Question

The pulley shown in the following figure has a radius of 20 cm and moment of inertia 0⋅2 kg-m². The string going over it is attached at one end to a vertical spring of spring constant 50 N/m fixed from below, and supports a 1 kg mass at the other end. The system is released from rest with the spring at its natural length. Find the speed of the block when it has descended through 10 cm. Take g = 10 m/s².

Answer 1

Given

Moment of inertia of the pully = I = 0.2 kg-m²

Radius of the pully = r = 0.2 m

Spring constant of the spring = k = 50 N/m

Mass of the block = m = 1 kg

g = 10 ms² and h = 0.1 m

On applying the law of conservation of energy, we get

\[mgh = \frac{1}{2}m v^2  + \frac{1}{2}k x^2  + \frac{1}{2}I \left( \frac{\omega}{r} \right)^2\]

On putting x = h = 0.1 m, we get

\[1 = \frac{1}{2} \times 1 \times  v^2  + \frac{1}{2} \times 0 . 2 \times \frac{v^2}{0 . 04} + \frac{1}{2} \times 50 \times 0 . 01\]

\[ \Rightarrow 1 = 0 . 5 v^2  + 2 . 5 v^2  + \frac{1}{4}\]

\[ \Rightarrow 3 v^2  = \frac{3}{4}  \]

\[ \Rightarrow  v^2  = \frac{1}{4}\]

\[ \Rightarrow v = \frac{1}{2} = 0 . 5\text{m/s}\]