Simple Harmonic Motion: PYQ Walkthrough (2)

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Question

A particle in SHM has amplitude 10 cm and period 2 s. Starting from x=+5x = +5 cm and moving in the positive direction, find the time to reach x=+10x = +10 cm. (NEET 2022 pattern.)

Solution — Step by Step

Take x(t)=Asin⁡(ωt+ϕ)x(t) = A\sin(\omega t + \phi) since we want a clean expression. With A=10A = 10 cm and ω=2π/T=π\omega = 2\pi/T = \pi rad/s:

x(t)=10sin⁡(πt+ϕ)x(t) = 10\sin(\pi t + \phi)

At t=0t = 0: x=5x = 5 cm and v>0v > 0 (moving in positive xx).

5=10sin⁡ϕ  ⟹  sin⁡ϕ=0.5  ⟹  ϕ=π/65 = 10\sin\phi \implies \sin\phi = 0.5 \implies \phi = \pi/6

(We pick π/6\pi/6 rather than 5π/65\pi/6 because velocity is positive there: cos⁡ϕ>0\cos\phi > 0.)

10=10sin⁡(πt+π/6)  ⟹  πt+π/6=π/210 = 10\sin(\pi t + \pi/6) \implies \pi t + \pi/6 = \pi/2

πt=π/3  ⟹  t=1/3 s\pi t = \pi/3 \implies t = 1/3\text{ s}

Final answer: t=13 s≈0.33 st = \dfrac{1}{3}\text{ s} \approx 0.33\text{ s}

Why This Works

The phase angle ωt+ϕ\omega t + \phi moves at constant rate ω\omega. At x=Ax = A, the phase is π/2\pi/2. We started at phase π/6\pi/6, so we need to advance by π/2−π/6=π/3\pi/2 - \pi/6 = \pi/3, taking time (π/3)/ω=1/3(\pi/3)/\omega = 1/3 s.

Phase reasoning is much faster than solving sin⁡−1\sin^{-1} equations from scratch.

Alternative Method

Use the phasor (reference circle) picture: the particle’s projection on the x-axis traces SHM, while a reference vector rotates uniformly. From phase π/6\pi/6 to π/2\pi/2 is an arc of π/3\pi/3 rad, taking T⋅(π/3)/(2π)=T/6=1/3T \cdot (\pi/3)/(2\pi) = T/6 = 1/3 s.

Common Mistake

Students take ϕ=5π/6\phi = 5\pi/6 (the other solution of sin⁡ϕ=0.5\sin\phi = 0.5). This corresponds to the particle moving in the negative direction at t=0t = 0 — opposite of what the problem states. Always check the velocity sign when choosing the phase.

For SHM, the average time from x=A/2x = A/2 to x=Ax = A is T/12T/12 if you start at maximum speed, or T/6T/6 if you start at x=A/2x = A/2. Memorise the standard fractions of TT — they appear in 3-4 questions per JEE paper.

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