Question

Figures 13.20 correspond to two circular motions. The radius of the circle, the period of revolution, the initial position, and the sense of revolution (i.e. clockwise or anti clockwise) are indicated on each figure.

Obtain the corresponding simple harmonic motions of the x-projection of the radius vector of the revolving particle P, in each case.

Answer 1

Time period, $T$ = $2 \,s$
Amplitude, $A$ = $3 \,cm$
At time, $t$ = $0$, the radius vector OP makes an angle $\frac{\pi}{2}$ with the positive x-axis, i.e., phase angle $\phi$ =$+\frac{\pi}{2}$
Therefore, the equation of simple harmonic motion for the the x-projection of OP, at time t, is given by the displacement equation:
$x$ = $A\cos\bigg[\frac{2\pi t}{T}+\phi\bigg]$

=$3\cos\bigg(\frac{2\pi t}{2}+\frac{\pi }{2}\bigg)$=$-3\sin\bigg(\frac{2\pi t}{2}\bigg)$
$\therefore$ $x$ =$-3\sin\pi \; t\; cm$
Time period, $T$ =$4 \,s$
Amplitude, $a$ = $2\,m$
At time $t$ = 0, OP makes an angle $\pi$ with the x-axis, in the anticlockwise direction.
Hence, phase angle, $\phi$ =$+\pi$
Therefore, the equation of simple harmonic motion for the x-projection of OP, at time t, given as:

$x$ =$a\cos\bigg(\frac{2\pi t}{T}+\phi\bigg)$= $2\cos\bigg(\frac{2\pi t}{4+\pi}\bigg)$

$\therefore$ $x$ =$-2\cos\bigg(\frac{\pi}{2} t\bigg)m$