NEET Physics — Simple Harmonic Motion
Periodic and Oscillatory Motion – Complete Notes, Revision, Important Questions & Downloads
This topic covers Types of Periodic Motion — the classification of all repeating motions into periodic vs. oscillatory, harmonic vs. non-harmonic. NEET tests this in two direct patterns: (1) identifying whether a given function (e.g., e^{−ωt}, sin ωt + cos ωt) represents periodic motion, and (2) distinguishing harmonic from non-harmonic oscillations using function form. A key fact: every SHM is periodic and oscillatory, but not every periodic motion is oscillatory (circular motion repeats but does not return via the same path). NEET 2022 asked students to identify a non-periodic function from a set of four — the trap was that e^{−ωt} looks regular but never repeats. One function that appears deceptive is sin ωt + cos ωt, which is still harmonic because it simplifies to √2 sin(ωt + π/4).
4 SubtopicsClassification TopicNCERT Ch 13
Expected QuestionsQ
0–1
Direct question on periodic vs. oscillatory classification appears roughly once every 2–3 years in NEET; more often embedded in function-identification MCQs.
Time Required⏱
1–2 hours
One focused reading session plus 10–15 classification practice problems to recognise harmonic vs. non-harmonic functions on sight.
Difficulty⚡
Easy
Pure classification and definition recall. The only difficulty is reading the function correctly — especially recognising that sums of harmonics at the same ω remain harmonic.
NRI USA Curriculum GapUS
Low–Medium
US AP Physics 1 introduces periodic motion and oscillation concepts, but does not explicitly test harmonic vs. non-harmonic classification by function form, nor does it emphasise the distinction that circular motion is periodic but not oscillatory.
4Subtopics
6+Practice Questions
4Free Downloads
1–2 hrsPrep Time
NEET Weightage — Periodic and Oscillatory Motion
Simple Harmonic Motion (Chapter 16)| NEET Year | Questions from this Topic | Bar | Marks |
|---|---|---|---|
| 2024 | 0 | 0 | |
| 2023 | 0 | 0 | |
| 2022 | 1 | 4 | |
| 2021 | 0 | 0 | |
| 2020 | 1 | 4 | |
| 2019 | 0 | 0 | |
| 6-Year Total (2019–2024) | 1–2 | 4–8 |
NEET frequently presents 4 mathematical functions and asks which is non-periodic — e^{−ωt} is the classic non-periodic decoy; it decreases monotonically and never repeats.
The statement 'every oscillatory motion is periodic, but every periodic motion need not be oscillatory' is a high-frequency one-liner in NEET assertion-reason questions.
Harmonic oscillation y = a sin ωt + b sin 2ωt is non-harmonic because it requires two sine functions — NEET tests this by asking whether a mixed-frequency expression represents SHM.
The statement 'every oscillatory motion is periodic, but every periodic motion need not be oscillatory' is a high-frequency one-liner in NEET assertion-reason questions.
Harmonic oscillation y = a sin ωt + b sin 2ωt is non-harmonic because it requires two sine functions — NEET tests this by asking whether a mixed-frequency expression represents SHM.
~0.3
Avg Questions / Year
8
Total Marks (6 yrs)
Direct
Pattern
Easy
Difficulty
Exam Strategy for Periodic and Oscillatory Motion
1
Memorise the three-tier hierarchy Periodic motion ⊃ Oscillatory motion ⊃ SHM. Circular motion is periodic but not oscillatory. Oscillatory motion that follows a single sine/cosine is harmonic SHM. Oscillatory motion requiring two or more harmonics is non-harmonic. Internalise this hierarchy — NEET's function-identification questions test exactly where each motion sits.
2
Learn to test functions for periodicity A function is periodic if f(t + T) = f(t) for some finite T. Check: sin ωt → periodic (T = 2π/ω). e^{−ωt} → fails test, non-periodic. log(ωt) → fails test, non-periodic. sin ωt + cos ωt → combine using √2 sin(ωt + π/4) → still periodic. Practise this test on at least 10 functions before the exam.
3
Recognise the harmonic/non-harmonic trap A single function a sin ωt or a cos ωt is harmonic. A sum a sin ωt + b sin 2ωt is non-harmonic because of the 2ω term — it requires two harmonics. The trap: students think any sine-based expression is harmonic. The check: are all frequency components identical (same ω)? If yes → harmonic. If different ω values → non-harmonic.
Study Materials — Periodic and Oscillatory Motion
PDF · Cheat Sheet · MCQ Set · PYQFull Notes
Complete notes covering Types of Periodic Motion: definitions of periodic motion, oscillatory motion, harmonic and non-harmonic oscillations, classification hierarchy, and examples from NEET context.
Download NotesFormula Sheet
Key relations: y = a sin ωt (harmonic), y = a sin ωt + b sin 2ωt (non-harmonic), periodicity test f(t+T) = f(t). Includes function-type quick identification table.
Download SheetMCQ Practice
15 classification MCQs: function-type identification, periodic vs. oscillatory matching, harmonic vs. non-harmonic determination. Covers NEET-style one-correct and assertion-reason formats.
Download MCQsPYQ
Solved Previous Year Questions from NEET (2014–2024) on periodic motion classification, function identification, and oscillation-type recognition. All questions year-tagged.
Download PYQsSubtopics in Periodic and Oscillatory Motion
2-Column TableRapid Revision — Periodic and Oscillatory Motion
Concept → Trap → Example1) Types of Periodic Motion
ClassificationEvery SHM motion is periodic but every periodic motion is not SHM. Harmonic: y = a sin ωt. Non-harmonic: y = a sin ωt + b sin 2ωt.
- Periodic motion repeats after a fixed time interval T; it can be rectilinear, closed curvilinear, or open curvilinear.
- Oscillatory motion is periodic motion confined within definite limits about a mean position — the body moves to and fro about the equilibrium.
- Harmonic oscillation requires only a single sine or cosine function; non-harmonic oscillation needs two or more harmonics at different frequencies.
Example (NEET-style)Testing e^{−ωt}: compute f(t + T) = e^{−ω(t+T)} ≠ e^{−ωt} for any finite T → not periodic. Testing sin ωt + cos ωt = √2 sin(ωt + π/4): repeats with T = 2π/ω → periodic and harmonic.
US Curriculum Gaps — Periodic and Oscillatory Motion
Students from the US curriculum may not have encountered these distinctions explicitly.AP Physics 1 does not classify harmonic vs. non-harmonic by function form
US AP Physics 1 (College Board) covers oscillations and simple harmonic motion but does not ask students to classify a mathematical function as harmonic vs. non-harmonic, nor to test f(t + T) = f(t) for periodicity on arbitrary expressions.
- NEET tests exact periodicity: can you factorise sin ωt + cos ωt into √2 sin(ωt + π/4) and identify it as harmonic?
- AP Physics 1 assumes oscillations are sinusoidal — it does not present non-harmonic examples or ask students to distinguish them.
- For NEET, practise identifying periodicity and harmonic character from the function expression, not just from physical description.
US curricula treat circular motion as a separate chapter, not contrasted with oscillatory motion
In US Physics courses (AP Physics 1, MIT OpenCourseWare 8.01), circular motion and oscillatory motion are taught in separate units. NEET explicitly contrasts them: circular motion is periodic but not oscillatory, making it a direct exam trap.
- NEET assertion-reason questions use 'circular motion is periodic but not oscillatory' as the assertion — US-trained students who haven't encountered this distinction will incorrectly call it oscillatory.
- The key differentiator: oscillatory motion requires a restoring force and confinement to a mean position; circular motion has centripetal force directed to the centre, not a mean equilibrium.
- Review this contrast explicitly before NEET — it is a reliable source of 1–2 marks across the 2019–2024 paper set.
NEET-style Practice Questions — Periodic and Oscillatory Motion
4 scenario-based questions1Which of the following functions of time represents a non-periodic motion? (NEET 2022 style)Function Identification
sin ωt + cos ωt
e^{−ωt}
sin(ωt + π/4)
sin ωt · cos ωt
A periodic function must satisfy f(t + T) = f(t) for a finite, non-zero T. Option (b): e^{−ωt} is a monotonically decaying exponential. Testing: e^{−ω(t+T)} = e^{−ωt} · e^{−ωT}. Since e^{−ωT} ≠ 1 for any finite T > 0, this function is non-periodic. Option (a): sin ωt + cos ωt = √2 sin(ωt + π/4), which repeats with period 2π/ω → periodic. Option (c): sin(ωt + π/4) is a shifted sine → periodic with T = 2π/ω. Option (d): sin ωt · cos ωt = (1/2)sin 2ωt, periodic with T = π/ω.
2A particle moves to and fro about the mean position x = 0 with displacement given by x(t) = 3 sin(2t) + 4 sin(3t) metres. This motion is best described as:Harmonic Classification
Simple harmonic, period π seconds
Non-harmonic oscillatory, period 2π seconds
Simple harmonic, period 2π/3 seconds
Periodic but not oscillatory
x(t) = 3 sin(2t) + 4 sin(3t) contains two harmonics at different angular frequencies (ω₁ = 2 rad/s, ω₂ = 3 rad/s). Because it requires two sine functions to express it, this is non-harmonic oscillation — it cannot be written as a single a sin(ωt + φ). It is oscillatory because the particle moves about x = 0 within defined limits. The overall period is the LCM of T₁ = π and T₂ = 2π/3, which is 2π seconds. Option (a) is wrong: it is not SHM. Option (c) is wrong: 2π/3 is the period of only the second component. Option (d) is wrong: the particle clearly oscillates about a fixed mean position.
3The motion of the hour hand of a clock is best classified as:Classification
Oscillatory and harmonic
Oscillatory but non-harmonic
Periodic but not oscillatory
Neither periodic nor oscillatory
The hour hand completes one revolution every 12 hours — clearly periodic with T = 12 hours. However, it is not oscillatory: oscillatory motion requires the body to move back and forth about a fixed mean equilibrium position with a restoring force. The hour hand moves continuously in one direction (clockwise) and never reverses direction or returns via the same path. Hence it is periodic but not oscillatory. This is the same reason Earth's revolution around the Sun (T = 1 year) is periodic but not oscillatory.
4A body executing SHM passes through the equilibrium position with maximum speed v₀. At a displacement equal to half its amplitude, the ratio of its kinetic energy to total energy is:Energy Application
1/4
3/4
1/2
√3/2
v_max = aω at the equilibrium (mean) position. Total energy E = (1/2)mω²a². At displacement y = a/2: KE = (1/2)mω²(a² − y²) = (1/2)mω²(a² − a²/4) = (1/2)mω²(3a²/4) = (3/4) × (1/2)mω²a² = (3/4)E. Therefore KE/E = 3/4. Option (a) 1/4 is wrong — that is the PE fraction at y = a/2, not KE. Option (c) 1/2 is the fraction when y = a/√2. Option (d) √3/2 confuses the velocity ratio with the energy ratio.
Practice Questions — Periodic and Oscillatory Motion
Click "Reveal Answer" after attempting1A particle moves along x-axis with displacement x(t) = A sin(ωt) + B sin(2ωt). If A = 2 m, B = 1 m, ω = π rad/s, what is the time period of the resultant motion?
1 s
2 s
0.5 s
4 s
👁 Reveal Answer
Option (b) 2 s. The two components have periods T₁ = 2π/ω = 2 s and T₂ = 2π/(2ω) = 1 s. The resultant period is LCM(2, 1) = 2 s. The motion is non-harmonic oscillatory because two different frequencies are present, but its overall period is the least common multiple of the individual periods.
2Which of the following is an example of oscillatory motion that is NOT simple harmonic? (Give the condition that makes it non-SHM.)
Simple pendulum for small angles (< 5°)
Spring-mass system on a frictionless surface
Simple pendulum with large angle (45°) oscillations
Horizontal spring system with F = −kx
👁 Reveal Answer
Option (c): large-angle pendulum oscillations. For SHM, the restoring force must be F ∝ −x. A simple pendulum's restoring torque is τ = −mgL sin θ. For small θ, sin θ ≈ θ, giving τ ≈ −mgLθ (SHM). For θ = 45°, sin 45° = 0.707 ≠ 0.785 (= 45° in radians), so sin θ ≠ θ and the proportionality fails. The motion is oscillatory but non-harmonic.
3log(ωt) is tested for periodicity. Determine whether it is periodic.
Periodic with period 2π/ω
Periodic with period 1/ω
Non-periodic because it is defined only for t > 0 and increases monotonically
Non-periodic because it oscillates without a fixed period
👁 Reveal Answer
Option (c). For a function to be periodic: f(t + T) = f(t) for some finite T > 0. Test: log(ω(t + T)) = log(ωt + ωT) ≠ log(ωt) for any finite T (logarithm is strictly increasing, never repeating). Additionally, log(ωt) is only defined for t > 0 and increases without bound — it never returns to a previous value.
4A mass-spring system and a pendulum are both set into motion at t = 0 with their respective amplitudes. Which statement about their motions is correct?
Both are periodic and oscillatory; both are SHM
Both are oscillatory; only the spring system is SHM
Both are SHM for small amplitudes; both periods depend on amplitude
The pendulum is periodic but not oscillatory; the spring is oscillatory
👁 Reveal Answer
Option (a) for small amplitude. Both the spring (F = −kx always) and the simple pendulum (τ ≈ −mgθ for small θ) execute SHM. Both are periodic, oscillatory, and harmonic. A critical NEET fact: neither period depends on amplitude in SHM (T_spring = 2π√(m/k), T_pendulum = 2π√(L/g)) — amplitude independence is a defining property of SHM tested directly in NEET.
Physics — Simple Harmonic Motion Revision Checklist
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Use this section for quick chapter tracking before mocks, part tests, and final NEET revision.
Frequently Asked Questions — Periodic and Oscillatory Motion
Notes · Downloads · Revision · Important QuestionsWhat is the difference between periodic motion and oscillatory motion?
Periodic motion is any motion that repeats after a regular time interval T, regardless of path — circular motion, revolution of Earth around the Sun, and a clock's hour hand are periodic. Oscillatory motion is a subset: it is periodic motion where the body moves back and forth about a fixed mean equilibrium position, confined within definite limits on either side. Every oscillatory motion is periodic, but periodic motions like uniform circular motion are not oscillatory because the body never reverses direction about a mean position.
What is harmonic oscillation and how does it differ from non-harmonic oscillation?
Harmonic oscillation is one that can be expressed as a single sine or cosine function: y = a sin ωt or y = a cos ωt. This is exactly SHM. Non-harmonic oscillation requires two or more harmonic functions at different frequencies: y = a sin ωt + b sin 2ωt. The key test: if the expression can be written as A sin(ωt + φ) with a single frequency ω, it is harmonic. If multiple frequencies are needed, it is non-harmonic.
How do I test if a function is periodic?
Apply the periodicity test: a function f(t) is periodic if there exists a finite T > 0 such that f(t + T) = f(t) for all t. Sine and cosine satisfy this: sin(ωt + ωT) = sin ωt when T = 2π/ω. The function e^{−ωt} fails: e^{−ω(t+T)} = e^{−ωt} · e^{−ωT} ≠ e^{−ωt} for any T > 0. The function log(ωt) is strictly increasing — it never repeats. Both are non-periodic.
Is circular motion oscillatory?
No. Circular motion is periodic (the particle returns to the same position after every revolution with period T = 2π/ω), but it is not oscillatory. Oscillatory motion requires the body to move back and forth about a fixed equilibrium position with a restoring force pulling it back. In circular motion, the centripetal force always points toward the centre — there is no equilibrium position about which the body reverses. This distinction is a common NEET assertion-reason trap.
Why is every SHM oscillatory but not every oscillatory motion SHM?
SHM requires F = −kx (restoring force directly proportional to displacement, directed toward equilibrium). This guarantees the motion is oscillatory (particle moves back and forth about equilibrium) and harmonic (displacement is a single sine/cosine function). A large-angle pendulum is oscillatory — it moves back and forth about equilibrium — but because sin θ ≠ θ at large angles, F is not exactly proportional to x, so it is not SHM. A particle executing y = a sin ωt + b sin 2ωt oscillates but requires two harmonics — it is oscillatory but not SHM.
What does 'amplitude independence of time period in SHM' mean and why does NEET test it?
In SHM, the period T depends only on the system's inertia and restoring constant, not on the oscillation amplitude. For a spring-mass system, T = 2π√(m/k); for a simple pendulum, T = 2π√(L/g). Doubling the amplitude doubles the maximum speed aω but leaves T unchanged. NEET tests this because students often expect larger oscillations to take longer. The trap question: 'A pendulum with amplitude 5 cm has period T. If amplitude is increased to 10 cm, the new period is _?' Answer: T (unchanged), provided the angle remains small enough for the small-angle approximation.
How is SHM related to uniform circular motion?
SHM on a line can be viewed as the projection of uniform circular motion (UCM) onto a diameter of the reference circle. If a particle P moves on a circle of radius a with angular velocity ω, its projection onto the y-axis gives y = a sin ωt — exactly the SHM displacement equation. This geometrical interpretation explains why ω (angular frequency of SHM) links directly to the angular velocity of the reference circle, and why the period T = 2π/ω equals the period of one full revolution of the reference circle.
What are the four examples of oscillatory motion given in NCERT for Class 11?
The textbook lists: (a) the motion of the pendulum of a wall clock, (b) the motion of a load attached to a spring when pulled and released, (c) the motion of liquid contained in a U-tube when compressed in one limb and released, and (d) a loaded piece of wood floating on a liquid surface when pressed down and released. All four are oscillatory because the body moves to and fro about a fixed mean position. NEET occasionally asks which of a given list of motions is oscillatory — recognise all four examples.
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