Physics Module 1.pdf on classical mechanics.

**Classical Mechanics: A Detailed Overview**

**Classical mechanics**, also known as Newtonian mechanics, is a branch of physics that deals with the motion of objects and the forces acting upon them. It is one of the most foundational areas of physics, primarily developed by Sir Isaac Newton in the 17th century. Though its principles have been superseded in certain domains by quantum mechanics and relativity, classical mechanics remains crucial for understanding the macroscopic world and serves as the bedrock for many modern physical theories.

In this comprehensive exploration of classical mechanics, we will cover key principles, laws, mathematical frameworks, and its applications in different fields of science and engineering.

### Historical Context and Development

Classical mechanics began with the works of early scientists like Galileo Galilei, who studied the motion of objects under gravity. However, it was **Isaac Newton** who unified the ideas of motion and force with his three **Laws of Motion** and the **Law of Universal Gravitation**. His seminal work, *Philosophiæ Naturalis Principia Mathematica* (1687), laid the foundation for understanding the motion of bodies under the influence of forces.

The theory of classical mechanics was expanded by many later scientists, including **Joseph-Louis Lagrange** and **William Rowan Hamilton**, who reformulated Newton’s equations into more abstract and mathematically elegant frameworks: **Lagrangian mechanics** and **Hamiltonian mechanics**. These developments were instrumental in extending the reach of classical mechanics to fields like celestial mechanics, fluid dynamics, and rigid body motion.

### Core Concepts in Classical Mechanics

#### 1. **Newton's Laws of Motion**
The three **Laws of Motion**, formulated by Newton, are the foundation of classical mechanics.

- **First Law (Law of Inertia)**: An object at rest will stay at rest, and an object in motion will stay in motion with the same speed and direction unless acted upon by a net external force. This law introduces the concept of inertia, which is the resistance of an object to changes in its state of motion.

- **Second Law (F = ma)**: The acceleration of an object is directly proportional to the net force acting on it and inversely proportional to its mass. This law provides the quantitative relationship between force, mass, and acceleration. Mathematically, it is expressed as:

\[
F = ma
\]
where \( F \) is the force acting on the object, \( m \) is the mass, and \( a \) is the acceleration.

- **Third Law (Action and Reaction)**: For every action, there is an equal and opposite reaction. This means that forces always occur in pairs; if object A exerts a force on object B, object B will exert an equal and opposite force on object A.

#### 2. **Kinematics**
Kinematics is the study of the motion of objects without considering the forces that cause the motion. It involves quantities like displacement, velocity.

1. MODULE 1
CLASSICAL MECHANICS

3. INTRODUCTION
Limitations of classical mechanics:

(a) In distances smaller than

we must apply principles of quantum mechanics or quantum field theory

(b) Velocities close to the velocity of light

we must apply relativistic corrections

Why to study classical mechanics

Range of Applicability : for sizes larger than atoms and smaller than the
solar system, and ordinary velocities, classical mechanics is an excellent
approximation

mv
h
4

4. Force
A force is a pull or push exerted on a body which changes or attempt
to change, the condition of the rest of a body, or its state of uniform
motion
When all the forces acting upon an object balance each other,
the object will be at equilibrium; it will not accelerate.
Resultant Force
The Resultant force is the sum of all forces acting on a body at a
specific instant. If the resultant force is zero remains in the state of
constant acceleration
Balanced and unbalanced forces

5. a set of criteria or stated values in relation to
which measurements or judgements can be made
FRAMES OF REFERENCE
Frames of Reference
Inertial Non-inertial
•Inertial Frame of Reference: The frame of reference
in which Newton’s laws are valid.
E.g: any frame which is at rest or in uniform motion
(one of the most common inertial frame of reference is
earth )
•Non-Inertial Frame of Reference: Any frame which
is accelerated can be termed as non-inertial frame of
reference

6. NEWTON’S FIRST LAW OF MOTION
A body remains in it’s state of rest
or uniform motion unless acted by
an external force.
Or
If the sum of all the force acting
on a particle is zero then and only
then the particle remains
unaccelearated
i.e. =0 if and only if =0


7. .
Newton’s Second Law of Motion
The rate of change of momentum of a
body is proportional to the applied
force and takes place in the direction
in which the force act

11. FREE BODY DIAGRAM

12. .
Approach Towards Problem Solving
(a) Identify the system
(b) Identify the forces
(c) Free body diagram
(d) Choose an axis and write equations
(e) Code the problem

13. KINEMATICS

14. GRAVITATION
The force of gravity is directly
proportional to the mass of
the earth, the mass of the
object and inversely
proportional to the distance
between the object and the
earths centre.

15. GRAVITY
G is the force of gravitation which exists between two
bodies of unit mass kept at a unit distance




Note :The value of G very small and hence
gravitational force is considered to be one of the
weakest force.
If they attract
Why they are
not sticking to
each other

16. Acceleration due to gravity is defined by

Variation in the Value of g m
F
a



Force of gravitation for an object at the surface of the earth at a distance h
from the earth surface is given by

mg
hR
GMm
F 



2 
2
0
2
1










R
h
g
hR
GM
m
F
g
Hence,

If h<<R







R
h
gg
o
2
1
NOTE: g is maximum at the earth surface. The value of g decrease with increasing
in height from the earth surface . The value of g decreases with the increase in
depth

17. •Mass is the measure of how much material in a
object

•Weight is the gravitational force exerted on that
mass when placed in gravitational field


•Mass will be a constant everywhere, whereas the
weight changes according to the place.

Weight and Mass

18. FRICTION
•When two bodies are in contact with
each other, the particles at the surface
of the two bodies exerts an
electromagnetic force between each
other.

•The force has a normal and horizontal
component

•The normal component is called
normal force or normal contact force

•Friction is the horizontal component of
the contact force

19. FRICTION
•Act always surfaces in contact

•Always act tangential on the surface

•Always resist relative motion

•Friction force =-applied force

•Independent of the surface area of the surface at
contact

Types of Friction

20. KINEMATIC FRICTION
•When two bodies are in contact and bodies in contact
slip over one another the friction generated is kinematic
friction

21. STATIC FRICTION
Limiting friction

22. WORK AND ENERGY
Energy is the capacity of a physical system to do
work

Work, energy transferred by a force
Or
Force multiplied by displacement of point of the
application of force in the direction of force.

23. WORK AND ENERGY
 Work =Force X
displacement
(Nm-Joule)
•UNIT in SI
Work and Energy:
Joule (Nm)=Kgm
-
2
s
-2

24. DIFFERENT FORMS OF ENERGY
Kinetic Energy Potential Energy
Motion Gravitational
Thermal energy Chemical
Sound Nuclear
Electromagnetic Elastic
Electric -

25. KINETIC ENERGY
Kinetic energy is the work done by a force F to
produce movement in an object from its initial
position to the final position

26. KINETIC ENERGY
A tangential force is required to make a linear
movement in a body or we need a tangential
component of force to produce kinetic energy
or change in kinetic energy

Change in Kinetic energy
=

Work – Energy theorem: Work done on a
particle by the resultant force=the change in
kinetic energy

27. POWER
Power: Rate of change of work

Unit: Watt or Joule/second



 rdFw . w
dt
d
p 
vFp.

28. CONSERVATIVE &
NON-CONSERVATIVE FORCE
If the work done by a force depends only on
the initial and final state and not on the path
taken, then it is called a conservative force.
Example of conservative force: Gravity and
coulomb force

Example of non-conservative force : force of
friction

29. POTENTIAL ENERGY
the energy possessed by a body by virtue of its
position relative to others, stresses within
itself, electric charge, and other factors.
Potential energy at a constant height mghw
Potential energy due to spring motion, Hooke’s law 2
2
1
kxw

30. LAW OF CONSERVATION OF MECHANICAL
ENERGY
For a system of conservative forces and if there is no
external force acting on the system, work done by the
system to change the potential energy of the body
from its initial position to final position will be
)( ifcif kkwuu  iiff ukku 
Upon rearranging
The total mechanical energy of a system remains
constant if the internal forces are conservative
and external forces do not work

31. ROTATIONAL MOTION
Axis of rotation: If each particle in a rigid body moves in a
circle and the centre of the circles lies in a straight line, then
the plane of all the circles traced by the particles are
perpendicular to the straight line. Then the body is said to be
rotating about a line and this line is called the axis of rotation.
E.g. Fan, Gas stove nob etc.

For all the particles exhibit circular motion . The particles near
to the line move less faster as compared to the particles away
from the straight line. Hence all the particles covers one
rotation at the same time

32. ROTATIONAL MOTION 
is the Angular position
• Angular velocity , dt
d

• SI unit of angular velocity is radian/seconds
• For uniform angular acceleration, t

33. ROTATIONAL MOTION
• If the body is not covering equal angles at
equal intervals of time, then the body is said
to be have a rotational acceleration 2
2
dt
d
dt
d 
 
• Angular acceleration
•If Angular acceleration is constant 


2
2
1
2
0
2
2
0
0



t
t

34. ROTATIONAL MOTION
Relation between linear motion and rotation 

ra
rv


Where, v is the linear velocity and a is the linear acceleration
For Linear acceleration =non-zero force
For Angular acceleration = non-zero torque

35. TORQUE
The force responsible to produce angular
momentum is called torque Fr
Case 1
If F is parallel to AB (axis of rotation )

Then, along the angle of rotation, there wont be any component
of torque due to force F
Case 2

If F and r intersects
0Fr

36. TORQUE
The force responsible to produce angular
momentum is called torque Fr
•Skew lines (Non-Parallel, non- intersecting lines)
•F Perpendicular to AB but F and AB won’t intersects
A
B
O
S
P
F
θ   sinsin rFrFFr  )(OSF 
Magnitude of force X common
perpendicular for AB and F
Skew lines but not perpendicular

 combination of case 1 and case 3

37. TORQUE 


N
i
iitotal rMI
1

The total external torque acting on a system can be
defined as the product of angular acceleration and the
momentum of inertia of the system I
ext

OR

38. 1.39 1.40 2.8 2.9 2.22