Question

A system of five solid spheres each of mass '\(m\)' and radius '\(r\)' are rotating about an axis AA' as shown in figure. Hence the moment of inertia of the system about the axis of rotation AA' is

• A. \(6 mr^2\)
• B. \(5 mr^2\)
• C. \(4 mr^2\)
• D. \(3 mr^2\)

Hint:

Spheres on the axis contribute \(\frac25mr^2\) each; the two touching the axis also need the parallel axis term.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
In the figure, three spheres have their centres on the axis AA' (top, one low, and the bottom one). The two spheres at the sides each touch the axis, so their centres are at distance \(r\) from it.

Step 2: Key Formula or Approach
Solid sphere about a diameter: \(\frac25mr^2\). For the side spheres use the parallel axis theorem: \(I=\frac25mr^2+mr^2=\frac75mr^2\).

Step 3: Detailed Explanation
Three spheres on the axis: \(3\times\frac25mr^2=\frac65mr^2\).
Two spheres at the side: \(2\times\frac75mr^2=\frac{14}{5}mr^2\).
\[ I=\frac65mr^2+\frac{14}5mr^2=\frac{20}{5}mr^2=4mr^2 \]

Final Answer:
The moment of inertia is \(4mr^2\), option (C). \[ \boxed{4mr^2\ \text{(C)}} \]