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)

$I_1=2\left(\frac{m{\ell }^2}{12}\right)+2(m){\left(\frac{\ell }{2}\right)}^2=\frac{2}{3}m{\ell }^2$
$I_2=0+2\left(\frac{{\left(m\ell \right)}^2}{3}\right)+{\left(m\ell \right)}^2=\left(\frac{5}{3}\right){\left(m\ell \right)}^2$
$I_3=4\left[\frac{m{\ell }^2}{3}si{n}^{2}45^o\right]=\frac{2}{3}m{\ell }^2=I_1$
NOTE : $I_1=I_3\left(\text{think why ?}\right)$