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Question #42636

Difficulty: 8/10 | 🟢 Free

The Question

A damped harmonic oscillator of mass $m = 1 \text{ kg}$ is subjected to a restoring force $F_k = -100x \text{ N}$ and a damping force $F_d = -2v \text{ N}$ , where $x$ is the displacement in meters and $v$ is the velocity in m/s. The oscillator is driven by an external force $F(t) = F_0 (\cos(10t) + \cos(8t)) \text{ N}$ , where $t$ is in seconds. In the steady state, the displacement of the particle is given by the superposition $x(t) = A_1 \cos(10t - \phi_1) + A_2 \cos(8t - \phi_2)$ . What is the value of $\tan(|\phi_1 - \phi_2|)$ ?

Options

A.
$\frac{9}{4}$
B.
$0$
C.
$\frac{4}{9}$
D.
$1$

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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

Superposition of SHMs|Driven Oscillations and Resonance|Approximation (Binomial)

Why This Question Is Tricky

Common Mistakes

This question targets the misconception that the phase of a driven harmonic oscillator in steady state is identical to the driving force or constant across different frequencies.

Students often assume $\phi_1 = \phi_2 = 0$ (leading to an incorrect phase difference of 0) or incorrectly apply the resonance condition to all frequencies.

By forcing the calculation of the frequency-dependent phase lag $\phi = \tan^{-1}[b\omega / (k - m\omega^2)]$ , it tests the understanding of dynamics-based superposition where the interference envelope is modulated by the distinct phase lags of each frequency component.

Quick Facts About This Question

Question Information

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