Question

A solid sphere and a ring have equal mass and equal radius of gyration. If sphere is rotating about its diameter and ring about an axis passing through centre and perpendicular to its plane, then the ratio of radius of sphere to that of ring is \( \sqrt{\frac{x}{2}} \) then the value of ' x ' is

• A. $2$
• B. $3$
• C. $5$
• D. $7$

Hint:

Radius of gyration: - Depends on mass distribution - $k = \sqrt{I/M}$

Answer 1

The Correct Option is C

Concept: Radius of gyration: \[ k = \sqrt{\frac{I}{M}} \] For different bodies:
• Solid sphere about diameter: \[ I_s = \frac{2}{5}MR_s^2 \Rightarrow k_s = \sqrt{\frac{2}{5}}\,R_s \]
• Ring about centre: \[ I_r = MR_r^2 \Rightarrow k_r = R_r \]

Step 1: Given condition.
\[ k_s = k_r \]

Step 2: Substitute expressions.
\[ \sqrt{\frac{2}{5}}\,R_s = R_r \]

Step 3: Find ratio.
\[ \frac{R_s}{R_r} = \sqrt{\frac{5}{2}} \]

Step 4: Compare with given form.
\[ \frac{R_s}{R_r} = \sqrt{\frac{x}{2}} \Rightarrow x = 5 \] Answer: $5$