Question

A pulley wheel of diameter $2cm$ has a $1m$ long cord strapped across its periphery. The wheel is initially at rest. If it is given an angular acceleration of $0.1rad/s^2$, then the time taken for the cord to unwind completely is

• A. $20s$
• B. $20\sqrt{3}s$
• C. $20\sqrt{5}s$
• D. $10\sqrt{1}0s$

Answer 1

The correct option is C $20\sqrt{5}s$

Initial angular velocity $\left({\omega }_0\right)=0rad/s$
Given, angular acceleration $\left(\alpha \right)=0.1rad/s^2$

Total angular displacement of the pulley wheel
$\theta =\frac{l}{r}=\frac{100}{1}=100rad$

Since. angular acceleration is uniform, we are free to use equation of motion.
$\theta =w_{0}t+\frac{1}{2}\alpha t^2$
$\Rightarrow 100=0+\frac{1}{2}\times \left(0.1\right)t^2$
$\Rightarrow 100=\frac{1}{20}{t}^2$
$\Rightarrow t^2=2000$
$\therefore t\approx 20\sqrt{5}s$