An empty bucket of mass $1 \mathrm{~kg}$ attached by a light cord passed over a pulley of a water well is…
An empty bucket of mass $1 \mathrm{~kg}$ attached by a light cord passed over a pulley of a water well is released from rest. If the pulley assembly is assumed to be a uniform solid cylinder of mass $8 \mathrm{~kg}$ and free to rotate about its axis without any friction, then the speed of the bucket as it hits the water $16 \mathrm{~m}$ below is (take, $g=10 \mathrm{~ms}^{-2}$ )
$4 \mathrm{~ms}^{-1}$
$8 \mathrm{~ms}^{-1}$
$16 \mathrm{~ms}^{-1}$
$20 \mathrm{~ms}^{-1}$
Solution
Torque on pulley $=I \alpha$
$
\begin{aligned}
& \Rightarrow \quad m g R=\frac{M R^2 \alpha}{2} \\
& \alpha=\frac{2 m g}{M R} \\
&
\end{aligned}
$
So, acceleration of rope, $a=R \alpha$
$
=R \cdot \frac{2 m g}{M R}=\frac{2 m g}{M}=\frac{2 \times 1 \times 10}{8}=\frac{10}{4} \mathrm{~ms}^{-2}
$
Speed of bucket as it falls by $16 \mathrm{~m}$ is
$
v=\sqrt{2 \times a \times s}=\sqrt{2 \times \frac{10}{4} \times 16}=\sqrt{80}=8.6 \approx 8 \mathrm{~ms}^{-1}
$