Question

A solid sphere of radius \(10\) cm is rotating about an axis which is at a distance \(15\) cm from its centre. The radius of gyration about this axis is \( \sqrt{n} \) cm. Find the value of \( n \).

Hint:

Radius of gyration directly reflects how mass is distributed about the axis—parallel axis theorem is essential here.

Answer 1

Correct Answer: 265

Step 1: Understanding the Concept:
The moment of inertia of a body about any axis can be found using the parallel axis theorem if the moment of inertia about a parallel axis passing through the center of mass is known. The radius of gyration \(k\) is the distance from the axis where the entire mass can be assumed to be concentrated to give the same moment of inertia.
Step 2: Key Formula or Approach: 1. Moment of inertia of a solid sphere about center: \(I_{cm} = \frac{2}{5}MR^2\).
2. Parallel axis theorem: \(I = I_{cm} + Md^2\).
3. Radius of gyration: \(I = Mk^2\).
Step 3: Detailed Explanation:
Given: \(R = 10\) cm, \(d = 15\) cm.
Using Parallel Axis Theorem: \[ I = \frac{2}{5} MR^2 + Md^2 \] \[ Mk^2 = M \left( \frac{2}{5} R^2 + d^2 \right) \] \[ k^2 = \frac{2}{5} (10)^2 + (15)^2 \] \[ k^2 = \frac{2}{5} (100) + 225 \] \[ k^2 = 40 + 225 = 265 \] Given the radius of gyration \(k = \sqrt{n}\), then: \[ (\sqrt{n})^2 = 265 \implies n = 265 \] Step 4: Final Answer:
The value of \(n\) is 265.