Mechanical Engineering Principles

LINEAR AND ANGULAR MOTION 129

Since a wheel is rotating at 573 revolutions per

minute, then in 80/60 minutes it makes

573 × 80

60

=764 revolutions.

Now try the following exercise

Exercise 53 Further problems on linear

and angular velocity

1. A pulley driving a belt has a diameter of
   360 mm and is turning at 2700/πrevolu-
   tions per minute. Find the angular velocity
   of the pulley and the linear velocity of the
   belt assuming that no slip occurs.
   [ω=90 rad/s,v= 16 .2m/s]


2. A bicycle is travelling at 36 km/h and the
   diameter of the wheels of the bicycle is
   500 mm. Determine the angular velocity
   of the wheels of the bicycle and the linear
   velocity of a point on the rim of one of
   the wheels. [ω=40 rad/s,v=10 m/s]

11.3 Linear and angular acceleration

Linear acceleration, a, is defined as the rate of
change of linear velocity with respect to time. For an
object whose linear velocity is increasing uniformly:

linear acceleration=

change of linear velocity

time taken

i.e a=

v 2 −v 1

t

( 11. 7 )

The unit of linear acceleration is metres per second
squared (m/s^2 ). Rewriting equation (11.7) withv 2 as
the subject of the formula gives:

v 2 =v 1 +at ( 11. 8 )

wherev 2 =final velocity andv 1 =initial velocity.
Angular acceleration,α, is defined as the rate
of change of angular velocity with respect to time.
For an object whose angular velocity is increasing

uniformly:

Angular acceleration=

change of angular velocity

time taken

i.e. α=

ω 2 −ω 1

t

( 11. 9 )

The unit of angular acceleration is radians per sec-

ond squared (rad/s^2 ). Rewriting equation (11.9) with

ω 2 as the subject of the formula gives:

ω 2 =ω 1 +αt ( 11. 10 )

whereω 2 =final angular velocity andω 1 =initial

angular velocity. From equation (11.6),v = ωr.

For motion in a circle having a constant radiusr,

v 2 =ω 2 randv 1 =ω 1 r, hence equation (11.7) can

be rewritten as:

a=

ω 2 r−ω 1 r

t

=

r(ω 2 −ω 1 )

t

But from equation (11.9),

ω 2 −ω 1

t

=α

Hence a=rα ( 11. 11 )

Problem 3. The speed of a shaft increases

uniformly from 300 revolutions per minute

to 800 revolutions per minute in 10s. Find

the angular acceleration, correct to 3

significant figures.

From equation (11.9),

α=

ω 2 −ω 1

t

Initial angular velocity,

ω 1 =300 rev/min= 300 /60 rev/s

=

300 × 2 π

60

rad/s,

final angular velocity,

ω 2 =

800 × 2 π

60

rad/s