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Question #42387

Difficulty: 8/10 | 🟢 Free

The Question

A uniform solid cube of mass $m$ and side length $a$ is placed on a rough inclined plane of angle $\theta$ . The coefficient of kinetic friction between the cube and the plane is $\mu$ . By means of external forces, the cube is manipulated such that its leading lower edge slides down the incline at a constant velocity $v_0$ , while the cube simultaneously topples forward about this edge at a constant angular velocity $\omega_0$ . Assume $\omega_0$ is small enough that the cube never loses contact with the incline. What is the magnitude of the work done by the friction force on the cube during the time it rotates from its initial flat position to an angle of $\pi/4$ ?

Options

A.
$\mu m \frac{v_0}{\omega_0} \left( \frac{\pi}{4} g \cos \theta - \frac{1}{2} a \omega_0^2 \right)$
B.
$\frac{\pi}{4} \mu m g \cos \theta \frac{v_0}{\omega_0}$
C.
$\mu m \frac{v_0}{\omega_0} \left( \frac{\pi}{4} g \cos \theta + \frac{1}{2} a \omega_0^2 \right)$
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

Work-Energy Theorem for Rotation|Toppling vs Sliding|Small Angle Approximation

Why This Question Is Tricky

Common Mistakes

This question targets the 'Lethal Collision' trap in rigid body dynamics where students incorrectly assume that friction does zero work during toppling, or they fail to account for the dynamic normal force.

When a body simultaneously translates and rotates, the work done by friction must be integrated over the displacement of the contact point ( $W = \int \vec{f} \cdot d\vec{x}_{contact}$ ).

Furthermore, the normal force is not simply $mg \cos \theta$ ; the center of mass undergoes centripetal acceleration about the pivot, which dynamically alters the normal force and consequently the frictional work.

Students who ignore the CM acceleration will incorrectly choose the constant normal force option, while those who assume pure rotation will incorrectly choose zero.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	System of Particles and Rotational Motion - Class 11 Part 1
Concept	Work-Energy Theorem for Rotation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free