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 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. Take Z axis to be perpendicular to the plane of the paper.

• A. The minimum moment of inertia (about Z-axis), $I_{zz}$ of the system is $5ml^2$.
• B. The magnitude of tension in the rod in subsequent motion is $\frac{2mv_0^2}{9l}$
• C. The ratio of moment of inertia about Z-axis at points A and B, $\frac{I_{zz}^A}{I_{zz}^B}=2$

Answer 1

Answer: (C)

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

Answer 2

Let's analyze each statement:

Statement A: Minimum Moment of Inertia

The minimum moment of inertia occurs about the center of mass (COM). The COM is located at a distance of $2l$ from mass $m$ and $l$ from mass $2m$. Therefore, the moment of inertia about the COM ($I_{zz}$) is:

$I_{zz} = m(2l)^2 + 2m(l)^2 = 4ml^2 + 2ml^2 = 6ml^2$

Since the statement claims the minimum moment of inertia is $5ml^2$, it is incorrect.

Statement C: Ratio of Moments of Inertia

The moment of inertia about point A ($I_{zz}^A$) is:

$I_{zz}^A = m(0)^2 + 2m(3l)^2 = 18ml^2$

The moment of inertia about point B ($I_{zz}^B$) is:

$I_{zz}^B = m(3l)^2 + 2m(0)^2 = 9ml^2$

The ratio is:

$\frac{I_{zz}^A}{I_{zz}^B} = \frac{18ml^2}{9ml^2} = 2$

Therefore, statement C is correct.

Statement B: Tension in the Rod

This is the most complex statement and requires careful consideration of the initial conditions and the constraint imposed by the rigid rod.

Initial Conditions and Constraints

The rod is rigid, and mass $2m$ is suddenly given a velocity $v_0$ towards the right. Let's assume the rod is initially aligned along the y-axis, and the velocity $v_0$ is along the x-axis. Let $v_A$ and $v_B$ be the velocities of mass $m$ and $2m$ respectively. Since the rod is rigid, the components of the velocities along the rod must be equal.

Let's define $v_A = (v_{Ax}, 0)$ and $v_B = (v_0, 0)$.

Conservation of Linear Momentum

Since the system is initially at rest, the total linear momentum is conserved in the x-direction:

$m v_{Ax} + 2m v_0 = 0 \implies v_{Ax} = -2v_0$

Angular Velocity

The angular velocity $\omega$ of the rod can be found using the relationship between the velocities and the length of the rod:

$\omega = \frac{v_0}{l}$

Tension Calculation

The tension in the rod provides the centripetal force required for the circular motion of the masses around the center of mass. The tension can be calculated using either mass:

$T = m \omega^2 (2l) = m (\frac{v_0}{l})^2 (2l) = \frac{2mv_0^2}{l}$

Or

$T = 2m \omega^2 (l) = 2m (\frac{v_0}{l})^2 (l) = \frac{2mv_0^2}{l}$

The tension is $\frac{2mv_0^2}{l}$, not $\frac{2mv_0^2}{9l}$. Therefore, statement B is incorrect.

Conclusion

Only statement C is correct.