Question

A circular ring and a solid sphere having same radius roll down on an inclined plane from rest without slipping. The ratio of their velocities when reached at the bottom of the plane is $\sqrt{\frac{\mathrm{x}}{5}}$ where $\mathrm{x}=$ _______.

Hint:

Use mechanical energy conservation to find the velocities of the ring and the solid sphere.

Answer 1

Correct Answer: 3 - 4

1. Mechanical energy conservation: \[ K_i + U_i = K_f + U_f \] \[ 0 + Mgh = \frac{1}{2} mv^2 \left(1 + \frac{k^2}{R^2}\right) + 0 \] \[ v = \sqrt{\frac{2gh}{1 + \frac{k^2}{R^2}}} \]
2. Ratio of velocities: \[ \frac{v_{\text{ring}}}{v_{\text{solid sphere}}} = \sqrt{\frac{1 + \frac{2}{5}}{1 + 1}} = \sqrt{\frac{7}{10}} \] \[ x = 3.5 \approx 4 \] Consequently, option (4) is the correct answer, with a value of 4.