Week 1

MATH 142

K. Arunakirinathar

August 1, 2007

K. Arunakirinathar : MATH 142, 1

General introduction to the course

I Grading

I Exams

I Objectives

I Expectations



I Grading

I Exams

I Objectives

I Expectations



I Grading

I Exams

I Objectives

I Expectations



I Grading

I Exams

I Objectives

I Expectations


Information

I Professors: Dr. K. Arunakirinathar (Aruna)

I Dr. M. Govender (Magen)

I Dr. S. Moopanar (Selvan)

I Office: 304 Desmond Clarence Building

I Office Hrs: Mon.9.00 12.00hrs or by appointment

I Phone: (031) 260 3830

I Email: [email protected]

I http://ols.ukzn.ac.za


Information






I Phone: (031) 260 3830




Information






I Phone: (031) 260 3830




Information






I Phone: (031) 260 3830




Grading

QuizzesTest 1Test 2Test 3Final Exam

13 × 20%−−−−−−−−−80%


General Policy

AttendanceAttendance and class participation are required. Quizzes that aregiven when you are not in class can not be made up.

In-class Quizzes and TestsIn-class Quizzes will be given periodically to assess understandingof the basic material covered in the course. Test will be given in afixed date to provide adequate time to demonstrate understandingof the course topics.


General Policy

AttendanceAttendance and class participation are required. Quizzes that aregiven when you are not in class can not be made up.

In-class Quizzes and TestsIn-class Quizzes will be given periodically to assess understandingof the basic material covered in the course. Test will be given in afixed date to provide adequate time to demonstrate understandingof the course topics.


Learning Objectives

1. To introduce and define the subject of Dynamics

2. To introduce Newton’s laws, and to understand thesignificance of these laws

3. To introduce work, power and Energy

4. To introduce the motion of rigid bodies


Learning Objectives






Learning Objectives






Learning Objectives






Learning Objectives






Definitions:Dynamics: the study of bodies in motion and the forces that

create the motion.

Kinematics: the study of motion of a body (i.e., position, velocityand acceleration).

Kinetics: a study of the relationship between the motion andforces acting on a body The study is based on theNewton’s Laws.

particle: a body of infinitely small size such that the spatialdistribution of the body is immaterial. A body offinite size can be treated as a particle if it purelytranslates without rotating (i.e., any rotationalenergy is negligibly small).

Momentum : Linear Momentum or quantity of motion is definedas the product of mass and its velocity. That is,p = m r or = m v.



create the motion.Kinematics: the study of motion of a body (i.e., position, velocity

and acceleration).

Kinetics: a study of the relationship between the motion andforces acting on a body The study is based on theNewton’s Laws.






and acceleration).Kinetics: a study of the relationship between the motion and

forces acting on a body The study is based on theNewton’s Laws.








particle: a body of infinitely small size such that the spatialdistribution of the body is immaterial.

A body offinite size can be treated as a particle if it purelytranslates without rotating (i.e., any rotationalenergy is negligibly small).

















Definitions

Force exerted on a body has two effects:

I The external effect, which is tendency to change the motionof the body or to develop resisting forces in the body

I The internal effect, which is the tendency to deform thebody.


Definitions





Definitions





Definitions

A force is a vector quantity that, when applied tosome rigid body, has a tendency to producetranslation (movement in a straight line) ortranslation and rotation of body. When problems aregiven, a force may also be referred to as a load orweight.

Characteristics of force are the magnitude,direction(orientation) and point of application.


Definitions

A force is a vector quantity that, when applied tosome rigid body, has a tendency to producetranslation (movement in a straight line) ortranslation and rotation of body. When problems aregiven, a force may also be referred to as a load orweight.

Characteristics of force are the magnitude,direction(orientation) and point of application.


Definitions

Scalar Quantity has magnitude only (not direction)and can be indicated by a point on a scale. Examplesare temperature, mass, time and dollars.

Vector Quantities have magnitude and direction.Examples are wind velocity, distance between topoints on a map and forces.


Definitions

Scalar Quantity has magnitude only (not direction)and can be indicated by a point on a scale. Examplesare temperature, mass, time and dollars.

Vector Quantities have magnitude and direction.Examples are wind velocity, distance between topoints on a map and forces.


Types of Forces(Loads)

1. Point loads -concentratedforces exerted atpoint or location

2. Distributed loads -a force appliedalong a length orover an area. Thedistribution can beuniform ornon-uniform.










Newton’s Laws

Newton’s First Law: In the absence of external forces, a particleoriginally at rest or moving with a constant velocitywill remain at rest or continue to move with aconstant velocity along a straight line.

– assumed that frame of reference (known as inertialframe of reference or Newtonian frame of reference)is fixed


Newton’s Laws

Newton’s First Law: In the absence of external forces, a particleoriginally at rest or moving with a constant velocitywill remain at rest or continue to move with aconstant velocity along a straight line.

– assumed that frame of reference (known as inertialframe of reference or Newtonian frame of reference)is fixed


Newton’s Laws

Newton’s Second Law: If an external force acts on a particle, theparticle will be accelerated in the direction of theforce, and the rate of change of the particle’smomentum will be directly proportional to theapplied force.

– F ∝ ddt (m r)

– F = k ddt (m r) ; k is positive constant,

– F = k {m r + m r}– F = k m r.

– The unit system chosen is such that k = 1.

– So, one unit of force (the Newton, N) produces unitacceleration (1 ms−2) when acting on unit mass (1kg).


Newton’s Laws


– F ∝ ddt (m r)


– F = k {m r + m r}– F = k m r.




Newton’s Laws


– F ∝ ddt (m r)


– F = k {m r + m r}– F = k m r.




Newton’s Laws


– F ∝ ddt (m r)


– F = k {m r + m r}

– F = k m r.




Newton’s Laws


– F ∝ ddt (m r)


– F = k {m r + m r}– F = k m r.




Newton’s Laws


– F ∝ ddt (m r)


– F = k {m r + m r}– F = k m r.




Newton’s Laws

Newton’s Third Law: The forces of action and reaction betweenbodies in contact have the same magnitude, sameline of action, and opposite sense.


Newton’s Laws

Newton’s Fourth Law or Law of universal gravitational attractionTwo particle of mass M and m are mutuallyattracted with equal and opposite forces F and −F .The magnitude of the force is proportional to theproduct of their gravitational masses and inverselyproportional to the square of the distance betweenthem.

– Having selected a system of units, we writeF = G M m

R2 ; where R is the distance between them

– G = 6.673× 10−11m3s−2kg−1 =3.44× 10−8ft4s−2lb−1


Newton’s Laws




– G = 6.673× 10−11m3s−2kg−1 =3.44× 10−8ft4s−2lb−1


Newton’s Laws




– G = 6.673× 10−11m3s−2kg−1 =3.44× 10−8ft4s−2lb−1


Newton’s Laws




– G = 6.673× 10−11m3s−2kg−1 =3.44× 10−8ft4s−2lb−1


concept of Massarises in two of the laws:

– Second Law ; inertial mass is considered to be ameasure of a particle resistance to acceleration,

– Fourth Law; gravitational mass is defined as theproperty of the particle that influences itsgravitational attraction.

– Newton assumed that these two concepts of masswere equivalent.

– Based on this assumption, an object in free fall willaccelerate at g .





















Units in Mechanics

Mechanics problems are generally stated in terms of consistentunits. This means that the four units(length, time, mass and force)must be selected in such a way that they are dimensionallyconsistent with Newtons second law. The desired units for lengthand time are established based upon the preference of theengineer. If the units for mass are specified, then the units of forcemust be derived from Newtons second law.

Dimensions British System Metric (SI units) Dimension

Length foot (ft) meter (m) LMass pound (lb) kilogram (kg) MTime second (s) second (s) TForce Poundal (lb-ft s−2 ) Newton ( kg-ms−2) MLT−2


Units in Mechanics

Mechanics problems are generally stated in terms of consistentunits. This means that the four units(length, time, mass and force)must be selected in such a way that they are dimensionallyconsistent with Newtons second law. The desired units for lengthand time are established based upon the preference of theengineer. If the units for mass are specified, then the units of forcemust be derived from Newtons second law.

Dimensions British System Metric (SI units) Dimension

Length foot (ft) meter (m) LMass pound (lb) kilogram (kg) MTime second (s) second (s) TForce Poundal (lb-ft s−2 ) Newton ( kg-ms−2) MLT−2


Example Problems

1. Two spherical bodies have masses of 60 kg and 80 kg,respectively. Determine the gravitational force of attractionbetween the spheres if the distance between centers is 600mm.

2. Using the fact that 1 in. = 25.4 mm, convert a speed of 75mph to units of meters/sec.

3. How many barrels of oil are contained in 100 kL of oil? Onebarrel is equal to 42 gal.

4. In the equation

y = y0 + v t +1

2at2,

y and y0 are distance, v is velocity, a is an acceleration, and tis time. Is the equation dimensionally homogeneous?


5. The elongation of a bar of uniform cross sectionsubjected to an axial force is given by the equation

δ =P L

A EWhat are dimensions of E if δ and L are lengths, P isa force and A is an area.


Equivalent Metric-English Measures

UNIT ENGLISH to METRIC METRIC to ENGLISH

Length 1 mile = 1.61 kilometers 1 kilometer = .62 mile1 yard = 0.914 meter 1 meter = 1.09 yard1 foot = 0.305 meter 1 meter = 3.28 feet1 inch = 2.54 centimeters 1 centimeter = 0.394 in

Volume 1 gallon = 3.79 liters 1 liter = 0.264 gallon1 quart = 0.946 liter 1 liter = 1.06 quarts

Weight 1 pound = 0.454 kilogram 1 kilogram = 2.2 pounds1 ounce = 28.35 grams 1 gram = 0.0353 ounce


Lecture Goals

I Normal and Tangential Coordinates

I Rectilinear Equations of Motion


Equations of Motion for a particleNewtons Second Law of motion may be written in the vector form:∑

F = ma

The vectors can be broken into components similar to equilibriumin rectangular coordinates:∑

(Fx i + Fy j + Fzk) = m(ax i + ay j + azk)

The scalar forms of this equation:∑Fx = max = mx∑Fy = may = my∑Fz = maz = mz



F = ma






F = ma





In terms of normal and tangential coordinates for planar motion asfollows:

∑Ft = mat = m

d v

dt∑Fn = man = m

v2

ρ


In terms of normal and tangential coordinates for planar motion asfollows: ∑

Ft = mat = md v

dt∑Fn = man = m

v2

ρ


In terms of normal and tangential coordinates for planar motion asfollows: ∑

Ft = mat = md v

dt∑Fn = man = m

v2

ρ


Solution strategy

I Draw the Free-Body Diagram.

I Write the equations of motion Solve for the velocities andaccelerations need or obtain the equations for the components.

I Solve for the forces


Solution strategy





Solution strategy





Definitions

Ideal strings, cords, ropes: If a string, cord or rope is ideal then

(i) it has negligible mass,(ii) it is perfectly flexible(iii) it is inextensible

Smooth surfaces: A smooth surfaces is one that can producereaction forces only at right angles to itself, on theobject which are in touch with it.


Definitions

Ideal strings, cords, ropes: If a string, cord or rope is ideal then(i) it has negligible mass,

(ii) it is perfectly flexible(iii) it is inextensible



Definitions

Ideal strings, cords, ropes: If a string, cord or rope is ideal then(i) it has negligible mass,(ii) it is perfectly flexible

(iii) it is inextensible



Definitions

Ideal strings, cords, ropes: If a string, cord or rope is ideal then(i) it has negligible mass,(ii) it is perfectly flexible(iii) it is inextensible



Definitions

Ideal strings, cords, ropes: If a string, cord or rope is ideal then(i) it has negligible mass,(ii) it is perfectly flexible(iii) it is inextensible



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Week 1