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: Diameter:
\(I_d=\dfrac25MR^2\). Since \(I=MK^2\), \(K_d^2=\dfrac25R^2\).

Step 2: Tangent (Parallel Axis Theorem):
\(I_t=I_d+MR^2=\dfrac25MR^2+MR^2=\dfrac75MR^2\). So \(K_t^2=\dfrac75R^2\).

Step 3: Ratio:
\[ \frac{K_d}{K_t}=\sqrt{\frac{2/5}{7/5}}=\sqrt{\frac27} \]

Step 4: Check the Options:
Options (A) and (B) give a ratio greater than 1, but \(K_t>K_d\) because the tangent is farther from the centre. Option (D) \(\sqrt{2/5}\) is just \(K_d/R\). So (C) is correct.

Final Answer:
The ratio is \((2/7)^{1/2}\), option (C). \[ \boxed{\text{(C) } \left(\frac{2}{7}\right)^{1/2}} \]