Question

A disk and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first ?

• A. Sphere
• B. Both reach at the same time
• C. Depends on their masses
• D. Disk

Answer 1

The Correct Option is A

Comparison of Acceleration for Disc and Sphere 

The formula for acceleration (\( a \)) of an object rolling down an incline is given by:

\(a = \frac{g \sin \theta}{1 + \frac{K^2}{R^2}}\)

Where: - \( g \) is the acceleration due to gravity, - \( \theta \) is the angle of the incline, - \( K \) is the radius of gyration of the object, - \( R \) is the radius of the object.

For a Disc

For a disc, the value of \( \frac{K^2}{R^2} \) is:

\(\frac{K^2}{R^2} = \frac{1}{2} = 0.5\)

For a Sphere

For a sphere, the value of \( \frac{K^2}{R^2} \) is:

\(\frac{K^2}{R^2} = \frac{2}{5} = 0.4\)

Comparison of Acceleration

From the formula for acceleration, we see that the value of \( \frac{K^2}{R^2} \) for the sphere is smaller than for the disc. Therefore, the acceleration of the sphere is greater than the acceleration of the disc:

\(a(\text{sphere}) > a(\text{disc})\)

Conclusion:

Since the acceleration of the sphere is greater than the acceleration of the disc, the sphere reaches the bottom of the incline first:

\(\therefore \text{sphere reaches first}\)