JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Rotational Mechanics

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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

\frac{2}{x}

. The value of

x

is ______

Answer: 5

Solution

For a ring about the axis through its centre perpendicular to its plane,

I = m R_{1}^{2}

, so its radius of gyration is

K_{1} = R_{1}

. For a solid sphere about a diameter,

I = \frac{2}{5} m R_{2}^{2}

, so

K_{2} = \frac{2}{5} R_{2}

. Equal radii of gyration give

R_{1} = \frac{2}{5} R_{2}

, so

\frac{R _{1}}{R _{2}} = \frac{2}{5}

. Comparing with

\frac{2}{x}

, the value of

x

5

JEE Main · 2023 · 06 Apr 2023 · Shift 2 · Integer answer

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