Engineering Mechanics

(^446) „„„„„ A Textbook of Engineering Mechanics
21.2. IMPORTANT TERMS
The following terms, which will be frequently used in this chapter, should be clerarly understood
at this stage :

1. Angular velocity. It is the rate of change of angular displacement of a body, and is expressed
   in r.p.m. (revolutions per minute) or in radian per second. It is, usually, denoted by ω
   (omega).


2. Angular acceleration. It is the rate of change of angular velocity and is expressed in
   radian per second per second (rad/s^2 ) and is usually, denoted by α. It may be constant or
   variable.


3. Angular displacement. It is the total angle, through which a body has rotated, and is
   usually denoted by θ. Mathematically, if a body is rotating with a uniform angular velocity
   (ω) then in t seconds, the angular displacement
   θ=ωt

21.3. MOTION OF ROTATION UNDER CONSTANT ANGULAR ACCELERATION

Consider a particle, rotating about its axis.

Let ω 0 =Initial angular velocity,

ω=Final angular velocity,

t = Time (in seconds) taken by the particle to change its velocity

from ω 0 to ω.

α = Constant angular acceleration in rad/s^2 , and

θ=Τotal angular displacement in radians.

Since in t seconds, the angular velocity of the particle has increased steadily from ω 0 to ω at

the rate of α rad/s^2 , therefore

ω=ω 0 + α t ...(i)

and average angular velocity^0

2

ω+ω

=

We know that the total angular displacement,

θ= Average velocity × Time =

0

2

t

⎛⎞ω+ω

⎜⎟×

⎝⎠

...(ii)

Substituting the value of ω from equation (i),

00 0 2

0

()2 1

222

tt

tttt

ω+ω+α ω+α

θ= ×= ×=ω + α ...(iii)

and from equation (i), we find that

t=ωω–^0

α

Substituting this value of t in equation (ii),

22

000 ––

22

⎛⎞⎛⎞ω+ω ω ω ω ω

θ=⎜⎟⎜⎟× =

⎝⎠⎝⎠αα

∴ ω=ω+αθ^2202 ...(iv)