A Classical Approach of Newtonian Mechanics

7 CIRCULAR MOTION 7.5 Non-uniform circular motion

s

s

=

.,

=

acts towards the centre of the circle. In other words,

T sin θ = m ω^2 r. (7.19)

Taking the ratio of Eqs. (7.18) and (7.19), we obtain

However, by simple trigonometry,

tan θ =

ω^2 r

.

g

(7.20)

Hence, we find

tan θ =

r

. (7.21)
h

ω =

g

. (7.22)
h

Note that if l is the length of the string then h = l cos θ. It follows that

ω

g

. (7.23)
l cos θ

For instance, if the length of the string is l = 0.2 m and the conical angle is

θ = 30 ◦ then the angular velocity of rotation is given by

ω

‚

9.81

0.2 × cos 30 ◦

= 7.526 rad./s. (7.24)

This translates to a rotation frequency in cycles per second of
ω
f = = 1.20 Hz. (7.25)
2 π

7.5 Non-uniform circular motion

Consider an object which executes non-uniform circular motion, as shown in
Fig. 61. Suppose that the motion is confined to a 2-dimensional plane. We can

specify the instantaneous position of the object in terms of its polar coordinates r