Mechanical Engineering Principles

128 MECHANICAL ENGINEERING PRINCIPLES

Angular velocity

The speed of revolution of a wheel or a shaft
is usually measured in revolutions per minute or
revolutions per second but these units do not form
part of a coherent system of units. The basis used
in SI units is the angle turned through (in radians)
in one second.
Angular velocity is defined as the rate of change
of angular displacementθ, with respect to timet,
and for an object rotating about a fixed axis at a
constant speed:

angular velocity=

angle turned through

time taken

i.e ω=

θ

t

( 11. 4 )

The unit of angular velocity is radians per second
(rad/s).
An object rotating at a constant speed ofnrevo-
lutions per second subtends an angle of 2πnradians
in one second, that is, its angular velocity,

ω= 2 πnrad/s ( 11. 5 )

From equation (11.1), s = rθ, and from equa-
tion (4),θ=ωt, hence

s=rωt or

s

t

=ωr

However, from equation (11.3),v=st,

hence v=ωr ( 11. 6 )

Equation (11.6) gives the relationship between lin-
ear velocity,v, and angular velocity,ω.

Problem 1. A wheel of diameter 540 mm is

rotating at (1500/π) rev/min. Calculate the

angular velocity of the wheel and the linear

velocity of a point on the rim of the wheel.

From equation (11.5), angular velocityω= 2 πn,
wherenis the speed of revolution in revolutions
per second, i.e.

n=

1500

60 π

revolutions per second.

Thus,angular velocity,

ω= 2 π

(

1500

60 π

)

=50 rad/s

The linear velocity of a point on the rim,v=ωr,

whereris the radius of the wheel, i.e.r= 0. 54 / 2

or 0.27 m. Thus,linear velocity,

v=ωr= 50 × 0. 27 = 13 .5m/s

Problem 2. A car is travelling at 64.8 km/h

and has wheels of diameter 600 mm.

(a) Find the angular velocity of the wheels

in both rad/s and rev/min.

(b) If the speed remains constant for

1.44 km, determine the number of

revolutions made by a wheel, assuming

no slipping occurs.

(a) 64.8km/h= 64. 8

km

h

× 1000

m

km

×

1

3600

h

s

=

64. 8

3. 6

m/s=18 m/s

i.e. the linear velocity,v,is18m/s

The radius of a wheel is( 600 / 2 )mm= 0 .3m.

From equation (11.6),v=ωr, henceω=v/r

i.e. theangular velocity,

ω=

18

0. 3

=60 rad/s

From equation (11.5), angular velocity,

ω= 2 πn,wherenis in revolutions per second.

Hence n = ω/ 2 π and angular speed of a

wheel in revolutions per minute is 60ω/ 2 π; but

ω=60 rad/s, hence

angular speed=

60 × 60

2 π

=573 revolutions

per minute (rpm)

(b) From equation (11.3), time taken to travel

1.44 km at a constant speed of 18 m/s is

1440 m

18 m/s

=80 s.