A Classical Approach of Newtonian Mechanics

11 OSCILLATORY MOTION 11.6 Uniform circular motion

'

I

s

Note that the reaction, R, at the peg does not contribute to the torque, since
its line of action passes through the pivot point. Combining the previous two
equations, we obtain the following angular equation of motion of the pendulum:

I θ ̈^ = −M g d sin θ. (11.29)

Finally, adopting the small angle approximation, sin θ θ, we arrive at the simple
harmonic equation:

I θ ̈^ = −M g d θ. (11.30)

It is clear, by analogy with our previous solutions of such equations, that the

angular frequency of small amplitude oscillations of a compound pendulum is

given by

ω =

‚

., M g d

. (11.31)

It is helpful to define the length

Equation (11.31) reduces to

L =

I

. (11.32)
M d

ω =

g
, (11.33)
L
which is identical in form to the corresponding expression for a simple pendulum.
We conclude that a compound pendulum behaves like a simple pendulum with

effective length L.

11.6 Uniform circular motion

Consider an object executing uniform circular motion of radius a. Let us set up a
cartesian coordinate system whose origin coincides with the centre of the circle,

and which is such that the motion is confined to the x-y plane. As illustrated in
Fig. 99 , the instantaneous position of the object can be conveniently parameter-

ized in terms of an angle θ.