Engineering Mechanics

Chapter 31 : Kinetics of Motion of Rotation „„„„„ 623

where F = Force acting on the body, and

l = Perpendicular distance between the point O and line of action of the
force (known as arm or leverage).
Notes:1. The units of torque depend upon the units of force and leverage. If the force is in N and
leverage in mm, then the unit of torque will be N-mm. Similarly, if the force is in kN
and leverage in m, then the unit of torque will be kN-m.

2. The magnitude of the moment of a force is numerically equal to that of the torque, if the
   force and the arm is the same. The term torque is used for the moment of a force in the
   motion of rotation.

31.3.WORK DONE BY A TORQUE

Consider a body pivoted at O. Let a tangential force P be applied at a distance r from the
pivot as shown in Fig. 31.1. As a result of the force, let the body rotate
through a small angle (θ) in radians.

We know that the length of the arc AB

=rθ

and work done by the force P in rotating the body from A to B

= P (AB) = P (rθ) = Pr (θ)
But (P × r) is equal to the torque (T). Therefore work done by a
torque is equal to the torque (T) multiplied by the angular displacement
(θ) in radians.

31.4.ANGULAR MOMENTUM

It is the total motion possessed by a rotating body and is expressed mathematically as :

Momentum =Mass moment of inertia × Angular velocity

=Iω

31.5.NEWTON’S LAWS OF MOTION OF ROTATION

Following are the three Newton’s Laws of Motion of Rotation :

1. Newton’s First Law of Motion of Rotation states, “Every body continues in its state of
   rest or of uniform motion of rotation about an axis, unless it is acted upon by some
   external torque”.


2. Newton’s Second Law of Motion of Rotation states, “The rate of change of angular
   momentum of a body is directly proportional to the impressed torque, and takes place in
   the same direction in which the torque acts”.


3. Newton’s Third Law of Motion of Rotation states, “To every torque, there is always an
   equal and opposite torque.”

31.6.MASS MOMENT OF INERTIA

In chapter 7 we have discussed the moment of inertia of plane figures such as T-section,
I-section, L-section etc. But in this chapter, we shall discuss the moment of inertia of solid bodies or
mass moment of inertia. The mass moment of inertia of a solid body about a line is equal to the
product of mass of the body and square of the distance from that line. Mathematically :

I=m 1 r 12 + m 2 r 22 + m 3 r 32 + ......

=Σ mr^2

Fig. 31.1. Work done

by a torque.