Question

The radius of gyration of a solid sphere of radius 'R' and mass 'M' about its diameter is \(K_d\) and that about a tangent of a solid sphere is \(K_t\). The ratio of \(K_d\) to \(K_t\) is

• A. \((\frac{7}{5})^{1/2}\)
• B. \((\frac{7}{2})^{1/2}\)
• C. \((\frac{2}{7})^{1/2}\)
• D. \((\frac{2}{5})^{1/2}\)

Hint:

Use I = (2/5)MR^2 about a diameter and the parallel axis theorem for the tangent.

Answer 1

The Correct Option is C

Step 1: Moments of Inertia:
Solid sphere: $I_{cm}=\tfrac25MR^2$. A tangent is at distance $R$ from the centre.

Step 2: Compare:
$\dfrac{I_d}{I_t}=\dfrac{2/5}{7/5}=\dfrac27$. Since $I=MK^2$ with the same mass, $\dfrac{K_d^2}{K_t^2}=\dfrac27$.

Step 3: Answer:
$\dfrac{K_d}{K_t}=\sqrt{2/7}$. Option (C).

Final Answer:
Option (C). \[ \boxed{\text{(C) } \left(\frac{2}{7}\right)^{1/2}} \]