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⚛️ NEET Physics❓ Question

Periodic vs oscillatory vs SHM: which of these functions is actually simple harmonic?

Sara Fernandes11th Sample

1 Sept 2026 · 1 min read

Between football practice and homework I just did a worksheet on Oscillations and got most of the 'which of these represent SHM' questions wrong, so I think I'm missing something simple.

The ones I got confused on:

  1. x=sin⁡ωt+cos⁡ωtx = \sin\omega t + \cos\omega t
  2. x=sin⁡ωt+sin⁡2ωtx = \sin\omega t + \sin 2\omega t
  3. x=e−ωtx = e^{-\omega t}
  4. x=sin⁡ωt⋅cos⁡ωtx = \sin\omega t \cdot \cos\omega t

I know the defining condition is a=−ω2xa = -\omega^2 x but checking it for each one by differentiating twice takes forever and I make slip-ups. Is there a faster way to decide? And NCERT keeps saying 'every SHM is periodic but not every periodic motion is SHM', also 'oscillatory'. What is the difference between periodic and oscillatory? The Earth going round the Sun is periodic but we never call it an oscillation, right?

Also, for (2), what would be its time period? I thought π/ω\pi/\omega because the second term repeats faster, but the answer key says something else.

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6 answers & replies

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  • [+]Meera Patel11thSample· 1mo ago

    A faster method: any function of the form Asin⁡(ωt+ϕ)A\sin(\omega t + \phi), a single sinusoid, is SHM. Sums of sines and cosines with the same ω\omega combine into a single sinusoid, so (1) is SHM:

    sin⁡ωt+cos⁡ωt=2sin⁡(ωt+π4)\sin\omega t+\cos\omega t = \sqrt2\sin\left(\omega t+\frac{\pi}{4}\right)

    Sums with different frequencies, like (2), are not a single sinusoid, so not SHM. For (4), use sin⁡ωtcos⁡ωt=12sin⁡2ωt\sin\omega t\cos\omega t = \tfrac12\sin2\omega t, which is SHM with angular frequency 2ω2\omega.

  • [+]Harsh Gupta12thSample· 1mo ago

    For (2): sin ωt has period 2π/ω, sin 2ωt has period π/ω, the sum repeats as soon as either one repeats, so period is the smaller one, π/ω. Also I'm a periodic function, I show up late every day 🙃 (same time, every day, very reliable). Anyway that's my answer and I'm sticking to it.

    • [+]Tushar PawarDropperSample· 1mo ago

      Harsh, 'as soon as either repeats' is wrong. The sum repeats only when both terms are back at the same values together. Check at t=π/ωt = \pi/\omega: sin⁡ωt\sin\omega t has become −sin⁡ωt-\sin\omega t, so the sum is different. Period must be the smallest TT that is a whole-number multiple of both individual periods: multiples of 2π/ω2\pi/\omega and π/ω\pi/\omega, so T=2π/ωT = 2\pi/\omega. It's the LCM idea.

      • [+]Sara Fernandes11thSample· 1mo agoAuthor

        thanks all! LCM idea makes sense, I'd have bet on the smaller one too (sorry Harsh, you're not alone 😂)

  • [+]Zoya Khan12thSample· 1mo ago

    On the vocabulary, since it confused me too: periodic just means the motion repeats after equal intervals of time (Earth's orbit, a clock hand, a satellite). Oscillatory means a to-and-fro motion about a mean position, like a pendulum. All oscillations are periodic, but not all periodic motions are oscillatory. SHM is the special oscillation where the restoring force is proportional to displacement and directed to the mean position. So circles inside periodic, then oscillatory inside that, and SHM inside that.

  • [+]Aman YadavDropperSample· 1mo ago

    (3) is not even periodic. It just decays and never repeats. Not oscillatory either, it never passes the mean position and comes back.

    Exam rule: if x(t)x(t) can be written as one sine or cosine of ωt\omega t, it is SHM. Otherwise don't waste time, mark it not SHM.

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