The following figure shows displacement versus time curve for a particle executing simple harmonic motion : Which one of the following statements is correct ?
- (a)Phase of the oscillating particle is same at t = 1 s and t = 3 s
- (b)Phase of the oscillating particle is same at t = 2 s and t = 8 s
- (c)Phase of the oscillating particle is same at t = 3 s and t = 7 s
- (d)Phase of the oscillating particle is same at t = 4 s and t = 10 s
Answer
Why
Correct — C, the phase is the same at t = 3 s and t = 7 s. The printed curve plots displacement against time with the time axis marked 0, 2, 4, 6, 8 and 10 seconds. The trace starts at zero, crosses zero again at 2, 4, 6, 8 and 10 s, peaks at about 1 s, 5 s and 9 s, and reaches its lowest points at about 3 s and 7 s. Three complete waves fit into the ten seconds shown.
Read the period straight off those repeats: successive crests sit at 1 s and 5 s, so T = 4 s.
Two instants are in the same phase only when they are a whole number of periods apart — the particle must be at the same displacement AND moving the same way. t = 3 s and t = 7 s differ by 4 s, exactly one period, and both sit at the bottom of a trough. So the phase is identical.
Why the others are wrong
- (a)Phase of the oscillating particle is same at t = 1 s and t = 3 s — These are 2 s apart, which is half a period. The particle is at a crest at 1 s and at a trough at 3 s — exactly out of phase, the opposite of the same phase.
- (b)Phase of the oscillating particle is same at t = 2 s and t = 8 s — The gap is 6 s, which is one and a half periods, not a whole number of them. Both instants have zero displacement, but at 2 s the particle is heading downwards and at 8 s it is heading upwards, so the phases are opposite.
- (d)Phase of the oscillating particle is same at t = 4 s and t = 10 s — Again a 6 s gap — one and a half periods. Equal displacement alone is not enough; the direction of motion is reversed at these two instants, so the phase differs.
Concept
In simple harmonic motion the phase fixes both where the particle is and which way it is travelling. The motion repeats exactly after one period T, so two instants carry the same phase only when they are separated by a whole number of periods: t₂ − t₁ = nT. A separation of T/2 puts the particle in antiphase — same speed, mirrored displacement, opposite direction.
The whole item reduces to reading the period off the graph and then testing each time gap against it. Count crest to crest, not crest to trough: here crests at 1 s and 5 s give T = 4 s, and a common slip is to take the 2 s crest-to-trough spacing as the period, which halves it. The second trap is judging phase by displacement alone — at 2 s and at 8 s the displacement is zero both times, yet the particle is moving in opposite directions, so the phases are not the same.
Key facts
- Same phase requires t₂ − t₁ = nT, a whole number of periods.
- From the figure, crests fall at 1 s, 5 s and 9 s, so T = 4 s.
- t = 3 s and t = 7 s are 4 s apart — exactly one period.
- A separation of T/2 gives antiphase, not the same phase.
- Equal displacement is not enough; the direction of motion must match too.
Only a gap that is a whole number of periods preserves phase — 3 s and 7 s, option (c).
Study next
Common traps
- Taking crest-to-trough spacing as the full period instead of half of it.
- Calling two instants 'in phase' because the displacement matches, without checking the direction of motion.
- Treating a gap of 1.5 T as equivalent to a whole period.
NDA prints an SHM displacement–time curve and offers pairs of instants — read the period from crest to crest, then keep only the pair separated by a whole number of periods.
Related PYQs
If T is the time period of an oscillating pendulum, which one of the following statements is NOT correct ?
- (a) The motion repeats after time T only once
- (b) T is the least time after which motion repeats itself
- (c) The motion repeats itself after nT, where n is a positive integer
- (d) T remains the same only for small angular displacements
Answer(a) The motion repeats after time T only once
The definition of the period itself — that T is the least time after which the motion repeats. That is precisely the property used here to decide which pair of instants shares a phase.
Practice
- practice — not a real PYQ
A particle in SHM has a period of 6 s. The phase at t = 2 s is the same as the phase at
- (a)t = 5 s
- (b)t = 8 s
- (c)t = 9 s
- (d)t = 11 s
Answer(b) t = 8 s — the instants must differ by a whole period, and 8 − 2 = 6 s = T. - practice — not a real PYQ
Two instants in a simple harmonic motion are separated by half a period. The particle at these instants has
- (a)the same phase
- (b)opposite phase
- (c)a phase difference of 90°
- (d)zero displacement at both
Answer(b) opposite phase — a gap of T/2 corresponds to a phase difference of 180°.