A body is said to move with simple harmonic motion if its acceleration is:
Try it first: pick an option, then read the explanation.
Asked in SSC JE 03/03/2017 (Morning)
Detailed Explanation
Simple harmonic motion is the to-and-fro motion of a mass on a spring, or a simple pendulum swinging through small angles. A restoring force always pulls the body back towards its mean position.
\[ a = -\omega^2 x \]
Here \(x\) is the displacement from the mean position and \(\omega\) is the angular frequency. The minus sign shows that the acceleration is opposite to the displacement, i.e. towards the mean position. Period \(T = 2\pi/\omega\).
Why “proportional to the distance from the point of reference and directed towards it” Is Correct
The defining condition of SHM is that the acceleration is directly proportional to the distance from the mean (reference) position and always directed towards it. That is option (c).
| Position | Displacement | Velocity | Acceleration |
|---|---|---|---|
| Mean position | 0 | Maximum (ωA) | 0 |
| Extreme position | ±A (amplitude) | 0 | Maximum (ω²A) |
Why the Other Options Are Wrong
- A. always directed away from the centre, at the point of reference: Acceleration directed away from the centre would push the body further out — not oscillation.
- B. proportional to the square of the distance from the point of reference: SHM needs acceleration proportional to the distance itself, not its square.
- D. None of these: Option (c) is the correct definition.
Key Point
SHM: a = −ω²x (proportional to displacement, towards mean position).
Key Concepts for Students
- Velocity is maximum at the mean position and zero at the extremes.
- Period of a simple pendulum: T = 2π√(L/g).
- Period of a mass–spring system: T = 2π√(m/k).
Practice more: this is question 9 in Force Analysis MCQs – Part 1. Try the whole set and check every answer instantly.