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Question #42027

Difficulty: 8/10 | 🟢 Free

The Question

A solid cylinder of mass $m$ and radius $R$ is projected with an initial translational velocity $v_0$ and zero angular velocity on a rough horizontal surface with a coefficient of kinetic friction $\mu_k$ . The cylinder experiences slipping before transitioning to pure rolling. Let $A_v$ be the area under the velocity-time ( $v-t$ ) graph and $A_\omega$ be the area under the angular velocity-time ( $\omega-t$ ) graph from $t=0$ until the exact moment pure rolling commences. Which of the following expressions correctly represents the total mechanical energy dissipated due to friction during this process?

Options

A.
$\mu_k m g (A_v - R A_\omega)$
B.
$\mu_k m g A_v$
C.
$\mu_k m g (A_v + R A_\omega)$
D.
$\frac{1}{2} \mu_k m g (A_v - R A_\omega)$

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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

Velocity-Time Slope-Area Duality|Pure Rolling Condition|Vector Cross-Products

Why This Question Is Tricky

Common Mistakes

This question tests the student's understanding of energy dissipation during the transition from slipping to pure rolling.

A common trap is assuming that the energy dissipated is simply the translational work done by friction ( $\mu_k m g A_v$ ), ignoring the positive work friction does on the rotational degree of freedom.

By forcing the evaluation of the Work-Energy Theorem through the geometric areas of the $v-t$ and $\omega-t$ graphs, it exposes the misconception that the 'no-slip' condition ( $v = \omega R$ ) applies during the kinetic phase.

The true dissipation depends strictly on the relative slip distance, represented by the difference between the translational displacement and the rotational arc length.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Motion in a Straight Line - Class 11 Part 1
Concept	Velocity-Time Slope-Area Duality
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free