Question

An empty bucket of mass $1\, kg$ 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 $8\, kg$ and free to rotate about its axis without any friction, then the speed of the bucket as it hits the water $16\, m$ below is
(take, $g=10\, ms ^{-2}$ )

• A. $4\, ms ^{-1}$
• B. $8\, ms ^{-1}$
• C. $16\, ms ^{-1}$
• D. $20\, ms ^{-1}$

Answer 1

Answer: (B)

Torque on pulley $=I \alpha$
$\Rightarrow m g R=\frac{M R^{2} \alpha}{2}$
$\alpha=\frac{2 m g}{M R}$
So, acceleration of rope, $a=R \alpha$
$=R \cdot \frac{2 m g}{M R}=\frac{2 m g}{M}$
$=\frac{2 \times 1 \times 10}{8}=\frac{10}{4} ms ^{-2}$
Speed of bucket as it falls by $16\, m$ is
$v=\sqrt{2 \times a \times s}=\sqrt{2 \times \frac{10}{4} \times 16}$
$=\sqrt{80}= 8.6 \approx 8\, ms ^{-1}$