Question

The $x$ and $y$ coordinates of a particle are $x=A\sin \left(\omega t\right)$ and $y=A\sin (\omega t+\frac{\pi }{2})$. The motion of the particle is ($x$ and $y$ are in the directions shown in the figure)

• A. circular clockwise
• B. circular anti-clockwise
• C. elliptical
• D. rectilinear

Answer 1

The correct option is A circular clockwise

At time = 0, (0,A)
$\begin{array}{cc}{x}_{\left(at0\right)}& =A\sin \left(\omega t\right)\\ & =A\sin 0\\ & =0\\ y_{\left(at0\right)}& =A\sin (\omega t+\frac{\pi }{2})\\ & =A\sin \frac{\pi }{2}\\ & =A\end{array}$

At time t = $\frac{\pi }{2\omega }$, (A,0)
$\begin{array}{cc}{x}_{\left(att\right)}& =A\sin (\omega \times \frac{\pi }{2\omega })\\ & =A\sin \frac{\pi }{2}\\ & =A\\ y_{\left(att\right)}& =A\sin (\omega \times \frac{\pi }{2\omega }+\frac{\pi }{2})\\ & =A\sin \pi \\ & =0\end{array}$

At time t = $\frac{\pi }{\omega }$, (0,-A)

$\begin{array}{cc}{x}_{\left(atgivent\right)}& =A\sin (\omega \times \frac{\pi }{\omega })\\ & =0\\ y_{\left(atgivent\right)}& =A\sin (\omega \times \frac{\pi }{2\omega }+\frac{\pi }{2})\\ & =A\sin \pi \\ & =0\end{array}$