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.
Correct — B, the force is directly proportional to the displacement from the mean position and acts in the opposite direction. This is the defining property of simple harmonic motion: the restoring force always points back towards the mean (equilibrium) position and grows in proportion to how far the oscillator has been displaced, written F = −kx (the minus sign shows the force opposes the displacement). This restoring force is what pulls the oscillator back and makes the motion repeat.
- (a)Force acting is directly proportional to the displacement from the mean position and is in same direction. — If the force acted in the same direction as the displacement it would push the oscillator further from the mean position, not restore it — the motion would run away rather than oscillate.
- (c)Acceleration of the oscillator is constant. — In SHM the acceleration is a = −ω²x, so it varies continuously with position — zero at the mean, maximum at the extremes — and is not constant.
- (d)The velocity of the oscillator is not periodic. — The velocity of an SHM oscillator repeats every cycle — maximum at the mean position, zero at the extremes — so it is very much periodic.
Simple harmonic motion is oscillation under a restoring force proportional to displacement and directed towards equilibrium: F = −kx. This gives acceleration a = −ω²x and sinusoidal position, velocity and acceleration that all vary periodically with time. A mass on a spring and a small-amplitude pendulum are the standard examples.
The two 'proportional to displacement' options differ only in direction — and the sign is everything. A restoring (opposite-direction) force is the hallmark of SHM; the same-direction version describes an unstable, non-oscillating system.
- Defining condition of SHM: F = −kx (restoring force proportional to displacement, directed towards equilibrium).
- Acceleration a = −ω²x — maximum at the extremes, zero at the mean position.
- Speed is maximum at the mean position and zero at the extreme positions.
- Examples: a mass on a spring; a simple pendulum for small angular displacements.
The restoring force opposes the displacement — option (b).
- Picking the 'same direction' option — the SHM force must oppose the displacement.
- Thinking acceleration is constant; it varies as −ω²x.
Asked as the defining property of SHM, or applied to a pendulum or spring's period, velocity and acceleration.
Consider the following statements: A simple pendulum is set into oscillation. Then I. The acceleration is zero when the bob passes through the mean position. II. In each cycle the bob attains a given velocity twice. III. Both acceleration and velocity of the bob are zero when it reaches its extreme position during its oscillation. IV. The amplitude of oscillation of the simple pendulum decreases with time. Which of these statements are correct?
- (a) I and II
- (b) III and IV
- (c) I, II and IV
- (d) II, III and IV
Answer(c) I, II and IV
Same concept — a simple pendulum is an SHM system; the statements test that acceleration is zero at the mean position (a = −ω²x) and that a real oscillation's amplitude decays, the same F = −kx physics asked here.
Which one of the following statements regarding simple pendulum is correct? Simple pendulum has a time period independent of amplitude:
- (a) only for small amplitudes because then the net force on its bob is independent of its displacement.
- (b) for any amplitude because the net force on the bob is always proportional to its displacement.
- (c) for any amplitude because the net force on the bob is independent of its displacement.
- (d) only for small amplitudes because then the net force on its bob is proportional to its displacement.
Answer(d) only for small amplitudes because then the net force on its bob is proportional to its displacement.
Same concept — a pendulum behaves as an SHM oscillator only for small amplitudes, where the restoring force is proportional to displacement, the exact defining property tested in this question.
NDA_GAT_2022_I_Q702022The time period of a 1 m long pendulum approximates to
- (a) 6 s
- (b) 4 s
- (c) 2 s
- (d) 1 s
Answer(c) 2 s
Related — the time period of a pendulum (an SHM system) depends on its length; recognising the pendulum as SHM rests on the same restoring-force idea.
- practice — not a real PYQ
For a body executing simple harmonic motion, the acceleration is maximum
- (a)at the mean position
- (b)at the extreme positions
- (c)midway between mean and extreme
- (d)everywhere equal
Answer(b) at the extreme positions — where displacement (and hence −ω²x) is greatest.
- practice — not a real PYQ
In simple harmonic motion the restoring force is
- (a)constant
- (b)proportional to displacement and in the same direction
- (c)proportional to displacement and in the opposite direction
- (d)inversely proportional to displacement
Answer(c) proportional to displacement and in the opposite direction — F = −kx.