Mechanics

MechanicsPhysics in English

Nankai University, CY Li

Mechanics

Kinematics， dynamics Newton’s three laws The law of universal gravitation The equilibrium of rigid bodies Moment of Inertia Work, energy and power Momentum



Some definitions

Mechanics studies the relationships among force, matter, and motion.

Kinematics is a part of mechanics dealing with the description of motion.

Dynamics is another part dealing with the relation of motion to its causes.



Some words

Mechanics Kinematics Dynamics Displacement Velocity Acceleration Friction Equilibrium

Projectile motions Free fall Falling objects inertia Impulse Momentum Universal gravitation Gravitational

acceleration



Kinematics in one dimension

Displacement, velocity and acceleration, Kinetic equations for a motion with

constant acceleration:

asvv

attvs

atvv

2 :disp vsv

: t vsdisp

: t vsv

20

2

221

0

0



Free fall motion

Objects in motion solely under the influence of gravity are said to be in free fall.

According to legend, Galileo dropped balls of different mass from the Leaning Tower of Pisa to help support his ideas that all objects free fall at the same rate, regardless of their mass.

Newton later took Galileo's ideas about mechanics and formalized them into his laws of motion.



Kinematics in two dimensions -projectiles.

A projectile is an object in motion after being given an initial velocity.

An object moves along a curved line. Projectile motions (ax=0, ay=g)

Horizontal projection Projection at an angle

A problem dealing with 2D motion can be considered as two 1D problems, that is, x part and y part of the motion.



Forces

Friction: A force that impedes movements. When two surfaces are in contact, the force of friction impedes sliding of the surfaces across each other. There is another type of friction force when a

solid object moves through a fluid or gas. Force of gravity & weight: It is the attraction

between the earth and other objects. This gravitational force is usually referred to as the weight of the object.



Ways to describe forces

A force acting on an object, A force on an object, An applied force on an object, A force applied to an object, A force on an object exerted by

something,



Newton’s first law (the law of inertia)

Every object continues in a state of rest, or of uniform motion in a straight line, unless it is compelled to change that state by forces acting on it.

The first equivalent statement The second equivalent statement



Newton’s first law

The second state: If there is no net force acting on an object, the object at rest remains at rest, or the object in motion continues to move with its constant velocity.

The third statement: If the net force acting on an object is zero, the object will either remain at rest or continue to move in a straight line with no change in its speed.



Newton’s first law

more statements: The state of motion of an object remains

unchanged whenever the net force acting on the object is zero.

If there is no net force on an object, the velocity of the object remains unchanged.

Inertia and mass Inertia: the natural tendency of an object to

remain at rest or in a motion at a constant speed along a straight line.



Equilibrium

When the state of motion of an object remains unchanged, the object is said to be in equilibrium.

An example of equilibrium, Assuming you are sitting at rest on a chair,

the net force on you is zero. However, your weight-the gravitational force exerted by the earth-is pulling you downward. This force is offsetted (balanced) by upward forces exerted on you by the chair and the floor.



A short video on equilibrium



An example

An ice cream vendor exerts a force of 40N to overcome friction and pushes his cart at a constant velocity. The cart has a mass of 150kg. Find the forces acting on the cart.

Solution:



Solution

Solution: There are four forces, two vertical ones (N

& w) and two horizontal ones (F & f),

N

F

w

f

NmskgmgwN

NFf

170)8.9)(150(

402



Newton’s second law

When there is a net force acting on an object, the object undergoes an acceleration in the same direction as the force. The force F needed to produce an acceleration a is:

where m is the mass of the object. An example

amF



An example

A child pushes a sled across a frozen pond with a horizontal force of 20 N. Assume friction is negligible. (a) If the sled accelerates at 0.5ms-2, what is its

mass? (b) Another child with a mass of 60kg sits on

the sled. What acceleration will the same force produce now?

Answer



Answer

(a) The sled does not move vertically, so its weight and normal force exerted by the ice cancel. Thus the net force on the sled is the 20 N force exerted by the child. From the second law, the mass is

(b) The total mass of the passenger and sled is 60kg+40kg=100kg. Thus the acceleration is now

kgms

kgms

ms

N

a

Fm 40

5.0

20

5.0

202

2

2

22.0100

20 mskg

N

m

Fa



The movement of a parachutist

With no parachute, it is a free-fall movement, a=g. Initially, v=0, a=g, With v increased, the resistance of air-resistance is

increased until F=0. F=0, v=44ms-1, terminal velocity.

With parachute open, there is air resistance (air drag) . Motion with an acceleration until it reaches the

terminal velocity: 7.44ms-1, Uniform motion.



Impulse and momentum

We can get another insight into the second law when the acceleration is replaced the velocity change per unit.

The product of force and time is called the impulse of the force. The product of mass and velocity is called the momentum.

If the force is not constant, the force can be plotted again time in a graph and the total impulse of the force can be estimated by measuring the area under the curve.

)( if vvmvmtF

t

vmamF



Newton’s third law

If one object exerts a force F on the second object, then the second one exerts an equal but opposite force –F on the first. Or, For every action, there is an equal and opposite reaction.

An example In a swimming pool, if you push a wall with your

legs, the wall exerts a force that propels your further into the pool.



Newton’s law of gravitation

All objects in the universe attract each other, forming a force pair.

Point particle: Objects of any shapes are so small compared to their separation that they may be considered as point particles.

According to Newton’s law of gravitation, every particle attracts every other particle in the universe with a force that is proportional to the masses of both particles and inversely proportional to the square of the distance between them.



The law of universal gravitation

If two spheres or particles have gravitational masses m & m’, and their centers are separated by a distance r, the forces between the two spheres have a magnitude which is proportional to their mass product and inversely proportional to the distance squared. where G is called the universal gravitational constant.

2

'

r

GmmF



Example

Using the measured values of the gravitational acceleration g, the gravitational constant G, and the radius of the earth Re, find the mass of the earth. (The radius of the earth is 6380km.)

Soultion



Solution

Reasoning: The gravitational force on an object of mass m at the surface of the earth is w=mg. It also follows the universal law of gravitation. Hence

Solution: Solving for Me, the mass m cancels out, and we get

2e

e

R

GmMmg

kgkgNm

mms

G

gRM e

e24

2211

2622

1098.51067.6

)1038.6)(8.9(

Documents

Mechanics