Question

A light rigid rod AB of length $3l$ has a point mass $m$ at end A and a point mass $2m$ at end B. It is kept on a smooth horizontal surface. Point C is the center of mass of the system. Initially, the system is at rest. The mass $2m$ is suddenly given a velocity $v_0$ towards right. Then:
[Take Z axis to be perpendicular to the plane of the paper.]

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• A. The minimum moment of inertia (about Z-axis), $I_{zz}$ of the system is $5m{l}^2$.
• B. The magnitude of tension in the rod in the subsequent motion is $\frac{2m{v}_0^2}{9l}$.
• C. The ratio of moment of inertia about Z - axis at points A and B, $\frac{(I_{zz}{)}_A}{(I_{zz}{)}_B}=2$
• D. Point C remains stationary during subsequent motion.

Answer 1

Answer: (B), (C)

The ratio of moment of inertia about Z - axis at points A and B, $\frac{(I_{zz}{)}_A}{(I_{zz}{)}_B}=2$


Position of COM of the system, $r_C=\frac{2m\left(0\right)+m\left(3l\right)}{3m}$
$\Rightarrow r_C=l$ (from $2m$ mass)

The moment of inertia about Z axis will be minimum about the centre of mass.

MOI about $C$, $(I_{zz}{)}_{min}=2m(l{)}^2+m(2l{)}^2=6m{l}^2$


From conservation of linear momentum,
$2m\times v_0=(m+2m)v_C\Rightarrow v_C\ne 0$

& from conservation of angular momentum about $C$,
$2m{v}_{0}l=6m{l}^2\omega$
$\therefore \omega =\frac{v_0}{3l}$

Tension in the rod provides the centripetal force for rotation.
$\Rightarrow T=2m{\omega }^{2}l=\frac{2m{v}_0^2}{9l}$

Also, $\frac{(I_{zz}{)}_A}{(I_{zz}{)}_B}=\frac{2m(3l{)}^2}{m(3l{)}^2}=2$

Therefore, options (b) and (c) are correct.