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 $t=0, O P$ 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 $O P$ 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 =4 s , $
$x(t) =A \cos \left(\frac{2 \pi}{4} t+\frac{\pi}{4}\right)$
which is a SHM of amplitude $A$, period $4 s$ and an initial phase $\frac{\pi}{4}$.