Question

The motion of a particle varies with time according to the relation $y=a\sin$$\omega$$t+a\cos$$\omega t$ . Then

• A. the motion is oscillatory but not SHM
• B. the motion is SHM with amplitude $a$
• C. the motion is SHM with amplitude $a$ $\sqrt{2}$
• D. the motion is SHM with amplitude $\frac{a}{2}$

Answer 1

Answer: (C)

the motion is SHM with amplitude $a$ $\sqrt{2}$

$y=asin\omega t+acos\omega t$
$=a\sqrt{2}sin(\omega t+\frac{R}{4})$
Clearly amplitude is $a\sqrt{2}$ and motion is SHM.
because $a=\frac{d^{2}y}{d{t}^2}=-a\sqrt{2}{\omega }^{2}sin(\omega t+\frac{R}{4})$
So, $a$ is proportion to ${\omega }^2$ ,which is the condition of SHM.