Question:
Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter.
The correct statement for this situation is
Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter.
The correct statement for this situation is
The correct statement for this situation is
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Use energy conservation with rolling: \(v^2 = \dfrac{2gh}{1 + I/(mR^2)}\); a smaller \(I/(mR^2)\) gives a larger speed.
Updated On: Nov 4, 2026
- All of them will have same velocity.
- The ring has greatest and the cylinder has the least velocity of the centre of mass at the bottom of the inclined plane.
- The sphere has the greatest and the ring has the least velocity of the centre of mass at the bottom of the inclined plane.
- The cylinder has the greatest and the sphere has the least velocity of the centre of mass at the bottom of the inclined plane.
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The Correct Option is C
Solution and Explanation
Energy conservation for rolling without slipping: \(mgh = \tfrac12 mv^2 + \tfrac12 I\omega^2 = \tfrac12 mv^2\left(1 + \dfrac{I}{mR^2}\right)\).
So \(v = \sqrt{\dfrac{2gh}{1 + I/(mR^2)}}\).
Ring: \(I/mR^2 = 1\), \(v = \sqrt{gh}\).
Solid cylinder: \(I/mR^2 = \tfrac12\), \(v = \sqrt{4gh/3}\).
Solid sphere: \(I/mR^2 = \tfrac25\), \(v = \sqrt{10gh/7}\).
Since \(1 \lt 4/3 \lt 10/7\) the sphere is fastest and the ring slowest.
Answer: (3) The sphere has the greatest and the ring has the least velocity of the centre of mass at the bottom of the inclined plane.
So \(v = \sqrt{\dfrac{2gh}{1 + I/(mR^2)}}\).
Ring: \(I/mR^2 = 1\), \(v = \sqrt{gh}\).
Solid cylinder: \(I/mR^2 = \tfrac12\), \(v = \sqrt{4gh/3}\).
Solid sphere: \(I/mR^2 = \tfrac25\), \(v = \sqrt{10gh/7}\).
Since \(1 \lt 4/3 \lt 10/7\) the sphere is fastest and the ring slowest.
Answer: (3) The sphere has the greatest and the ring has the least velocity of the centre of mass at the bottom of the inclined plane.
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