Physics Q2 – NEET-UG 2023

Home Physics Rotational MotionQuestion #442

The ratio of radius of gyration of a solid sphere of mass

M

and radius

R

about its own axis to the radius of gyration of the thin hollow sphere of same mass and radius about its axis is :

Clue 1 of 3: Radius of Gyration Concept

The radius of gyration is the effective distance of the mass from the axis of rotation for calculating the moment of inertia.

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Ninja Strategy•Qualitative Comparison

Recognize that a solid sphere has its mass more concentrated towards the center than a hollow sphere, so its moment of inertia and radius of gyration must be smaller. This eliminates any ratio greater than 1.

🧠 Full Solution Path

Step 1: Formulate the Squares of Radii of Gyration✦ Active

The radius of gyration, $k$ , is defined by the relation $I = M k^{2}$ , which implies $k^{2} = I / M$ . For a solid sphere, the moment of inertia is $I_{s o l i d} = \frac{2}{5} M R^{2}$ , so its squared radius of gyration is $k_{s o l i d}^{2} = \frac{2}{5} R^{2}$ . For a thin hollow sphere, the moment of inertia is $I_{h o l l o w} = \frac{2}{3} M R^{2}$ , so its squared radius of gyration is $k_{h o l l o w}^{2} = \frac{2}{3} R^{2}$ .

Step 2: Calculate the Ratio of the Squares○ Expand

The ratio of the squares of the radii of gyration is:

\frac{k_{s o l i d}^{2}}{k_{h o l l o w}^{2}} = \frac{\frac{2}{5} R^{2}}{\frac{2}{3} R^{2}} = \frac{2}{5} \times \frac{3}{2} = \frac{3}{5}

💡 Teacher's Secret Hint

Note: This question is known to be ambiguous. The literal ratio of radii is $\sqrt{3} : \sqrt{5}$ . The options match the ratio of the squares of the radii, $k_{s o l i d}^{2} : k_{h o l l o w}^{2}$ , which is the same as the ratio of the moments of inertia, $I_{s o l i d} : I_{h o l l o w}$ .

Step 3: Interpret the Result○ Expand

The calculated ratio is 3:5. Although the question asks for the ratio of the radii of gyration ( $k_{s o l i d} : k_{h o l l o w}$ ), the options correspond to the ratio of their squares ( $k_{s o l i d}^{2} : k_{h o l l o w}^{2}$ ). This is a common ambiguity in such problems, and 3:5 is the intended answer.

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Physics Question 2 – NEET-UG 2023

Clue 1 of 3: Radius of Gyration Concept

🧠 Full Solution Path