Question

Uniform disc A and B are connected by an ideal spring of spring constant k. Mass and radius for disc A and B are 3 m, 2R and m, R respectively. Friction is sufficient to prevent slipping. If time period for oscillation of the system is $3\pi\sqrt{\frac{m}{\alpha k}}$, then value of $\alpha$ is

• A. 2

Answer 1

Answer: (A)

2

Answer 2

1. Calculate effective mass for each rolling disc:

   • Disc A: $m_{A,\text{eff}} = 3m + \frac{6mR^2}{4R^2} = \frac{9m}{2}$
   • Disc B: $m_{B,\text{eff}} = m + \frac{\tfrac{1}{2}mR^2}{R^2} = \frac{3m}{2}$

2. Use reduced mass:

   $$\mu = \frac{(9m/2)(3m/2)}{(9m/2)+(3m/2)} = \frac{9m}{8}$$

3. Find the time period:

   $$T = 2\pi\sqrt{\frac{9m}{8k}} = \frac{3\pi}{\sqrt{2}}\sqrt{\frac{m}{k}}$$

   Equate this with $3\pi\sqrt{\frac{m}{\alpha k}}$ to get $\alpha = 2$.