Defining characteristic of simple harmonic motion (SHM) In simple harmonic motion, the instantaneous acceleration of the particle is directly proportional to which quantity?

Introduction / Context:
Simple harmonic motion (SHM) underlies oscillations of springs, pendulums (small angles), and many vibration problems. Its defining feature links acceleration directly to displacement and fixes the motion’s sinusoidal nature.

Given Data / Assumptions:

• One-dimensional motion about a stable equilibrium (mean position).
• Linear restoring mechanism (e.g., Hooke’s law spring).
• No damping or external forcing.

Concept / Approach:

For SHM, acceleration a is proportional to displacement x and is directed towards the equilibrium: a = −ω^2 x, where ω is the angular frequency. The negative sign indicates the restoring (opposite) direction, causing oscillatory motion.

Step-by-Step Solution:

Verification / Alternative check:

Solution x(t) = A sin(ω t + φ) differentiates to a(t) = −ω^2 x(t), confirming proportionality and phase opposition.

Why Other Options Are Wrong:

(a) 'Rate of change of velocity' is acceleration itself, not a proportionality statement; (c) would imply linear drag-like behavior; (d) lacks magnitude information; (e) contradicts the SHM definition.

Common Pitfalls:

Omitting the direction (restoring sense); confusing SHM with uniform circular motion’s projection without noting the acceleration law.

Final Answer:

Displacement from the mean position (directed towards it)