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Question #42611

Difficulty: 8/10 | 🟢 Free

The Question

A block of mass $m = 1 \text{ kg}$ is attached to a spring of force constant $k = 100 \text{ N m}^{-1}$ and undergoes weakly damped oscillations. The damping force is given by $F_d = -b(t)v$ , where $v$ is the instantaneous velocity and the damping coefficient varies with time as $b(t) = b_0(1 + \alpha t)$ . Given $b_0 = 0.2 \text{ kg s}^{-1}$ and $\alpha = 0.1 \text{ s}^{-1}$ . Assuming the fractional loss of energy per cycle is very small, at what time $t$ will the total mechanical energy of the oscillator decrease to $1/e$ of its initial value?

Options

A.
$10(\sqrt{2} - 1) \text{ s}$
B.
$5(\sqrt{3} - 1) \text{ s}$
C.
$5 \text{ s}$
D.
$10 \text{ s}$

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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

Energy Partitioning|Damped Oscillation Equation|Integration

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to derive the energy decay of a damped oscillator from first principles rather than relying on the memorized formula $E(t) = E_0 e^{-bt/m}$ .

The trap lies in the time-varying damping coefficient $b(t)$ .

Students who rote-memorize the standard solution will either incorrectly use $b_0$ (yielding 5 s) or erroneously substitute $b(t)$ directly into the exponent as $E_0 e^{-b(t)t/m}$ (yielding $5(\sqrt{3}-1)$ s).

The correct approach requires setting up the non-homogeneous differential energy balance $\frac{dE}{dt} = -b(t)\langle v^2 \rangle$ and integrating the time-dependent dissipation function.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Energy Partitioning
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free