Simple harmonic motion (SHM) property check: ‘‘The acceleration of a particle executing SHM is inversely proportional to its displacement from the mean position.’’ Is this statement correct?

Introduction / Context

Recognizing the defining proportionality of acceleration in simple harmonic motion is essential for analyzing oscillators (mass–spring systems, pendulums for small angles, etc.). Mixing up proportionalities leads to incorrect phase and frequency predictions.

Given Data / Assumptions

• Undamped, single-degree-of-freedom SHM.
• Displacement x measured from the mean (equilibrium) position.
• Angular natural frequency ω_n constant.

Concept / Approach

In SHM, acceleration is directly proportional to displacement and acts towards the mean position (restoring), i.e., opposite in direction to the displacement: a = −ω_n²x. The ‘‘inversely proportional’’ statement is therefore incorrect.

Step-by-Step Solution

Verification / Alternative check

Solutions x = X sin(ω_n t + φ) give a = −ω_n² X sin(ω_n t + φ), confirming proportionality with negative sign.

Why Other Options Are Wrong

• Yes: contradicts the fundamental SHM law.
• Only near small amplitudes: SHM itself is a small-amplitude linear model; even then the relation is direct, not inverse.
• True only at resonance: resonance concerns forced response; the proportionality law is kinematic and independent of resonance.
• True if damping is present: damping adds a velocity-proportional term; it does not change the proportionality a ∝ −x in the homogeneous part.

Common Pitfalls

• Confusing inverse proportionality with the ‘‘opposite sign’’ (restoring acceleration).
• Mixing up velocity-proportional damping with displacement-proportional stiffness.

Final Answer

No