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Periodic and Non-Periodic Motion

Distinguish between periodic and non-periodic motions, and recognize examples such as exponential functions.

Why this shows up in the exam

Identifying the nature of motion is a common NEET question.

How NEET tests this

Direct recall

Learn the idea

Periodic motion repeats exactly after a fixed time interval (the period). Any motion that never repeats for any non‑zero interval is non‑periodic – the exponential decay e⁻ᵞᵗ is the textbook example.

🧠 Memory hook: Periodic = repeats; Non‑periodic = never repeats – like a one‑way street that only goes down, never loops back (e⁻ᵞᵗ).

Get this one clearly and it pays off every single time it shows up in the paper. 🎯

Formulas & facts to keep ready

  • Periodic motion: f(t+T)=f(t) for some T>0
  • Period T = 2π/ω where ω is angular frequency
  • Sin ωt, cos ωt and any linear combination with the same ω are periodic with the same T
  • A sum of sinusoids with different ω may be non‑periodic
  • Exponential function e⁻ᵞᵗ never satisfies f(t+T)=f(t) for any T≠0

How to approach it

  1. 1Identify the mathematical form of the given function
  2. 2Check if there exists a non‑zero T such that replacing t by t+T leaves the expression unchanged
  3. 3For trigonometric terms use the relation sin(ω(t+T))=sin(ωt+ωT) and require ωT=2πk
  4. 4If no such T exists, the motion is non‑periodic

Worked example — watch it click

Identify the function which represents a non-periodic motion.

  • ✅e⁻ᵧᵗ
  • B)sin ωt
  • C)sin ωt + cos ωt
  • D)sin (ωt + π/4)

The concept behind this problem

The worked example asks you to apply the definition f(t+T)=f(t) and see which listed function fails to satisfy it, which is exactly what distinguishes periodic from non‑periodic motion.

Step by step

  1. 1Periodic motion repeats after a fixed time interval.
  2. 2(a) e⁻ᵧᵗ: exponential decay, never repeats - NON-PERIODIC.
  3. 3(b) sin ωt: periodic with period 2π/ω.
  4. 4(c) sin ωt + cos ωt = √2 sin(ωt + π/4): periodic with period 2π/ω.
  5. 5(d) sin(ωt + π/4): periodic with period 2π/ω.
  6. 6Only option (a) represents non-periodic motion.

Watch out

A common mistake is to label e⁻ᵞᵗ as non‑periodic because its amplitude falls, rather than because the function never repeats for any time interval.

Common slip-ups that cost marks

  • •Assuming any sinusoidal‑like graph is periodic without checking the frequency
  • •Ignoring the possibility of a phase shift – it does not change the period
  • •Mistaking a decaying amplitude as the sole reason for non‑periodicity (the key is lack of repetition)

🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.

Practise it

These are real questions from past NEET papers that test this exact idea.

Question 1 of 1NEET 2022

Identify the function which represents a non-periodic motion.

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