Question

(a) Define simple harmonic motion. Show that the motion of (point) projection of a particle performing uniform circular motion, on any diameter, is simple harmonic.
(b) Can a simple pendulum be used in an artificial satellite? Give the reason

Answer 1

(a) Simple harmonic motion(SHM) :

SHM of a particle is a linear periodic motion or oscillatory motion along a line where the restoring force is directly proportional to the displacement of particle and acts in the direction opposite to that of displacement.

(b) SHM as a projection of circular motion:

Consider a particle P moving on a circle of radius R, with a constant angular speed $\omega$. Assuming the two dimensional co-ordinate system at the center of the circle with center as origin and two perpendicular diameters along x-axis and y-axis respectively.

Suppose particle Pis on x-axis at t=0. During motion of particle at time t, the radius OP will make an angle $\theta$ with x-axis,

$\theta =\omega t$

Dropping the perpendiculars on x-axis and y-axis we get the x and y co-ordinates of the particle P at time t as,

$x=OX$

$x=OP\cos \omega t$

$x=R\cos \omega t$ ..........(1)

And

$y=OY$

$y=OP\sin \omega t$

$y=R\sin \omega t$ ..........(2)

From equations (1) and (2), we can state that the base of perpendiculars X and Y executes a SHM on x-axis and y-axis respectively. The amplitude of SHM is R and the angular frequency is $\omega$.

Thus, the projection of a particle performing UCM, on any diameter, is SHM.

(c) An artificial satellite is like a freely falling body hence, effect of gravity inside the satellite is zero. As the time period(of oscillation) T, of simple pendulum is,

$T\propto \sqrt{\frac{l}{g}}$

where, l is length of pendulum and g is gravitational acceleration(gravity).

Since, g is zero, the simple pendulum will oscillate with infinite time period, which is simply not possible. Hence, simple pendulums cannot be used in artificial satellites.