KCET · Physics · Rotational Motion

Three bodies a ring \( (R) \), a solid cylinder \( (C) \) and a solid sphere \( (S) \) having same mass and same radius roll down the inclined plane without slipping. They start from rest, if \( v_{R}, v_{c} \) and \( v_{s} \) are velocities of respective bodies on reaching the bottom of the plane, then

A \( v_{R}=v_{C}=v_{S} \)
B \( v_{R}>v_{C}>v_{S} \)
C \( v_{R} < v_{C} < v_{S} \)
D \( v_{R}=v_{C}>v_{S} \)

Verified Solution

Answer & Solution

Correct Answer

(C) \( v_{R} < v_{C} < v_{S} \)

Step-by-step Solution

Detailed explanation

The velocity of a body rolling without slipping down oninclined plane is given as
\(v=\sqrt{\frac{2 g h}{1+K^{2} / R^{2}}}\)
Here, \( K \) is the radius of gyration.
For ring, \( K=R \)
\(\Rightarrow v_{R}=\sqrt{g h}\)
For solid sphere, \( K=\sqrt{\frac{2}{5}} R \)
\(\Rightarrow v_{s}=\sqrt{\frac{10 g h}{7}}\)
For solid cylinder, \( K=\frac{R}{\sqrt{2}} \)
\(\Rightarrow v_{c}=\sqrt{\frac{4 g h}{3}}\)

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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