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

 To determine which object, the disk or the sphere, reaches the bottom of the inclined plane first, we need to consider the physics of rolling motion and energy conservation.

1. Concept of Rolling Motion: Both the disk and the sphere roll without slipping. This means the motion includes both translational and rotational components.
2. Energy Conservation during Rolling: The potential energy at the top of the incline is converted into translational and rotational kinetic energy at the bottom.
   • The total mechanical energy at the top is given by: 
     \(E_{\text{initial}} = mgh\), where \(m\) is the mass, \(g\) is gravity, and \(h\) is the height.
   • At the bottom of the incline, the energy is:
     \(E_{\text{final}} = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2\), where \(v\) is the linear velocity and \(I\omega^2\) is the rotational kinetic energy. Here, \(I\) is the moment of inertia and \(\omega\) is the angular velocity.
3. Moment of Inertia:
   • For a disk, \(I_{\text{disk}} = \frac{1}{2}mr^2\)
   • For a sphere, \(I_{\text{sphere}} = \frac{2}{5}mr^2\)
4. Roll Dynamics and Acceleration:
   • Acceleration \(a\) of a rolling object down an incline is proportional to \(\frac{g \sin \theta}{1 + \frac{I}{mr^2}}\).
   • Thus, \(a_{\text{disk}} = \frac{g \sin \theta}{1 + \frac{1}{2}}\) and \(a_{\text{sphere}} = \frac{g \sin \theta}{1 + \frac{2}{5}}\).
5. Conclusion: Due to its smaller moment of inertia, for the same translational and rotational energy distribution, the sphere accelerates faster and will reach the bottom of the incline first.

Answer: Sphere