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 261).
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 264).
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:
Answer: C. Acceleration a = −ω²x, which means a = Aω² cos(ωt + φ + π). The phase of acceleration exceeds that of displacement by π. (NCERT Class 11 Physics Chapter 13, page 261.)
Why A is wrong: A is wrong because a phase difference of 0 would mean acceleration is in phase with displacement, but a = −ω²x shows they are opposite in sign — a half-cycle apart.
Why B is wrong: B is wrong because π/4 rad (45°) has no standard relationship in SHM phase analysis. The only phase gaps that appear are π/2 (velocity–displacement) and π (acceleration–displacement).
Why D is wrong: D is wrong because π/2 is the phase difference between velocity and displacement, not between acceleration and displacement. This is the most common swap (trap: confusing the v–x and a–x phase gaps).
A particle in SHM is at the mean position at t = 0 and moves toward the positive direction. Which expression correctly represents its displacement?
Answer: B. At t = 0, x = 0 eliminates the cosine forms (A and C give x(0) = ±A) — step one. Between the two remaining sine forms, B and D, x = A sin(ωt) gives v(0) = Aω > 0 (positive direction), while x = −A sin(ωt) gives v(0) = −Aω < 0 — step two, needed to choose between them. So B is correct. This corresponds to an initial phase φ = −π/2 in the cosine convention. (NCERT Class 11 Physics Chapter 13, page 261.)
Why A is wrong: A is wrong because x = A cos(ωt) gives x(0) = A, meaning the particle starts at the positive extreme, not at the mean position.
Why C is wrong: C is wrong because x = −A cos(ωt) gives x(0) = −A, meaning the particle starts at the negative extreme, not at the mean position.
Why D is wrong: D is wrong because x = −A sin(ωt) gives x(0) = 0 (correct) but v(0) = −Aω < 0, meaning the particle initially moves in the negative direction, not the positive.
In SHM, velocity leads displacement by:
Answer: B. v = −Aω sin(ωt + φ) = Aω cos(ωt + φ + π/2). Comparing with x = A cos(ωt + φ), the phase of velocity exceeds that of displacement by π/2. (NCERT Class 11 Physics Chapter 13, page 261.)
Why A is wrong: A is wrong because π rad is the phase difference between acceleration and displacement, not velocity and displacement (trap: confusing the a–x and v–x phase gaps).
Why C is wrong: C is wrong because π/4 rad does not correspond to any standard SHM phase relationship. The relevant phase differences are π/2 (v leads x) and π (a leads x).
Why D is wrong: D is wrong because a 2π phase difference is equivalent to zero phase difference — the two quantities would be completely in phase, which is not the case for velocity and displacement in SHM.
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:
Answer: C. Phase of particle 1 is (ωt + 0) and phase of particle 2 is (ωt + π/3). Phase difference = (ωt + π/3) − (ωt + 0) = π/3 rad. The phase difference between two SHMs of the same ω is simply the difference in their initial phases. (NCERT Class 11 Physics Chapter 13, page 261.)
Why A is wrong: A is wrong because π/6 would be half of the actual phase constant difference. There is no reason to halve π/3 here — the phase difference is read directly from the arguments.
Why B is wrong: B is wrong because 2π/3 = π − π/3 is the supplement of the phase difference, as if one of the cosines had its sign flipped; the difference is read directly from the arguments.
Why D is wrong: D is wrong because π would imply the two particles are in antiphase (exactly opposite positions at all times). With a π/3 phase gap, they are not opposite — particle 2 simply reaches each phase point earlier than particle 1 by π/3 rad.
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?
Answer: A. v = −Aω sin(ωt + φ) = Aω cos(ωt + φ + π/2). The phase of velocity is always (ωt + φ + π/2), regardless of the particle's current position. At x = +A, the displacement phase satisfies cos(ωt + φ) = 1, so (ωt + φ) = 0, 2π, … and the velocity phase is π/2, 2π + π/2, … — but the general expression remains (ωt + φ + π/2). (NCERT Class 11 Physics Chapter 13, page 261.)
Why B is wrong: B is wrong because (ωt + φ) is the phase of the displacement, not the velocity. Velocity has an additional π/2 lead.
Why C is wrong: C is wrong because (ωt + φ + π) is the phase of the acceleration, not the velocity. This is the a–x phase relationship being confused with the v–x relationship.
Why D is wrong: D is wrong because (ωt + φ − π/2) would mean velocity lags displacement by π/2. In SHM, velocity leads displacement by π/2 — the sign is positive, not negative.
A particle starts SHM from the positive extreme position. Its initial phase φ in the equation x = A cos(ωt + φ) is:
Answer: B. At t = 0, x(0) = A cos(φ). For the particle to be at the positive extreme, x(0) = A, so cos(φ) = 1, giving φ = 0. (NCERT Class 11 Physics Chapter 13, page 261.)
Why A is wrong: A is wrong because φ = π/2 gives x(0) = A cos(π/2) = 0, meaning the particle starts at the mean position — not the positive extreme.
Why C is wrong: C is wrong because φ = π gives x(0) = A cos(π) = −A, placing the particle at the negative extreme.
Why D is wrong: D is wrong because φ = −π/2 gives x(0) = A cos(−π/2) = 0, starting the particle at the mean position (moving toward positive x). This is the sine-form starting condition, not the positive-extreme condition.
In SHM, at the instant when displacement is at its maximum positive value, the acceleration is:
Answer: D. a = −ω²x. When x = +A (maximum positive), a = −ω²A, which is the maximum magnitude of acceleration, directed in the negative direction. Displacement and acceleration are π out of phase — when one is at positive maximum, the other is at negative maximum. (NCERT Class 11 Physics Chapter 13, page 261.)
Why A is wrong: A is wrong because acceleration is zero at the mean position (x = 0), not at the extremes. At the extremes, |a| is maximum (trap: confusing where velocity is zero with where acceleration is zero).
Why B is wrong: B is wrong because a = −ω²x is always opposite in sign to x. When x is at its positive maximum, a must be at its negative maximum — the π phase difference ensures they are always opposite.
Why C is wrong: C is wrong because there is no factor of 1/2 in a = −ω²x. At x = A, the acceleration magnitude is ω²A, which is its full maximum — not half.
Two SHMs are represented by x₁ = 5 sin(4πt + π/6) cm and x₂ = 5 cos(4πt) cm. The phase difference of x₁ relative to x₂ (phase of x₁ minus phase of x₂) is:
Answer: A. Step one: put both expressions in the same trig form before comparing phases. Convert x₂ to sine form using cos θ = sin(θ + π/2): x₂ = 5 cos(4πt) = 5 sin(4πt + π/2), so its phase is (4πt + π/2). Step two, using that converted phase: phase of x₁ minus phase of x₂ = (4πt + π/6) − (4πt + π/2) = π/6 − π/2 = −π/3 rad. The second step depends on the phase produced in the first — comparing before converting would compare unlike functions. (NCERT Class 11 Physics Chapter 13, page 261.)
Why B is wrong: B is wrong because it is the correct magnitude with the sign reversed — this comes from computing (phase of x₂ − phase of x₁) instead of the asked order (phase of x₁ − phase of x₂).
Why C is wrong: C is wrong because it skips the conversion step entirely: it takes the phase of x₂ as 4πt (its raw cosine argument) and subtracts π/6 − 0, without first shifting x₂ into sine form to match x₁.
Why D is wrong: D is wrong because it is just the sin-to-cos conversion constant (π/2) itself, used as the final answer while dropping the π/6 initial-phase term of x₁ altogether.
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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.)
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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 Class 11 Physics, Ch. 13, p. 260Velocity 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 Class 11 Physics, Ch. 13, p. 267Displacement 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 |
More in Oscillations and Waves: 6 exam traps and mistakes · 9 formulas · 5 question patterns from its other lessons.
uses pi 2 instead of pi
Confuses v-x phase with a-x phase
Page numbers are the ones printed in the current NCERT textbook (2023 rationalised edition), unless marked pre-2023. The books are free at ncert.nic.in.
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