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Question #42357

Difficulty: 8/10 | 🟢 Free

The Question

A very long uniform plank of mass $M$ lies on a smooth horizontal surface. A small block of mass $m$ is initially at rest at the geometric center of the plank. At $t=0$ , an internal mechanism causes the block to move along the longitudinal axis of the plank such that its velocity relative to the plank is $v_{rel}(t) = c t$ , where $c$ is a positive constant. Simultaneously, an external time-varying force of magnitude $F(t) = kt$ (where $k$ is a positive constant) is applied to the geometric center of the plank. The direction of this force is continuously adjusted so that it always remains perpendicular to the longitudinal axis of the plank. Find the magnitude of the angular velocity of the plank as $t \to \infty$ .

Options

A.
$\frac{k}{2Mc}$
B.
$\frac{k}{4Mc}$
C.
$\frac{k}{2(M+m)c}$
D.
0

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Intuition_Breaker

Train your brain to resist "obvious" answers that are designed to deceive your intuition.

Concepts Behind This Question

System vs Particle Displacement|Relative Velocity in 2D|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly identify the center of mass of a dynamic system and apply the torque-angular momentum relation ( $\vec{\tau}_{cm} = d\vec{L}_{cm}/dt$ ) in a non-inertial, rotating frame.

A common 'lever arm' trap is calculating the torque about the plank's geometric center, which yields zero torque and incorrectly implies the plank does not rotate (Option D).

Another trap is failing to account for the time-varying moment of inertia of the system as the block moves, or incorrectly integrating the relative velocity (Option B).

By asking for the asymptotic angular velocity, the problem elegantly forces the student to realize that both the angular momentum and the moment of inertia scale as $t^4$ , leading to a constant terminal angular velocity where the mass of the block surprisingly cancels out.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	System vs Particle Displacement
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free