Question

Four rods of equal length $l$ and mass $m$ each from a square as shown in the figure. Moment of Inertia about three axes $1,2\text{and}3$ are say $I_{1}, \, I_{2}$ and $I_{3}.$ Then, match the following

	Table-1		Table-2
$(A)$	$I_{1}$	$(P)$	$\frac{4}{3}m\ell ^{2}$
$(B)$	$I_{2}$	$(Q)$	$\frac{2}{3}m\ell ^{2}$
$(C)$	$I_{3}$	$(R)$	$\frac{1}{2}m\ell ^{2}$
		$(S)$	$\text{None}$

• A. $\text{(A)-Q, (B)-S, (C)-Q}$
• B. $\text{(A)-S, (B)-Q, (C)-Q}$
• C. $\text{(A)-Q, (B)-Q, (C)-S}$
• D. $\text{(A)-R, (B)-Q, (C)-S}$

Answer 1

Answer: (A)

$I_{1}=2\left(\frac{m \ell ^{2}}{12}\right)+2\left(\right.m\left.\right)\left(\frac{\ell }{2}\right)^{2}=\frac{2}{3}m\ell ^{2}$
$I_{2}=0+2\left(\frac{\left(m\ell \right)^{2}}{3}\right)+\left(m\ell \right)^{2}=\left(\frac{5}{3}\right)\left(m\ell \right)^{2}$
$I_{3}=4\left[\frac{m \ell ^{2}}{3} sin^{2} 45^{ο}\right]=\frac{2}{3}m\ell ^{2}=I_{1}$
NOTE : $I_{1}=I_{3}\left(\text{think why ?}\right)$