Question

The radius of circle, the period of revolution, initial position and sense of revolution are indicated in the figure.

y - projection of the radius vector of rotating particle P is

• A. $y\left(t\right)=-3cos2\pi t,$ where y is in $m$
• B. $y\left(t\right)=4sin\left(\frac{\pi t}{2}\right),$ where y is in $m$
• C. $y\left(t\right)=3cos\left(\frac{3\pi t}{2}\right),$ where y is in $m$
• D.
  $y\left(t\right)=3cos\left(\frac{\pi t}{2}\right),$ where y is in $m$

Answer 1

The correct option is D

$y\left(t\right)=3cos\left(\frac{\pi t}{2}\right),$ where y is in $m$
When the particle is performing uniform circular motion, the y-projection of the radius performs SHM.

The y-projection is given by

$y=Asin(\omega t+\varphi )$

Here, amplitude $A=3m$ (radius of the circle)

$\omega =\frac{2\pi }{T}=\frac{\pi }{2}$
(given $T=4s$)

So, $y=3sin(\frac{\pi }{2}t+\varphi )$

At $t=0$, $y=3sin\varphi =3$

$\Rightarrow \varphi =\frac{\pi }{2}$

So the final equation of motion of the y-projection is

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