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Question #42872

Difficulty: 8/10 | 🟢 Free

The Question

A small block of mass $m = 1 \text{ kg}$ is placed on a long horizontal plank of mass $M = 3 \text{ kg}$ . The plank is attached to a fixed wall by a horizontal ideal spring of force constant $k = 400 \text{ N/m}$ . The coefficient of static friction between the block and the plank is $\mu_s = \frac{\sqrt{3}}{2}$ . The plank rests on a frictionless horizontal floor. Let $x=0$ be the equilibrium position of the plank. At $t = 0$ , the plank is at $x = -0.05 \text{ m}$ and is moving with a velocity $v = -\frac{\sqrt{3}}{2} \text{ m/s}$ . Assuming the block does not slip on the plank initially, find the exact time $t$ at which the block first begins to slip. (Take $g = 10 \text{ m/s}^2$ )

Options

A.
$\frac{\pi}{60} \text{ s}$
B.
$\frac{\pi}{20} \text{ s}$
C.
$\frac{\pi}{30} \text{ s}$
D.
$\frac{\pi}{12} \text{ s}$

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Phase Constant (Epoch)|Simple Pendulum Small Angle Approximation|Static Friction Threshold

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to synthesize SHM kinematics with non-constant restorative forces and static friction thresholds.

The primary trap lies in the phase constant determination: students often ignore the negative sign of the initial velocity, incorrectly assuming the block is moving towards the mean position.

This yields a false phase and leads directly to the distractor $t = \pi/20$ s.

Additionally, students may misidentify the slipping condition by equating the spring force directly to the friction on the block ( $kx = \mu_s mg$ ) instead of using the correct pseudo-force/acceleration relation ( $m\omega^2 x = \mu_s mg$ ), or they might assume slipping only occurs exactly at the extreme amplitude, leading to the distractor $t = \pi/30$ s.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Phase Constant (Epoch)
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free