Question

Consider a thin uniform square sheet made of a rigid material. If its side is '$a$', mass $m$ and moment of inertia $I$ about one of its diagonals, then :

• A. $I > \frac{ma^{2}}{12}$
• B. $\frac{ma^{2}}{24} < I < \frac{ma^{2}}{12}$
• C. $ I = \frac{ma^{2}}{12}$
• D. $ I = \frac{ma^{2}}{24}$

Answer 1

The Correct Option is B

To determine the moment of inertia \( I \) of a thin uniform square sheet about one of its diagonals, we need to use the concept of moment of inertia for a square plate about its center and relate it with its diagonal.

Step 1: Determine the formula for moment of inertia of a square sheet about an axis through its center

The moment of inertia \( I_c \) of a square sheet about an axis passing through its center and perpendicular to the plane is given by:

I_c = \frac{1}{6}ma^2

where \( m \) is the mass of the square sheet, and \( a \) is the side of the square.

Step 2: Use the perpendicular axis theorem

The perpendicular axis theorem states that for a planar object, the moment of inertia about an axis perpendicular to the plane (here the center) is the sum of the moments of inertia about two perpendicular axes in the plane. If we let \( I_x \) and \( I_y \) be the moments of inertia about the two in-plane perpendicular axes passing through the center of the sheet, then:

I_c = I_x + I_y

For a square plate, \( I_x = I_y \), and thus:

I_x = I_y = \frac{1}{12}ma^2

Step 3: Use the parallel axis theorem to find the moment of inertia about the diagonal

The diagonal of the square sheet can be considered as another axis in the plane. For the moment of inertia about the diagonal, we calculate considering the two components:

Using geometry and symmetry, the moment of inertia about the diagonal \( I_d \) can be calculated as:

I_d = \frac{1}{2}(I_x + I_y) = \frac{1}{2}\left( \frac{1}{12}ma^2 + \frac{1}{12}ma^2 \right) = \frac{1}{24}ma^2

Upon analyzing this, we can state:

\frac{ma^2}{24} < I < \frac{ma^2}{12}

This option fits the analysis from geometrical and physical interpretation.