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Question #42619

Difficulty: 8/10 | 🟢 Free

The Question

A particle of mass $m = 0.1 \text{ kg}$ is subjected to a net restoring force such that its displacement $x$ (in meters) from the origin is the superposition of two collinear simple harmonic motions given by $x_1(t) = 2 \sin(10 t)$ and $x_2(t) = 4 \sin(10 t + \frac{\pi}{3})$ , where $t$ is in seconds. What is the time-averaged potential energy of the particle over a large number of oscillations?

Options

A.
$70 \text{ J}$
B.
$50 \text{ J}$
C.
$90 \text{ J}$
D.
$140 \text{ J}$

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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

Mean vs. RMS Values|Phase Constant (Epoch)|Superposition of Potentials

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the principle of superposition to non-linear quantities like potential energy.

A common trap is assuming that the time-averaged potential energy of a superimposed state is simply the algebraic sum of the individual time-averaged potential energies ( $\langle U \rangle = \langle U_1 \rangle + \langle U_2 \rangle = 50 \text{ J}$ ).

This incorrectly ignores the cross-product term $2A_1 A_2 \cos(\Delta \phi)$ that arises when squaring the net displacement $x(t) = x_1(t) + x_2(t)$ .

By forcing the calculation of the time-averaged energy, students must recognize that while displacements superimpose linearly, energy depends on the RMS value of the net displacement, requiring proper phasor addition before averaging.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Oscillations - Class 11 Part 2
Concept	Mean vs. RMS Values
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free