Question

Consider a uniform square plate of side \({}'a'\) and mass \({}'m'\). The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is

• A. \(\frac{5ma^{2}}{6}\)
• B. \(\frac{2ma^{2}}{3}\)
• C. \(\frac{7ma^{2}}{12}\)
• D. \(\frac{ma^{2}}{12}\)

Hint:

Find the moment of inertia about the central perpendicular axis and shift it to the corner with the parallel axis theorem.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept
For a square plate of side \(a\) and mass \(m\), the moment of inertia about the perpendicular axis through the centre is \(I_{c}=\frac{ma^{2}}{6}\).

Step 2: Calculation
The corner is half a diagonal from the centre: \(d=\frac{a\sqrt{2}}{2}\), so \(d^{2}=\frac{a^{2}}{2}\). By the parallel axis theorem \(I=I_{c}+md^{2}=\frac{ma^{2}}{6}+\frac{ma^{2}}{2}=\frac{2ma^{2}}{3}\).

Step 3: Compare the options
The value \(\frac{2ma^{2}}{3}\) matches the second option.

Final Answer:
The moment of inertia about the corner is \(\frac{2ma^{2}}{3}\). \[ \boxed{\text{(B) }\dfrac{2ma^{2}}{3}} \]