In SHM, displacement follows x(t) = A cos(ωt + φ). The quantity (ωt + φ) is the phase of the oscillation at time t. The constant φ is the initial phase (or phase constant) — it fixes where in its cycle the particle sits at t = 0 (NCERT Class 11 Physics Chapter 13, page 3).
Why phase matters for NEET: the exam tests whether you can state the phase relationship between displacement, velocity, and acceleration without hesitation. Velocity v = −Aω sin(ωt + φ) can be rewritten as Aω cos(ωt + φ + π/2). Acceleration a = −ω²x = −Aω² cos(ωt + φ), which equals Aω² cos(ωt + φ + π). The phase relationships are therefore:
A common confusion: mixing up the π/2 (velocity–displacement) and π (acceleration–displacement) phase gaps. If the question asks for the phase difference between acceleration and displacement, the answer is π — not π/2. Conversely, if it asks for velocity relative to displacement, the answer is π/2 — not π.
Another point tested: the initial phase φ determines the starting condition. If x(0) = A (particle at positive extreme), then φ = 0 with x = A cos(ωt). If x(0) = 0 and the particle moves toward positive x, then x = A sin(ωt), equivalently A cos(ωt − π/2), so φ = −π/2. NCERT notes that the choice between sine and cosine form is a convention tied to the value of φ (Class 11 Physics Chapter 13, page 6).
Watch-out: when two SHM oscillators have different phase constants φ₁ and φ₂, their phase difference is (φ₁ − φ₂), a constant. NEET questions may present two particles on the same spring system at different starting positions and ask for the phase difference — it is simply the difference in their initial phases.
Select an option to see the explanation. Wrong answers show why your choice was tempting — and name the exact trap it exploits.
In simple harmonic motion, the phase difference between displacement and acceleration is:
A particle in SHM is at the mean position at t = 0 and moves toward the positive direction. Which expression correctly represents its displacement?
In SHM, velocity leads displacement by:
Two particles execute SHM with the same amplitude and angular frequency. Particle 1 has displacement x₁ = A cos(ωt) and particle 2 has displacement x₂ = A cos(ωt + π/3). The phase difference between them is:
A particle in SHM has displacement x = A cos(ωt + φ). At the instant when the particle is at x = +A, what is the phase of the velocity?
A particle starts SHM from the positive extreme position. Its initial phase φ in the equation x = A cos(ωt + φ) is:
In SHM, at the instant when displacement is at its maximum positive value, the acceleration is:
Two SHMs are represented by x₁ = 5 sin(4πt) cm and x₂ = 5 cos(4πt) cm. The phase difference (x₂ relative to x₁) is:
Pattern: Phase difference between displacement, velocity, and acceleration in SHM (PYQ pattern observed 2020, 2021, 2023, 2024).
Given
A particle executes SHM: x = 3.0 cos(2πt + π/4) cm. Find the phase difference between its velocity and acceleration.
Required
Phase difference between velocity and acceleration.
Concept
In SHM, velocity leads displacement by π/2, and acceleration leads displacement by π. Therefore acceleration leads velocity by π − π/2 = π/2. Equivalently, velocity leads acceleration by −π/2, or acceleration leads velocity by π/2.
Formula
x = A cos(ωt + φ) v = Aω cos(ωt + φ + π/2) → phase of v is (ωt + φ + π/2) a = Aω² cos(ωt + φ + π) → phase of a is (ωt + φ + π) Phase difference = phase of a − phase of v = (ωt + φ + π) − (ωt + φ + π/2) = π/2.
Substitution
Here φ = π/4 and ω = 2π, but the specific values cancel — the phase difference between v and a is always π/2, independent of φ, ω, or A.
Calculation
Phase difference (a relative to v) = π/2 rad = 90°. Note: the amplitude 3.0 cm, angular frequency 2π rad/s, and initial phase π/4 are given numerical values (exact as stated in the problem) and do not affect the phase relationship.
Final answer
The phase difference between velocity and acceleration is **π/2 rad** (acceleration leads velocity by π/2).
Common trap
Computing π (the acceleration–displacement phase gap) instead of π/2 (the acceleration–velocity phase gap). Always identify which two quantities the question asks about. The three fixed phase gaps in SHM are: v leads x by π/2, a leads x by π, a leads v by π/2.
Similar NEET-style question
"For a particle in SHM, the phase difference between its acceleration and velocity is (a) 0, (b) π/4, (c) π/2, (d) π." (Answer: π/2.) ---
Oscillation in which the restoring force is directly proportional to displacement from equilibrium and directed back toward equilibrium: F = -k x. Equation: a = -ω² x. Solution: x(t) = A cos(ω t + φ).
-- NCERT, p. 3Velocity leads displacement by π/2: v = -A ω sin(ωt+φ). Acceleration leads displacement by π: a = -A ω² cos(ωt+φ) = -ω² x. KE max at x=0; PE max at x=±A.
-- NCERT, p. 6When two waves of nearly equal frequencies superpose, amplitude oscillates at the difference frequency.
| Symbol | Quantity | SI Unit |
|---|---|---|
| f_beat | beat frequency | Hz |
| f1, f2 | superposed frequencies | Hz |
Period of simple pendulum of length L. Holds for small amplitudes (sin theta ~ theta).
| Symbol | Quantity | SI Unit |
|---|---|---|
| T | period | s |
| L | pendulum length | m |
| g | gravity | m/s^2 |
Displacement in simple harmonic motion. Velocity = -A*omega*sin(omega*t+phi); a = -omega^2 * x.
| Symbol | Quantity | SI Unit |
|---|---|---|
| A | amplitude | m |
| omega | angular frequency | rad/s |
| phi | phase | rad |
| T | period | s |
| f | frequency | Hz |
Total mechanical energy is constant. Oscillates between KE (max at x=0) and PE (max at x=±A).
| Symbol | Quantity | SI Unit |
|---|---|---|
| E | total energy | J |
| k | spring constant | N/m |
| A | amplitude | m |
| m | mass | kg |
| omega | angular frequency | rad/s |
Period of horizontal spring with mass m, spring constant k. Independent of amplitude.
| Symbol | Quantity | SI Unit |
|---|---|---|
| T | period | s |
| m | mass | kg |
| k | spring constant | N/m |
Pipe closed at one end has only odd harmonics: f, 3f, 5f, ...
| Symbol | Quantity | SI Unit |
|---|---|---|
| f_n | n-th harmonic | Hz |
| v | sound speed | m/s |
| L | pipe length | m |
Pipe open at both ends has all harmonics. Same formula as string.
| Symbol | Quantity | SI Unit |
|---|---|---|
| f_n | n-th harmonic | Hz |
| v | sound speed | m/s |
| L | pipe length | m |
Allowed frequencies on string fixed at both ends. n=1 fundamental; harmonics 2f, 3f, ...
| Symbol | Quantity | SI Unit |
|---|---|---|
| f_n | n-th harmonic | Hz |
| v | wave speed on string | m/s |
| L | string length | m |
| n | harmonic number | - |
Speed of sound in gas. Adiabatic index gamma, pressure P, density rho. Increases with sqrt(T).
| Symbol | Quantity | SI Unit |
|---|---|---|
| v | speed of sound | m/s |
| gamma | adiabatic index | - |
| P | pressure | Pa |
| rho | density | kg/m^3 |
Speed of transverse wave on string under tension T, linear mass density mu.
| Symbol | Quantity | SI Unit |
|---|---|---|
| v | wave speed | m/s |
| T | tension | N |
| mu | linear mass density | kg/m |
These are the exact patterns that cause wrong answers in NEET. Each trap includes when it triggers and how to avoid it.
Category: Overthinking
Student writes T as depending on bob mass. Simple pendulum T = 2π√(L/g); independent of m.
Question changes pendulum bob mass and asks for new period.
Mass cancels in derivation (gravitational mass = inertial mass). Mass changes the bob's KE and PE proportionally; period unaffected.
Category: Overthinking
Student claims SHM period depends on amplitude. For ideal SHM (Hooke's law spring or simple pendulum at small angle), period is INDEPENDENT of amplitude.
Question gives changes in amplitude and asks for new period.
T = 2π√(m/k) (spring) or 2π√(L/g) (pendulum, small angle) — neither depends on A. Only at large pendulum angles does T pick up a small amplitude correction.
Category: Similar Terms
Student includes even harmonics in a closed-end pipe. Closed pipe has only ODD harmonics (f, 3f, 5f, ...).
Question describes pipe closed at one end (e.g. resonance tube).
Open both ends: all harmonics, f_n = nv/(2L). Closed one end: odd only, f_n = (2n-1)v/(4L). Fundamental of closed pipe is HALF that of open pipe of same L.
Root cause: concept gap
Simple pendulum T = 2π√(L/g) — independent of mass. Equivalence of inertial and gravitational mass cancels m.
Root cause: concept gap
Ideal SHM: T = 2π√(m/k) (spring) or 2π√(L/g) (pendulum, small angle) — no amplitude dependence. Doubling amplitude does not change period.
Root cause: concept gap
Closed-end pipe has only ODD harmonics (f, 3f, 5f, ...). Open-both-ends pipe has all (f, 2f, 3f, ...). Reason: closed end has displacement node and pressure antinode.
treats beat as sum
Adds frequencies instead of subtracting
expects mass dependence
Assumes T depends on bob mass
treats closed pipe like open
Includes even harmonics in closed pipe
uses pi 2 instead of pi
Confuses v-x phase with a-x phase
forgets 2pi factor
Drops 2*pi
uses amplitude in period
Believes T depends on amplitude
uses linear tension scaling
Treats v ∝ T not sqrt(T)
Test yourself on this topic with real past-paper questions:
Practice this topic →Get a structured 30-day Mechanics plan and a complete formula booklet — delivered to your inbox instantly.