Answer to Question #340453 in Differential Equations for Rajasri

Question

E xplain the general m otion of a Sim ple P endulum .

Answer 1

A simple pendulum consists of a heavy metallic (brass) sphere with a hook (bob) suspended from a rigid stand, with clamp by a weightless inextensible and perfectly flexible thread through a slit cork, capable of oscillating in a single plane, without any friction, with a small amplitude (less than 150). There is no ideal simple pendulum. In practice, we make a simple pendulum by tying a metallic spherical bob to a fine cotton stitching thread.

The spherical bob may be regarded by as a point mass at its centre G. The distance between the point of suspension S and the centre G of the spherical bob is to be regarded as the effective length of the pendulum. The effective length of a simple pendulum, L = l + h + r. Where l is the length of the thread, h is length of hook, r is radius of bob.

The simple pendulum produces Simple Harmonic Motion (SHM) as the acceleration of the pendulum bob is directly proportional to its displacement from the mean position and is always directed towards it. The time period (T) of a simple pendulum for oscillations of small amplitude, is given by the relation,

$T=2\pi\sqrt{L/g}$

Where, g = value of acceleration due to gravity and L is the effective length of the pendulum.

T²= (4π²/g) X L or T² = KL (K= constant)

and, g = 4π²(L/T²)

If T is plotted along the Y-axis and L along the X-axis, we should get a parabola. If T² is plotted along the Y- axis and L along the X-axis, we should get a straight line passing through

the origin.