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MCQEasyJEE 2023Simple Harmonic Motion (SHM)

JEE Physics 2023 Question with Solution

A particle executes S.H.M. of amplitude AA along the x-axis. At t=0t = 0, the position of the particle is x=−A1 ⁣\/ ⁣2x = -A^{1}\!\/\!2, and it moves along the positive x-axis. The displacement of the particle in time tt is given as x=Asin⁡(ωt+δ)x = A \sin(\omega t + \delta). The value of δ\delta will be:

  • A

    π4\frac{\pi}{4}

  • B

    π2\frac{\pi}{2}

  • C

    π3\frac{\pi}{3}

  • D

    π6\frac{\pi}{6}

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: A particle executes SHM with amplitude AA and displacement

x=Asin⁡(ωt+δ)x = A \sin(\omega t + \delta)

At t=0t = 0, the particle is at the given initial position and moves along the positive x-axis.

Find: The value of δ\delta.

From the extracted solution working:

cos⁡θ=A2A=12\cos \theta = \frac{A}{2A} = \frac{1}{2}

So,

θ=π3\theta = \frac{\pi}{3}

The phase difference is then written as

δ=π2−π3\delta = \frac{\pi}{2} - \frac{\pi}{3}

Hence,

δ=π6\delta = \frac{\pi}{6}

Therefore, the working in the solution gives δ=π6\delta = \frac{\pi}{6}. This corresponds to option D.

Discrepancy noted: the solution says "The Correct Option is B", but the actual step-by-step working concludes δ=π6\delta = \frac{\pi}{6}, which matches option D. By the worked solution, the defensible answer is D.

Common mistakes

  • Using the option label shown in the solution without checking the actual working. This is wrong because the header says B while the derivation concludes δ=π6\delta = \frac{\pi}{6}. Always trust the worked steps over a mismatched label.

  • Ignoring the direction of motion at t=0t = 0. In SHM, the sign of velocity helps determine the correct phase constant. The initial position alone may not fix the correct quadrant for the phase.

  • Confusing the phase in a sine form with that in a cosine form. This is wrong because converting between sin⁡\sin and cos⁡\cos introduces a phase shift. Keep the given form x=Asin⁡(ωt+δ)x = A\sin(\omega t + \delta) consistent throughout.

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