Which one of the following statements regarding motion is correct?
- (a)All the periodic motions are necessarily simple harmonic
- (b)All the simple harmonic motions are necessarily periodic motions
- (c)There is no co-relation between the simple harmonic motions and the periodicity of motion
- (d)The relation between the simple harmonic motion and periodic motion depends upon the mass of object undergoing the motion
Correct — B, All the simple harmonic motions are necessarily periodic motions. Simple harmonic motion is defined by a restoring force proportional to the displacement and directed back towards the mean position, and that condition forces the displacement to follow a sine or cosine of time. A sine repeats after a fixed interval, so every simple harmonic motion repeats itself at regular intervals and is periodic. The reverse does not hold: a motion can repeat without the force being proportional to displacement.
- (a)All the periodic motions are necessarily simple harmonic — The Earth's orbit repeats every year and the bouncing of a ball repeats too, but neither has a restoring force proportional to displacement, so neither is simple harmonic. Periodic is the wider category.
- (c)There is no co-relation between the simple harmonic motions and the periodicity of motion — There is a definite relation, and it runs one way. Simple harmonic motion sits inside the set of periodic motions as a special case.
- (d)The relation between the simple harmonic motion and periodic motion depends upon the mass of object undergoing the motion — Mass affects the time period — for a spring the period is 2π√(m/k) — but it has no say in whether a motion counts as simple harmonic. That is settled by the form of the force.
Periodic motion is any motion that repeats after a fixed interval. Oscillatory motion is periodic motion that goes to and fro about a mean position. Simple harmonic motion is the special oscillation whose restoring force obeys F = −kx, which makes the displacement a pure sine of time. Each of the three is a subset of the one before it.
The item is a set-inclusion question dressed as physics, and drawing the three nested categories on the margin of the question paper settles it in seconds. Circular motion at constant speed is the cleanest counter-example to option (a) — it repeats every revolution, so it is periodic, but there is no to-and-fro and no restoring force pulling the body back to a mean position. A ball bouncing on a hard floor is another: periodic, oscillatory even, but the force during contact is nothing like proportional to displacement.
- Simple harmonic motion is defined by F = −kx, a restoring force proportional to displacement and directed towards the mean position.
- That condition makes the displacement vary as a sine or cosine of time, which repeats, so every simple harmonic motion is periodic.
- Uniform circular motion is periodic but not simple harmonic; its projection on a diameter, though, is simple harmonic.
- For a spring-mass system the time period is 2π√(m/k); for a simple pendulum making small oscillations it is 2π√(l/g) and does not involve the bob's mass.
- A pendulum stops being simple harmonic once the swing is wide, because sin θ is then no longer close enough to θ.
Every row below the first is also everything above it. That one-way inclusion is the whole question.
- Reversing the inclusion and claiming every periodic motion is simple harmonic.
- Treating uniform circular motion as simple harmonic — its projection is, the motion itself is not.
- Bringing mass into the definition when mass only affects the period, not the classification.
As a definition-boundary item. The examiner reverses a true statement and hopes the reversal reads as familiar.
Which one of the following statements is true for a simple harmonic oscillator?
- (a) Force acting is directly proportional to the displacement from the mean position and is in same direction.
- (b) Force acting is directly proportional to the displacement from the mean position and is in opposite direction.
- (c) Acceleration of the oscillator is constant.
- (d) The velocity of the oscillator is not periodic.
Answer(b) Force acting is directly proportional to the displacement from the mean position and is in opposite direction.
The defining condition itself, which is what makes the inclusion here one-way. A motion has to satisfy that force law to be simple harmonic, and most repeating motions do not.
A particle is executing simple harmonic motion. Which one of the following statements about the acceleration of the oscillating particle is true ?
- (a) It is always in the opposite direction to velocity
- (b) It is proportional to the frequency of oscillation
- (c) It is minimum when the speed is maximum
- (d) It decreases as the potential energy increases
Answer(c) It is minimum when the speed is maximum
Where the sine description leads next. Because acceleration tracks displacement, it falls to zero at the mean position at the very moment the speed peaks.
- practice — not a real PYQ
Which one of the following motions is periodic but NOT simple harmonic?
- (a)A mass oscillating on a light spring
- (b)The bob of a simple pendulum swinging through a small angle
- (c)The hands of a working clock
- (d)A loaded test tube bobbing vertically in water
Answer(c) The hands of a working clock — they repeat every twelve hours, but nothing pulls them back towards a mean position.
- practice — not a real PYQ
In simple harmonic motion, the acceleration of the particle is
- (a)constant in magnitude and direction
- (b)proportional to displacement and directed towards the mean position
- (c)proportional to displacement and directed away from the mean position
- (d)zero at the extreme positions
Answer(b) proportional to displacement and directed towards the mean position — that is the defining condition, written as a = −ω²x.