Question

The magnitude of the displacement of a particle moving in a circle of radius $a$ with constant angular speed $\omega$ varies with time $t$ as

• A. $2asin\omega t$
• B. $2asin\frac{\omega t}{2}$
• C. $2acos\omega t$
• D. $2acos\frac{\omega t}{2}$

Answer 1

The correct option is B $2asin\frac{\omega t}{2}$

In time $t$, particle has rotated an angle $\theta =\omega t$,



From $△ORQ$
$OR=a\cos \theta ,QR=a\sin \theta$
$PR=OP-OR=a-a\cos \theta$

Displacement of the particle
$S=PQ=\sqrt{Q{R}^2+P{R}^2}$
$\Rightarrow S=\sqrt{(a\sin \omega t{)}^2+(a-a\cos \omega t{)}^2}$
$\Rightarrow S=a\sqrt{(\sin \omega t{)}^2+(\cos \omega t{)}^2+1-2\cos \omega t}$
$\Rightarrow S=a\sqrt{2-2\cos \omega t}$
$\Rightarrow S=a\sqrt{2(1-\cos \omega t})$
$\Rightarrow S=a\sqrt{4(\sin \frac{\omega t}{2}{)}^2}$​​​​​​​​​​​​​​
$\Rightarrow s=2a\sin \frac{\omega t}{2}$
Hence, the correct answer is option (b).