Simple Harmonic Motion (SHM) – Acceleration–Displacement Relation Is the acceleration of a particle executing SHM proportional to its displacement from the mean position (with appropriate sign)?

Introduction / Context:
Simple harmonic motion (SHM) is a foundational model for vibrations of mechanical systems, oscillations of pendulums (small angles), and electrical LC circuits. A hallmark of SHM is a specific relationship between acceleration and displacement.

Given Data / Assumptions:

• Motion is undamped and unforced.
• Displacement x is measured from the mean (equilibrium) position.
• Angular frequency is ω (a constant for a given system).

Concept / Approach:
The defining equation of SHM is a = -ω^2 x. This means the acceleration magnitude is proportional to the displacement magnitude, and the negative sign indicates it is always directed toward the mean position (restoring). Thus the statement “acceleration is proportional to displacement” is correct, with the implied opposite direction.

Step-by-Step Solution:
Write the SHM differential equation: d^2x/dt^2 + ω^2 x = 0. Rearrange to acceleration form: a = d^2x/dt^2 = -ω^2 x. Identify proportionality: |a| ∝ |x| with a restoring (negative) sense. Therefore, the statement is true, acknowledging the restoring sign.

Verification / Alternative check:
From x(t) = A cos(ω t + φ), differentiate twice: v = -A ω sin(ω t + φ), a = -A ω^2 cos(ω t + φ) = -ω^2 x. The relationship holds for all t.

Why Other Options Are Wrong:
False: contradicts the defining equation of SHM. “Only for large amplitudes”: SHM model strictly assumes small amplitudes where linear restoring force applies. Damping or circular motion conditions are not required for the basic SHM proportionality.

Common Pitfalls:
Forgetting the negative sign and calling it “proportional in the same direction”. Applying SHM to systems with non-linear restoring forces where proportionality fails.

Final Answer:
True