanisms, from Chambers' Cyclopedia, 1728. Simple machines provide a

"vocabulary" for understanding more complex machines.

The idea that a machine can be broken down into simple movable elements led Archimedes to

define the lever, pulley and screw as simple machines. By the time of the Renaissance this list

increased to include the wheel and axle, wedge and inclined plane.

7.1.1.2Engines

An engine or motor is a machine designed to convert energy into useful mechanical motion.

Heat engines, including internal combustion engines and external combustion engines (such as

steam engines) burn a fuel to create heat, which is then used to create motion. Electric motors

convert electrical energy into mechanical motion, pneumatic motors use compressed air and

others, such as wind-up toys use elastic energy. In biological systems, molecular motors like

myosins in muscles use chemical energy to create motion.

7.1.1.3 Electrical

Electrical means operating by or producing electricity, relating to or concerned with electricity.

In other words it means using, providing, producing, transmitting or operated by electricity.







Industrial revolution

The 'Industrial Revolution' was a period from 1750 to 1850 where changes in agriculture,

manufacturing, mining, transportation, and technology had a profound effect on the social,

economic and cultural conditions of the times. It began in the United Kingdom, then

subsequently spread throughout Western Europe, North America, Japan, and eventually the rest

of the world.

Starting in the later part of the 18th century, there began a transition in parts of Great Britain's

previously manual labour and draft-animal–based economy towards machine-based

manufacturing. It started with the mechanisation of the textile industries, the development of

iron-making techniques and the increased use of refined coal.

Mechanization and automation

A water-powered mine hoist used for raising ore. This woodblock is from De re metallica byGeorg Bauer (Latinized name Georgius Agricola, ca. 1555) an early mining textbook that

contains numerous drawings and descriptions of mining equipment.

Mechanization or mechanisation (BE) is providing human operators with machinery that assists

them with the muscular requirements of work or displaces muscular work. In some fields,

mechanization includes the use of hand tools. In modern usage, such as in engineering or



economics, mechanization implies machinery more complex than hand tools and would not

include simple devices such as an un-geared horse or donkey mill. Devices that cause speed

changes or changes to or from reciprocating to rotary motion, using means such as gears, pulleys

or sheaves and belts, shafts, cams and cranks, usually are considered machines. After

electrification, when most small machinery was no longer hand powered, mechanization was

synonymous with motorized machines.

Automation is the use of control systems and information technologies to reduce the need for

human work in the production of goods and services. In the scope of industrialization,

automation is a step beyond mechanization. Whereas mechanization provides human operators

with machinery to assist them with the muscular requirements of work, automation greatly

decreases the need for human sensory and mental requirements as well. Automation plays an

increasingly important role in the world economy and in daily experience.

Automata

The Digesting Duck by Jacques de Vaucanson, hailed in 1739 as the first automaton capable of

digestion

An automaton (plural: automata or automatons) is a self-operating machine. The word is

sometimes used to describe a robot, more specifically an autonomous robot. An alternative

spelling, now obsolete, is automation.

7.2 mechanical advantage

Mechanical advantage is a measure of the force amplification achieved by using a tool,

mechanical device or machine system. Ideally, the device preserves the input power and simply

trades off forces against movement to obtain a desired amplification in the output force. The

model for this is the law of the lever. Machine components designed to manage forces and

movement in this way are called mechanisms.

An ideal mechanism transmits power without adding to or subtracting from it. This means the

ideal mechanism does not include a power source, and is frictionless and constructed from rigid

bodies that do not deflect or wear. The performance of a real system relative to this ideal is

expressed in terms of efficiency factors that take into account friction, deformation and wear

A simple machine has an applied force that works against a load force. If there are no friction

losses, the work done on the load is equal to the work done by the applied force. This allows an



increase in the output force at the cost of a proportional decrease in the distance moved by the

load. The ratio of the output force to the input force is the mechanical advantage of the machine.

If the simple machine does not dissipate or absorb energy, then its mechanical advantage can be

calculated from the machine's geometry. For example, the mechanical advantage of a lever is

equal to the ratio of its lever arms. A simple machine with no friction or elasticity is often called

an ideal machine.

For an ideal simple machine the rate of energy in, or power in, equals the rate of energy out, or

power out, that is

Because power is the product of a force and the velocity of its point of application, the applied

force times the velocity the input point moves, vin, must be equal to the load force times the

velocity the load moves, vout, given by

So the ratio of output to input force, the mechanical advantage, of a frictionless machine is equal

to the "velocity ratio"; the ratio of input velocity to output velocity:

(Ideal Mechanical Advantage)

In the screw, which uses rotational motion, the input force should be replaced by the torque, and

the velocity by the angular velocity the shaft is turned.

7.3 velocity ratio and efficiency of a machine their relationship

7.3.1 Speed ratio

The requirement for power input to an ideal mechanism to equal power output provides a simple

way to compute mechanical advantage from the input-output speed ratio of the system.

The power input to a gear train with a torque TA applied to the drive pulley which rotates at an

angular velocity of ωA is

P=TAωA.





power P is constant through the machine and force times velocity into the machine equals the

force times velocity out, that is

The ideal mechanical advantage is the ratio of the force, or effort, out of the machine relative tothe force or effort into the machine, that is

The constant power relationship provides yields a formula for this ideal mechanical advantage in

terms of the speed ratio,

The speed ratio of a machine can be calculated from its physical dimensions. The assumption of

constant power thus allows use of the speed ratio to determine the maximum value for the

mechanical advantage.

7.3.4 Actual mechanical advantage

The actual mechanical advantage (AMA) is the mechanical advantage determined by physical

measurement of the input and output forces. Actual mechanical advantage takes into account

energy loss due to deflection, friction, and wear.

The AMA of a machine is calculated as the ratio of the measured force output to the measured

force input,

where the input and output forces are determined experimentally.

The ratio of the experimentally determined mechanical advantage to the ideal mechanical

advantage is the efficiency η of the machine,



7.4 law of machine

Machines which are used to lift a load are governed by the "Law of machines", which states that

the effort to be applied on the machine (p) is related to the weight (w) which it can lift as -

p = mw + c

Where m and c are positive constants which are characteristics of the machine.

There is a scientific consensus that perpetual motion in an isolated system violates either the first

law of thermodynamics, the second law of thermodynamics, or both. The first law of

thermodynamics is essentially a statement of conservation of energy. The second law can be

phrased in several different ways, the most intuitive of which is that heat flows spontaneously

from hotter to colder places; the most well known statement is that entropy tends to increase (see

entropy production), or at the least stay the same; another statement is that no heat engine (an

engine which produces work while moving heat from a high temperature to a low temperature)

can be more efficient than a Carnot heat engine.

In other words:

1. In any isolated system, one cannot create new energy (first law of thermodynamics)

2. The output power of heat engines is always smaller than the input heating power. The rest

of the energy is removed as heat at ambient temperature. The efficiency (this is the

produced power divided by the input heating power) has a maximum, given by the

Carnot efficiency. It is always lower than one

3. The efficiency of real heat engines is even lower than the Carnot efficiency due to

irreversible processes.

The statements 2 and 3 only apply to heat engines. Other types of engines, which convert e.g.

mechanical into electromagnetic energy, can, in principle, operate with 100% efficiency.

Machines which comply with both laws of thermodynamics by accessing energy from

unconventional sources are sometimes referred to as perpetual motion machines, although they

do not meet the standard criteria for the name. By way of example, clocks and other low-power

machines, such as Cox's timepiece, have been designed to run on the differences in barometric

pressure or temperature between night and day. These machines have a source of energy, albeit

one which is not readily apparent so that they only seem to violate the laws of thermodynamics.

Machines which extract energy from seemingly perpetual sources - such as ocean currents - are

indeed capable of moving "perpetually" until that energy source runs down. They are not



considered to be perpetual motion machines because they are consuming energy from an external

source and are not isolated systems.

7.5 simple machines (lever, wheel and axle, pulleys, jacks winch crabs only)

7.5.1 lever

A lever is a machine consisting of a beam or rigid rod pivoted at a fixed hinge, or fulcrum. It is

one of the six simple machines identified by Renaissance scientists. The word comes from the

French lever, "to raise", cf. a levant. A lever amplifies an input force to provide a greater output

force, which is said to provide leverage. The ratio of the output force to the input force is the

ideal mechanical advantage of the lever.

7.5.1.1 Early use

The earliest remaining writings regarding levers date from the 3rd century BC and were provided

by Archimedes. "Give me a place to stand, and I shall move the Earth with it"[note 1]

is a remark of

Archimedes who formally stated the correct mathematical principle of levers (quoted by Pappus

of Alexandria).

It is assumed that in ancient Egypt, constructors used the lever to move and uplift obelisks

weighing more than 100 tons.

7.5.1.2 Force and levers

A lever in balance

A lever is a beam connected to ground by a hinge, or pivot, called a fulcrum. The ideal lever

does not dissipate or store energy, which means there is no friction in the hinge or bending in the

beam. In this case, the power into the lever equals the power out, and the ratio of output to input



force is given by the ratio of the distances from the fulcrum to the points of application of these

forces. This is known as the law of the lever.

The mechanical advantage of a lever can be determined by considering the balance of moments

or torque, T, about the fulcrum,

where M1 is the input force to the lever and M 2 is the output force. The distances a and b are the

perpendicular distances between the forces and the fulcrum.

The mechanical advantage of the lever is the ratio of output force to input force,

This relationship shows that the mechanical advantage can be computed from ratio of the

distances from the fulcrum to where the input and output forces are applied to the lever.

7.5.1.3 Classes of levers

Three classes of levers

Levers are classified by the relative positions of the fulcrum and the input and output forces. It is

common to call the input force the effort and the output force the load or the resistance. This

allows the identification of three classes of levers by the relative locations of the fulcrum, the

resistance and the effort:



• Class 1: Fulcrum in the middle: the effort is applied on one side of the fulcrum and the

resistance on the other side, for example, a crowbar or a pair of scissors.

• Class 2: Resistance in the middle: the effort is applied on one side of the resistance and

the fulcrum is located on the other side, for example, a wheelbarrow, a nutcracker, a

bottle opener or the brake pedal of a car. Mechanical advantage is greater than 1.

• Class 3: Effort in the middle: the resistance is on one side of the effort and the fulcrum is

located on the other side, for example, a pair of tweezers or the human mandible.

Mechanical advantage is less than 1.

These cases are described by the mnemonic "fre 123" where the fulcrum is in the middle for the

1st class lever, the resistance is in the middle for the 2nd class lever, and the effort is in the

middle for the 3rd class lever.

7.5.1.4 The law of the lever

The lever is a movable bar that pivots on a fulcrum attached to or positioned on or across a fixed

point. The lever operates by applying forces at different distances from the fulcrum, or pivot.

As the lever pivots on the fulcrum, points farther from this pivot move faster than points closer to

the pivot. The power into and out of the lever must be the same. Power is the product of force

and velocity, so forces applied to points farther from the pivot must be less than when applied to

points closer in.

If a and b are distances from the fulcrum to points A and B and if force F A applied to A is the

input force and FB exerted at B is the output, the ratio of the velocities of points A and B is given

by a/b, so the ratio of the output force to the input force, or mechanical advantage, is given by



This is the law of the lever, which was proven by Archimedes using geometric reasoning. It

shows that if the distance a from the fulcrum to where the input force is applied (point A) is

greater than the distance b from fulcrum to where the output force is applied (point B), then the

lever amplifies the input force. If the distance from the fulcrum to the input force is less than

from the fulcrum to the output force, then the lever reduces the input force. Recognizing the

profound implications and practicalities of the law of the lever, Archimedes has been famously

attributed with the quotation "Give me a place to stand and with a lever I will move the whole

world."

The use of velocity in the static analysis of a lever is an application of the principle of virtual

work.

7.5.1.5 Virtual Work and the Law of the Lever

A lever is modeled as a rigid bar connected to a ground frame by a hinged joint called a fulcrum.

The lever is operated by applying an input force FA at a point A located by the coordinate vector

rA on the bar. The lever then exerts an output force FB at the point B located by rB. The rotation

of the lever about the fulcrum P is defined by the rotation angle θ in radians.

This is an engraving from Mechanics Magazine published in London in 1824.

Let the coordinate vector of the point P that defines the fulcrum be rP, and introduce the lengths

which are the distances from the fulcrum to the input point A and to the output point B,

respectively.



Now introduce the unit vectors eA and eB from the fulcrum to the point A and B, so

The velocity of the points A and B are obtained as

where eA and eB

are unit vectors perpendicular to eA and eB, respectively.

The angle θ is the generalized coordinate that defines the configuration of the lever, and the

generalized force associated with this coordinate is given by

where FA and F B are components of the forces that are perpendicular to the radial segments PA

and PB. The principle of virtual work states that at equilibrium the generalized force is zero, that

is

Thus, the ratio of the output force FB to the input force F A is obtained as

which is the mechanical advantage of the lever.

This equation shows that if the distance a from the fulcrum to the point A where the input force

is applied is greater than the distance b from fulcrum to the point B where the output force is

applied, then the lever amplifies the input force. If the opposite is true that the distance from the

fulcrum to the input point A is less than from the fulcrum to the output point B, then the lever

reduces the magnitude of the input force.

This is the law of the lever, which was proven by Archimedes using geometric reasoning.

7.5.2 wheel and axle



The wheel and axle is one of six simple machines identified by Renaissance scientists drawing

from Greek texts on technology. The wheel and axle is generally considered to be a wheel

attached to an axle so that these two parts rotate together in which a force is transferred from one

to the other. In this configuration a hinge, or bearing, supports the rotation of the axle.

Hero of Alexandria identified the wheel and axle as one of five simple machines used to lift

weights. This is thought to have been in the form of the windlass which consists of crank or

pulley connected to a cylindrical barrel that provides mechanical advantage to wind up a rope

and lift a load such as a bucket from a well.

This system is a version of the lever with loads applied tangentially to the perimeters of the

wheel and axle, respectively, that are balanced around the hinge, which is the fulcrum. The

mechanical advantage of the wheel and axle is the ratio of the distances from the fulcrum to the

applied loads, or what is the same thing the ratio of the radial dimensions of the wheel and axle.

A windlass, a well known application of the wheel and axle.

7.5.2.1 Mechanical advantage

The simple machine called a wheel and axle refers to the assembly formed by two disks, or

cylinders, of different diameters mounted so they rotate together around the same axis. Forces

applied to the edges of the two disks, or cylinders, provide mechanical advantage. When used as

the wheel of a cart the smaller cylinder is the axle of the wheel, but when used in a windlass,

winch, and other similar applications (see medieval mining lift to left) the smaller cylinder may

be separate from the axle mounted in the bearings.

Assuming the wheel and axle does not dissipate or store energy, the power generated by forces

applied to the wheel must equal the power out at the axle. As the wheel and axle system rotates



around its bearings, points on the circumference, or edge, of the wheel move faster than points

on the circumference, or edge, of the axle. Therefore a force applied to the edge of the wheel

must be less than the force applied to the edge of the axle, because power is the product of force

and velocity.

Let a and b be the distances from the center of the bearing to the edges of the wheel A and the

axle B. If the input force FA is applied to the edge of the wheel A and the force F B at the edge of

the axle B is the output, then the ratio of the velocities of points A and B is given by a/b, so the

ratio of the output force to the input force, or mechanical advantage, is given by

The mechanical advantage of a simple machine like the wheel and axle is computed as the ratio

of the resistance to the effort. The larger the ratio the greater the multiplication of force (torque)

created or distance achieved. By varying the radii of the axle and/or wheel, any amount of

mechanical advantage may be gained. In this manner, the size of the wheel may be increased to

an inconvenient extent. In this case a system or combination of wheels (often toothed, that is,

gears) are used. As a wheel and axle is a type of lever, a system of wheels and axles is like a

compound lever.

7.5.1.2 Ideal mechanical advantage

The ideal mechanical advantage of a wheel and axle is calculated with the following formula:

7.5.1.3 Actual mechanical advantage

The actual mechanical advantage of a wheel and axle is calculated with the following formula:

where

R = resistance force, i.e. the weight of the bucket in this example.

Eactual = actual effort force, the force required to turn the wheel.



7.5.3 pulleys

A pulley is a wheel on an axle that is designed to support movement of a cable or belt along its

circumference. Pulleys are used in a variety of ways to lift loads, apply forces, and to transmit

power.

A pulley is also called a sheave or drum and may have a groove between two flanges around its

circumference. The drive element of a pulley system can be a rope, cable, belt, or chain that runs

over the pulley inside the groove.

Hero of Alexandria identified the pulley as one of six simple machines used to lift weights.

Pulleys are assembled to form a block and tackle in order to provide mechanical advantage to

apply large forces. Pulleys are also assembled as part of belt and chain drives in order to transmit

power from one rotating shaft to another.

7.5.3.1 Block and tackle

Various ways of rigging a tackle.

A set of pulleys assembled so that they rotate independently on the same axle form a block. Two

blocks with a rope attached to one of the blocks and threaded through the two sets of pulleys

form a block and tackle.

A block and tackle is assembled so one block is attached to fixed mounting point and the other is

attached to the moving load. The mechanical advantage of the block and tackle is equal to the

number of parts of the rope that support the moving block.

In the diagram on the right the mechanical advantage of each of the block and tackle assemblies

shown is as follows:



• Gun Tackle: 2

• Luff Tackle: 3

• Double Tackle: 4

• Gyn Tackle: 5

• Threefold purchase: 6

7.5.3.2 Rope and pulley systems

Pulley in oil derrick

A hoist using the compound pulley system yielding an advantage of 4. The single fixed pulley is

installed on the hoist (device). The two movable pulleys (joined together) are attached to the

hook. One end of the rope is attached to the crane frame, another to the winch.



A rope and pulley system -- that is, a block and tackle -- is characterised by the use of a single

continuous rope to transmit a tension force around one or more pulleys to lift or move a load—

the rope may be a light line or a strong cable. This system is included in the list of simple

machines identified by Renaissance scientists.

If the rope and pulley system does not dissipate or store energy, then its mechanical advantage is

the number of parts of the rope that act on the load. This can be shown as follows.

Consider the set of pulleys that form the moving block and the parts of the rope that support this

block. If there are p of these parts of the rope supporting the load W, then a force balance on the

moving block shows that the tension in each of the parts of the rope must be W/p. This means the

input force on the rope is T=W/p. Thus, the block and tackle reduces the input force by the factor

p.

•

A gun tackle has a single pulley in both the fixed and moving blocks with two rope parts

supporting the load W.

•

Separation of the pulleys in the gun tackle show the force balance that results in a rope

tension of W/2.

•



A double tackle has two pulleys in both the fixed and moving blocks with four rope parts

supporting the load W.

•

Separation of the pulleys in the double tackle show the force balance that results in a rope

tension of W/4.

7.5.3.3 How it works

The simplest theory of operation for a pulley system assumes that the pulleys and lines areweightless, and that there is no energy loss due to friction. It is also assumed that the lines do not

stretch.

In equilibrium, the forces on the moving block must sum to zero. In addition the tension in the

rope must be the same for each of its parts. This means that the two parts of the rope supporting

the moving block must each support half the load.

•

Fixed pulley

•

Diagram 1: The load F on the moving pulley is balanced by the tension in two parts of the

rope supporting the pulley.



•

Movable pulley

•

Diagram 2: A movable pulley lifting the load W is supported by two rope parts with

tension W/2.

These are different types of pulley systems:

• Fixed: A fixed pulley has an axle mounted in bearings attached to a supporting structure.

A fixed pulley changes the direction of the force on a rope or belt that moves along its

circumference. Mechanical advantage is gained by combining a fixed pulley with a

movable pulley or another fixed pulley of a different diameter.

• Movable: A movable pulley has an axle in a movable block. A single movable pulley issupported by two parts of the same rope and has a mechanical advantage of two.

• Compound: A combination of fixed and a movable pulleys forms a block and tackle. A

block and tackle can have several pulleys mounted on the fixed and moving axles, further

increasing the mechanical advantage.

•

Diagram 3: The gun tackle "rove to advantage" has the rope attached to the moving

pulley. The tension in the rope is W/3 yielding an advantage of three.



•

Diagram 3a: The Luff tackle adds a fixed pulley "rove to disadvantage." The tension in

the rope remains W/3 yielding an advantage of three.

The mechanical advantage of the gun tackle can be increased by interchanging the fixed and

moving blocks so the rope is attached to the moving block and the rope is pulled in the direction

of the lifted load. In this case the block and tackle is said to be "rove to advantage."[10]

Diagram 3

shows that now three rope parts support the load W which means the tension in the rope is W/3.

Thus, the mechanical advantage is three.

By adding a pulley to the fixed block of a gun tackle the direction of the pulling force is reversed

though the mechanical advantage remains the same, Diagram 3a. This is an example of the Luff

tackle.

7.5.3.4 Free body diagrams

The mechanical advantage of a pulley system can be analyzed using free body diagrams which

balance the tension force in the rope with the force of gravity on the load. In an ideal system, the

massless and frictionless pulleys do not dissipate energy and allow for a change of direction of a

rope that does not stretch or wear. In this case, a force balance on a free body that includes the

load, W, and n supporting sections of a rope with tension T, yields:

The ratio of the load to the input tension force is the mechanical advantage of the pulley system,

Thus, the mechanical advantage of the system can be determined by counting the number of

sections of rope supporting the load.

7.5.3.5 Belt and pulley systems



Flat belt on a belt pulley

Belt and pulley system

Cone pulley driven from above by a line shaft

A belt and pulley system is characterised by two or more pulleys in common to a belt. Thisallows for mechanical power, torque, and speed to be transmitted across axles. If the pulleys are

of differing diameters, a mechanical advantage is realised.

A belt drive is analogous to that of a chain drive, however a belt sheave may be smooth (devoid

of discrete interlocking members as would be found on a chain sprocket, spur gear, or timing



belt) so that the mechanical advantage is approximately given by the ratio of the pitch diameter

of the sheaves only, not fixed exactly by the ratio of teeth as with gears and sprockets.

In the case of a drum-style pulley, without a groove or flanges, the pulley often is slightly convex

to keep the flat belt centred. It is sometimes referred to as a crowned pulley. Though once widely

used in factory line shafts, this type of pulley is still found driving the rotating brush in upright

vacuum cleaners. Agricultural tractors built up to the early 1950s generally had a belt pulley. It

had limited use as the tractor and equipment being powered needed to be stationary. It has thus

been replaced by other mechanisms, such as power take-off and hydraulics.

7.5.4 jacks winch crabs

7.5.4.1 Winch Crab Single Purchase

Fitted with heavy cast iron wall brackets. The grooved wheel is of 25 cm diameter and gears are

machine cut. This apparatus is used for experiments in efficiency of mechanical advantage.

Weights are not included.

7.5.4.2 Winch Crab Double Purchase



Experimental type. Same as above but with double set of gearing arrangement. Without weights.

Review Questions

1. Explain Concept of machine.

2. Describe t he mechanical advantage.

3. What do you mean by velocity ratio and efficiency of a machine their relationship?

4. Describe law of machine.

5. What is lever?

6. Define wheel and axle.

7. Describe pulleys.

8. What do you mean by jacks winch crabs?



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