Question

A solid sphere of radius R and mass M is rotating about its diameter. The moment of intertia of the solid sphere rotating about an axis at a distance \(\frac{R}{3}\) from the centre and parallel to that diameter is

• A. \(\frac{11}{15}MR^2\)
• B. \(\frac{13}{20}MR^2\)
• C. \(\frac{23}{45}MR^2\)
• D. \(\frac{47}{45}MR^2\)

Hint:

At the top, the water stays if gravity alone can supply the centripetal force.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The critical position is the top of the circle. The water does not fall if at least the weight provides the centripetal force: \(mg \le m\omega^2r\).

Step 2: Minimum condition:
At the limiting case \(mg = m\omega^2r\), so \(\omega = \sqrt{\frac gr}\). The period is
\[ T = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac rg} \]

Step 3: Meaning:
The period must be at most this value. Option (A) has g over r inside the root, which is the inverse of what we need, and (C) and (D) have wrong dimensions for a time.

Final Answer:
The required period is \(2\pi\sqrt{\frac rg}\), option (B). \[ \boxed{2\pi\sqrt{\frac{r}{g}}} \]