Question

Figures given 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

(a) Time period, $t=2s$
Amplitude, $A=3cm$
At time, $t=0$, the radius vector OP makes an angle $\pi /2$ with the positive x-axis, i.e., phase angle $\varphi =+\pi /2$
Therefore, the equation of simple harmonic motion for the x-projection of OP, at the time t, is given by the displacement equation:

$x=Acos[\frac{2\pi t}{T}+\varphi ]$

$=3cos(\frac{2\pi t}{2}+\frac{\pi }{2})=-3sin\left(\frac{2\pi t}{2}\right)$

$\therefore x=-3sin\left(\pi t\right)cm$

(b) Time Period, $t=4s$

Amplitude, $a=2m$

At time $t=0$, OP makes an angle π with the x-axis, in the anticlockwise direction, Hence, phase angle $\varphi =+\pi$

Therefore, the equation of simple harmonic motion for the x-projection of OP, at the time t, is given as:

$x=acos[\frac{2\pi t}{T}+\varphi ]$

$=2cos(\frac{2\pi t}{4}+\pi )$

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