Class notes Ece102

LOYOLA INSTITUTE OF TECHNOLOGY
PALANCHUR, CHENNAI 600123




(COMMON TO ALL B.E / B.TECH STUDENTS)

REGULATION – 2021




NAME: _______________________________________________

BRANCH: _____________________________________________

SECTION: _____________________________________________

,
, 1. Mechanics


Objective:
 To make the students effectively to achieve an understanding of mechanics.
Syllabus:
Multiparticle dynamics: Center of mass (CM) - CM of continuous bodies - motion of the CM -
kinetic energy of system of particles. Rotation of rigid bodies: Rotational kinematics - rotational
kinetic energy and moment of inertia - theorems of M .I - moment of inertia of continuous bodies
- M.I of a diatomic molecule - torque - rotational dynamics of rigid bodies - conservation of
angular momentum - rotational energy state of a rigid diatomic molecule - gyroscope - torsional
pendulum - double pendulum - Introduction to nonlinear oscillations.


1. Introduction
Mechanics is a branch of physics which deals with the motion of bodies under the action of
forces. In elementary mechanics, most of the bodies are assumed to be rigid. But in actual
practice, no body is perfectly rigid. When a stationery body is acted upon by some external
forces, then the body may start to rotate (or) move about any point. If the body doesn’t move (or)
rotate then it is said to be in equilibrium.
We know that the rigid body is the combination of many particles i.e., multiparticle. Let us
discuss the basic definitions relate to mechanics
(1) Angular displacement
Definition
The change in position of the particle moving in a circular path with respect to an angle (d) is
called angular displacement.
Proof A Rd

Let us consider a particle of mass m moving in a circular path of radius ‘R’ R B
with respect to the center of the circle O. At t=0 sec, the particle is located 2
at the point A and after time interval t , it reaches the point B as shown in 1
O
figure.
W.K.T., the angular displacement of a particle is the change in angular
Position between two points A and B, which can be measured by the angle
(2-1) between the radius vector of these two positions A and B.
∴ the angle between A and B is d = (2 - 1).
Angular displacement d = (2 - 1). (unit: Radian)
We can write arc length as AB = l,
Then the relation between angular displacement (d) and linear displacement (l) is given by its
arc length as l = R d.




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, (2) Angular velocity
The rate of change of angular displacement is called angular velocity
i.e., Angular velocity () = d /dt. (unit : Rad s-1)
The relation between angular velocity () and linear velocity (v) is given by v = r .
(3) Angular acceleration
The rate of change of angular velocity is called angular acceleration.
i.e., Angular acceleration () = d/dt (or) d2 / dt2. (Unit: Rad s-2).
(4) Angular momentum
The moment of inertia times of angular velocity of the particle is called angular momentum.
i.e., Angular momentum L = I  (Unit : kgm2s-1)
(5) Inertia
It is the tendency of an object to maintain its state of rest or of uniform motion along the same
direction. Inertia is a resisting capacity of an object to alter its state of rest and motion (direction
and /or magnitude).


1.1. Multiparticle dynamics (Dynamics in a system of particles)
We know dynamics is the study of motion of bodies under the action of forces. Multiparticle
dynamics (dynamics in a system of particles) is the study of motion in respect of a group of
particles in which the separation between the particles will be very small i.e., the distance
between the particles will be negligible.
Explanation
In dynamics, we study the physical parameters by considering an object as a point mass and its
shape and size is ignored. But, in real world problems, object will execute rotational and
translational motion. For example, if we kick the football, it has both translational and rotational
motions. As both the motion depends on the size and shape of the object, both cannot be ignored,
even it is negligible. Thus, the study of rotational and translator motion with respect to the system
of particles is called multi-particle dynamics.


1.2. Centre of mass
We know that mass is the measure of the body’s resistance to change the motion (or) it is measure
of inertia of the body. It is a scalar quantity and it is constant.
(i) A system consists of many particles with different masses and different position from the
reference point.
(ii) The mass of the system is equal to the sum of the mass of each particle in the system.




RR/ PHY/
Department VCET
of Physics Loyola Institute of Technology Page
Page 22