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Question #42490

Difficulty: 8/10 | 🟢 Free

The Question

A block of mass $m = 1 \text{ kg}$ is attached to a spring of force constant $k = 4 \text{ N/m}$ and is subjected to a damping force $F_d = -bv$ , where $b = 2 \text{ kg/s}$ and $v$ is the velocity of the block. The block is initially at the equilibrium position $x = 0$ and is given an initial velocity $v_0 = 2\sqrt{3} \text{ m/s}$ in the positive $x$ -direction. What is the maximum potential energy stored in the spring during the first oscillation?

Options

A.
$6 e^{-\frac{2\pi}{3\sqrt{3}}} \text{ J}$
B.
$6 \text{ J}$
C.
$8 \text{ J}$
D.
$8 e^{-\frac{\pi}{\sqrt{3}}} \text{ J}$

Check Your Answer

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

Mechanical Energy Conservation|Damped Oscillation Equation|Dimensional Consistency and Transcendental Functions

Why This Question Is Tricky

Common Mistakes

This question tests the 'Lethal Collision' trap where students incorrectly apply the conservative work-energy theorem ( $E = \frac{1}{2}mv^2 = \frac{1}{2}kA^2$ ) to a damped harmonic oscillator.

By ignoring the work done by the non-conservative damping force ( $F = -bv$ ) during the transit from equilibrium to the first turning point, they erroneously select 6 J.

Furthermore, assuming the peak of the sine function ( $\omega_d t = \pi/2$ ) coincides with the turning point ignores the phase-lag introduced by the exponential decay term, trapping students who fail to differentiate the full transcendental function to find the true maximum displacement.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Work, Energy, and Power - Class 11 Part 1
Concept	Mechanical Energy Conservation
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free