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Question #43663

Difficulty: 8/10 | 🟢 Free

The Question

A uniform straight rod of mass $M=3m$ and length $2R$ is in free space. A small ring of mass $m$ can slide without friction along the rod. Initially, the ring is held at a distance of $R/2$ from the center of the rod, and the entire system is rotating with an angular velocity $\omega_0$ about an axis passing through the system's center of mass and perpendicular to the rod. The ring is then released. What is the areal velocity of the ring relative to the center of the rod at the instant the ring reaches the end of the rod?

Options

A.
$\frac{19}{56} R^2 \omega_0$
B.
$\frac{1}{8} R^2 \omega_0$
C.
$\frac{5}{16} R^2 \omega_0$
D.
$\frac{7}{32} R^2 \omega_0$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Reduced Mass Effect|Kepler’s Second Law (Areal Velocity)|Non-Uniform Circular Motion|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the conservation of angular momentum to a distributed two-body system and distinguishes it from Kepler's Second Law for central forces.

A common lethal trap is assuming that the areal velocity of the ring is conserved ( $\frac{dA}{dt} = \text{constant}$ ), which is only true for a purely central force problem where the entire angular momentum is orbital ( $L = \mu r^2 \omega$ ).

Here, the rod's spin angular momentum couples with the orbital angular momentum of the reduced mass.

As the ring slides outwards, the rod exerts a tangential normal force on it, transferring angular momentum from the rod's spin to the ring's orbit.

Students who blindly apply Kepler's areal velocity law or forget the rod's intrinsic moment of inertia will incorrectly conclude the areal velocity remains $\frac{1}{8} R^2 \omega_0$ .

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Atoms - Class 12
Concept	Reduced Mass Effect
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free