Question

Figure depicts a circular motion. The radius of the circle, period of revolution, initial position and the sense of revolution are indicated on the figure. The simple harmonic motion of the $X$-projection of the radius vector of the rotating particle $P$ is

• A. $A\cos \left(\frac{2\pi }{4}t+\frac{\pi }{4}\right)$
• B. $A\cos \left(\frac{\pi }{4}t+\frac{\pi }{4}\right)$
• C. $A\cos \left(\frac{\pi }{3}t+\frac{\pi }{2}\right)$
• D. None of these

Answer 1

Answer: (A)

At $t=0,OP$ makes an angle of $45^{\circ }=(\pi /4)rad$ with the (positive direction of) $X$-axis. After time $t$, it covers an angle $\frac{2\pi }{T}t$ in the anti-clockwise sense and makes an angle of $\frac{2\pi }{T}t+\frac{\pi }{4}$ with the $X$-axis.

$\therefore$ The projection of $OP$ on the $X$-axis at time $t$ is given by
$x(t)=A\cos \left(\frac{2\pi }{T}t+\frac{\pi }{4}\right)$
For
$T=4s,$
$x(t)=A\cos \left(\frac{2\pi }{4}t+\frac{\pi }{4}\right)$
which is a SHM of amplitude $A$, period $4s$ and an initial phase $\frac{\pi }{4}$.