Question

A thin wire of length \( I \) and uniform linear mass density \( \rho \) is bent in the form of a square. Its moment of inertia about an axis along one of its edge is (A) \( \frac{2 \rho}{3}\left(\frac{I}{4}\right)^{3} \) (B) \( \frac{3 \rho}{2}\left(\frac{I}{4}\right)^{3} \) (C) \( \frac{5 \rho}{3}\left(\frac{I}{4}\right)^{3} \) (D) \( \frac{1 \rho}{3}\left(\frac{I}{4}\right)^{3} \)

Answer 1

Correct option is (C) \(\frac {5\rho}3 \left(\frac { l}{4} \right)^3\)

\(I_1 = I_2 = (m). \left(\frac l4\right)^2\)

\(= \frac {(\rho. \frac l4). (\frac l4)^2}3\)

\(= \frac{\rho.l^3}{3 \times 4^3}\)

\(= \frac{\rho l^3}{3 \times 64}\)

\(I_3 = \int r^2 .dm\)

For each dm mass on all wire

r = same = \(\frac l4\)

\(I_3 = \left[\frac l4\right]^2 \int dm\)

\(I_3 = \frac {l^2}{16} \times (8m)\)

\(= \frac {l^2}{16} \times \left[\rho . \frac l4 \right]\)

\(I_3 = \frac {\rho l^3}{64}\)

\(I = I_1 + I_2+ I_3\)

\(= \frac {\rho l^3}{3 \times 64} + \frac {\rho l63}{3 \times 64} + \frac {\rho l^3}{64}\)

\(= \frac {\rho l ^3}{64}\left[ \frac 23 + 1\right]\)

\(= \frac 53 \left(\frac {\rho l^3}{64} \right)\)

\(= \frac {5\rho}3 \left(\frac { l}{4} \right)^3\)