Question

A pulley of $10cm$ diameter is wrapped with a cord of length $2m$ around its periphery. If pulley is initially at rest and is given an angular acceleration of $0.5rad/s^2$, then total angular displacement of the pulley for the cord to unwind completely and the time taken by the cord to unwind completely are

• A. $20rad,\sqrt{10}s$
• B. $20rad,4\sqrt{10}s$
• C. $40rad,\sqrt{10}s$
• D. $40rad,4\sqrt{10}s$

Answer 1

The correct option is D $40rad,4\sqrt{10}s$

Initial angular velocity $\left({\omega }_0\right)=0rad/s$
Angular acceleration $\left(\alpha \right)=0.5rad/s^2$

Total angular displacement for the cord to unwind completely $\theta =\frac{l}{r}=\frac{200}{5}=40rad$
Since the angular acceleration is uniform, we can use
$\theta ={\omega }_{0}t+\frac{1}{2}\alpha t^2$
$40=\frac{1}{2}\times 0.5t^2$ ($\because {\omega }_0=0$)
$\Rightarrow t^2=160$
$\Rightarrow t=4\sqrt{10}s$