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?

Difficulty: Easy

Correct Answer: No

Explanation:


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 ωn 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 = −ωn2x. The ‘‘inversely proportional’’ statement is therefore incorrect.


Step-by-Step Solution

1) Write the SHM equation of motion: m ẍ + k x = 0 → ẍ + (k/m) x = 0.2) Identify ωn2 = k/m; hence acceleration ẍ = −ωn2 x.3) The magnitude |a| = ωn2|x| shows direct proportionality in magnitude and a 180° phase difference in sign.


Verification / Alternative check

Solutions x = X sin(ωn t + φ) give a = −ωn2 X sin(ωn 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

More Questions from Theory of machines

Discussion & Comments

No comments yet. Be the first to comment!
Join Discussion