A ring and a solid sphere rotating about an axis passing through their centers have same radii of gyration. The axis of rotation is perpendicular to plane of ring. The ratio of radius of ring to that of sphere is √(2/x). The value of x is ___________.

Solution

Physics sir · 4 steps

Write the radius of gyration for each body from its moment of inertia, then set the two equal.

1. Radius of gyration. Write I = Mk², where k is the radius of gyration. So k = √(I/M) for each body.
2. Ring. About the axis through its centre and perpendicular to its plane, every bit of mass is at distance Rᵣ. So I = MRᵣ² and k_(ring) = Rᵣ.
3. Solid sphere. About a diameter, I = 2/5MRₛ². So kₛₚₕₑᵣₑ = Rₛ√(2/5).

   Rᵣ = Rₛ√(2/5) ⇒ Rᵣ/Rₛ = √(2/5)

4. So. Compare this with √(2/x). That gives x = 5.

The common slip: 3. Using the hollow-sphere value 2/3MR² instead of the solid-sphere value 2/5MR² gives 3, not 5.

The idea to keep. The radius of gyration is √(I/M), and the masses cancel out. Equal k means you compare only the numerical factors in front of R².

Answer from the NTA final answer key. Solution by PrepWiser.