Question

A solid sphere and a hollow sphere of the same mass and of the same radius are rolled on an inclined plane. Let the time taken to reach the bottom by the solid sphere and the hollow sphere be \( t_1 \) and \( t_2 \), respectively, then:

• A. \( t_1>t_2 \)
• B. \( t_1 = 2t_2 \)
• C. \( t_1 = t_2 \)
• D. \( t_1<t_2 \)

Hint:

When comparing rolling objects, recall that objects with less moment of inertia relative to their mass and radius will accelerate faster and reach the bottom of the incline in less time.

Answer 1

The Correct Option is D

The time taken by an object to roll down an incline depends on its moment of inertia. The solid sphere has a smaller moment of inertia compared to the hollow sphere, which allows it to accelerate more quickly and reach the bottom in less time.

Using the equation for rolling motion, the time taken for the solid sphere to reach the bottom will be less than the time taken by the hollow sphere, as the solid sphere has a greater rotational inertia compared to the hollow one.

Final Answer: \( t_1 < t_2 \).

Answer 2

Step 1: Understand the problem setup.
We are given a solid sphere and a hollow sphere, both with the same mass and radius. These spheres are rolled on an inclined plane. The time taken to reach the bottom of the incline by the solid sphere and the hollow sphere are denoted as \( t_1 \) and \( t_2 \), respectively. We need to compare \( t_1 \) and \( t_2 \).

Step 2: Analyze the motion of the spheres.
When an object rolls down an incline, its motion involves both translational and rotational kinetic energy. The total energy at the start is converted into kinetic energy as the object rolls down.
The equation for the motion of a rolling object is given by:
\[ mgh = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2, \] where:
- \( m \) is the mass of the object,
- \( g \) is the acceleration due to gravity,
- \( h \) is the height from which the object rolls,
- \( v \) is the velocity of the center of mass,
- \( I \) is the moment of inertia of the object,
- \( \omega \) is the angular velocity.

Step 3: Moment of inertia for solid and hollow spheres.
For a solid sphere, the moment of inertia is:
\[ I_{\text{solid}} = \frac{2}{5}mr^2. \] For a hollow sphere, the moment of inertia is:
\[ I_{\text{hollow}} = \frac{2}{3}mr^2. \] The greater the moment of inertia, the more the rotational energy, which means less energy is available for translational motion, resulting in a slower acceleration.

Step 4: Compare the times.
The acceleration of an object rolling down the incline depends on the ratio of the translational kinetic energy to the rotational kinetic energy. The object with the smaller moment of inertia will accelerate faster and take less time to reach the bottom.
Since the solid sphere has a smaller moment of inertia compared to the hollow sphere, it will accelerate faster and reach the bottom in less time.

Final Answer:
The time taken by the solid sphere to reach the bottom is less than that of the hollow sphere, so \( t_1 < t_2 \).