Oscillations and SHM: Tricky Questions Solved (1)

easy • 2 min read
Tags Oscillations

Question

A particle executes SHM with amplitude A=5 cmA = 5\text{ cm} and time period T=2 sT = 2\text{ s}. Find (a) the maximum velocity, (b) the maximum acceleration, and (c) the velocity when the displacement is 3 cm3\text{ cm} from the mean position.

Solution — Step by Step

For SHM, ω=2πT=2π2=π rad/s\omega = \dfrac{2\pi}{T} = \dfrac{2\pi}{2} = \pi\text{ rad/s}. Every SHM formula has ω\omega in it, so getting this right first saves us from re-doing later steps.

vmax⁡=Aω=0.05×π=0.05π m/s≈0.157 m/sv_{\max} = A\omega = 0.05 \times \pi = 0.05\pi\text{ m/s} \approx 0.157\text{ m/s}.

This is the speed at the mean position, where all energy is kinetic.

amax⁡=Aω2=0.05×π2≈0.493 m/s2a_{\max} = A\omega^2 = 0.05 \times \pi^2 \approx 0.493\text{ m/s}^2.

This occurs at the extremes, where velocity is zero and the restoring force is largest.

Use v=ωA2−x2v = \omega\sqrt{A^2 - x^2}:

v=π(0.05)2−(0.03)2=π0.0016=0.04π m/s≈0.126 m/sv = \pi\sqrt{(0.05)^2 - (0.03)^2} = \pi\sqrt{0.0016} = 0.04\pi\text{ m/s} \approx 0.126\text{ m/s}

vmax⁡≈0.157 m/sv_{\max} \approx 0.157\text{ m/s}, amax⁡≈0.493 m/s2a_{\max} \approx 0.493\text{ m/s}^2, v(x=3 cm)≈0.126 m/sv(x=3\text{ cm}) \approx 0.126\text{ m/s}.

Why This Works

SHM is the projection of uniform circular motion on a diameter. So ω\omega from circular motion translates directly: position is Acos⁡(ωt)A\cos(\omega t), velocity peaks at AωA\omega, acceleration peaks at Aω2A\omega^2. The phase relation v=ωA2−x2v = \omega\sqrt{A^2 - x^2} comes from energy conservation: KE + PE = constant.

Alternative Method

Energy approach: total energy E=12mω2A2E = \tfrac{1}{2}m\omega^2 A^2. At xx, KE =E−12mω2x2= E - \tfrac{1}{2}m\omega^2 x^2, so v=ωA2−x2v = \omega\sqrt{A^2 - x^2}. Same formula, derived differently — useful when the question gives you energies instead of ω\omega.

Common Mistake

Mixing units. Amplitude is given in cm but ω\omega is in rad/s — if you forget to convert AA to metres before multiplying, you will be off by a factor of 100. Always convert to SI first, especially in NEET MCQs where one of the wrong options uses cm.

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