Mechanics

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.




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,Hence, if the mass of the entire particles of the system is concentrated at a particular point, that
point is called centre of mass of the system.
1.3. Centre of mass in a one dimensional system
The system consists of many particles with different positions and different masses. If the mass
of the entire particle in the system is concentrated at a particular point, then that point is called
centre of mass of the system.
Explanation
Let us consider a fulcrum placed along the x axis which is not at equilibrium position as shown
in figure.
m1 m2
mn-1 mn
x2
x1 xn-1
xn

Let the position of masses m1, m2, m3, ….., mn-1, mn be at a distance of x1,x2,……..,xn-1,xn
respectively from the fulcrum. The tendency of a mass to rotate with respect to origin or
supporting point is called moment of mass.
The moment of mass for an elemental mass mn with respect to the fulcrum can be written as
mnxn. If the moments on both sides are equal, then the system is sai to be in equilibrium.
Therefore, total moments with respect to the fulcrum shall be written as
N
m1x1  m2 x2  .......  mn xn   mi xi  0 (1)
i 1


If the total moment is equal to zero, then the centre of mass will lie at the supporting point (or)
fulcrum and the system is said to be in equilibrium. If the fulcrum is placed at the unbalanced
position, then it is shifted to a balanced position (say of distance X) to reach the equilibrium
position.
Under equilibrium condition,
n n

 mi xi   mi X  0
i 1 i 1

n n
(or)  mi xi   mi X
i 1 i 1

n n
(or)  mi xi  X  mi
i 1 i 1

n

m x i i
(or) X  i 1
n
(2)
m
i 1
i




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, n
Where m x
i 1
i i is the moment of system and

n

m
i 1
i is the mass of the system


Thus, the system should be move to a distance of X metres in order to attain the balanced position
of the system.
The distanced moved to obtain equilibrium position (or) so called the centre of mass in a one
dimensional system is given by

m1x1  m2 x2  ....
X (3)
m1  m2  .....

1.4. Centre of mass in three dimensional system
To find the centre of mass in a three dimensional system, let us consider a three dimensional
system in which let m1, m2, m3,…. be the masses placed at position vectors r1(x1,y1, z1),
r2(x2,y2,z2),…. Respectively from the origin ‘O’ as shown in figure

y
m1 r1(x1, y1, z1)
m2 r2(x2, y2, z2)
Rcm
m3 r3(x3, y3, z3)


O x


z

Here,
(i) The centre of mass along the x – axis,
n

m1 x1  m2 x2  .... m x i i
X   i 1
m1  m2  ..... n

m
i 1
i



(ii) The centre of mass along y-axis,
n

m y  m2 y2  .... m y i i
Y 1 1  i 1
m1  m2  ..... n

m
i 1
i



(iii) The centre of mass along z-axis,



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