Question

One ring, one solid sphere, and one solid cylinder are rolling down on the same inclined plane starting from rest. The radius of all three are equal. The object that reaches down with maximum velocity is

• A. Solid cylinder
• B. Solid sphere
• C. Ring
• D. Solid sphere and Ring

Hint:

The object with the smallest moment of inertia for a given mass and radius will have the largest velocity at the bottom of an inclined plane.

Answer 1

The Correct Option is B

The velocity of a rolling object at the bottom of an inclined plane is given by:

\[ v = \sqrt{\frac{2gh}{1 + \frac{I}{mr^2}}}, \] 

where:

• g is the acceleration due to gravity.
• h is the height of the inclined plane.
• I is the moment of inertia.
• m is the mass.
• r is the radius.

For different objects:

• Ring: \( I = mr^2 \) → \( \frac{I}{mr^2} = 1 \)
• Solid sphere: \( I = \frac{2}{5} mr^2 \) → \( \frac{I}{mr^2} = \frac{2}{5} \)
• Solid cylinder: \( I = \frac{1}{2} mr^2 \) → \( \frac{I}{mr^2} = \frac{1}{2} \)

Comparison:

Since \( \frac{2}{5} < \frac{1}{2} < 1 \), the solid sphere has the smallest value of \( \frac{I}{mr^2} \) and thus the largest velocity.

Conclusion: The solid sphere reaches the bottom with maximum velocity.