Question

Four point masses m, 2m, 3m and 4m are kept at the corners A, B, C and D respectively of a square ABCD of side '\(b\)'. The moment of inertia of the system about an axis perpendicular to the plane of the square and passing through the point D is

• A. \(12 mb^2\)
• B. \(10 mb^2\)
• C. \(8 mb^2\)
• D. \(5 mb^2\)

Hint:

Add \(mr^2\) for each mass, with \(r\) measured from the axis through D.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
For point masses, \(I=\sum m_ir_i^2\), where \(r_i\) is the perpendicular distance from the axis. The axis is perpendicular to the square through corner D.

Step 2: Key Formula or Approach
Distances from D: to A and C it is \(b\) (adjacent corners), to B it is the diagonal \(b\sqrt2\), to D it is 0.

Step 3: Detailed Explanation
A (mass \(m\)): \(m b^2\).
B (mass \(2m\)): \(2m(b\sqrt2)^2=4mb^2\).
C (mass \(3m\)): \(3mb^2\).
D (mass \(4m\)): \(0\).
\[ I=mb^2+4mb^2+3mb^2=8mb^2 \]

Final Answer:
The moment of inertia is \(8mb^2\), option (C). \[ \boxed{8mb^2\ \text{(C)}} \]