Defining simple harmonic motion (SHM) via acceleration A body is said to perform simple harmonic motion (SHM) when which of the following conditions on its acceleration is satisfied?

Introduction / Context:
Simple harmonic motion models many oscillatory systems near equilibrium: mass–spring, small-angle pendulum, and torsional oscillators. The defining feature is the linear restoring action that pulls the system back toward equilibrium.

Given Data / Assumptions:

• Small displacements so linearization is valid.
• Negligible damping and forcing.
• Motion measured from equilibrium position.

Concept / Approach:

The kinematic definition of SHM is that acceleration a is proportional to displacement x and directed toward equilibrium: a = −ω² x, where ω is the natural angular frequency. This sign convention (negative) ensures acceleration opposes the displacement, providing restoring behavior and sinusoidal solutions.

Step-by-Step Solution:

Verification / Alternative check:

Small-angle pendulum: restoring torque ≈ −m g L θ, equation becomes θ̈ + (g/L) θ = 0, matching SHM form with ω = √(g/L).

Why Other Options Are Wrong:

(a) Direction away from equilibrium is antirestoring. (b) Square of displacement gives strongly nonlinear behavior, not SHM. (d) Inverse dependence is not SHM. (e) is false because (c) is correct.

Common Pitfalls:

Confusing direction of acceleration; overlooking that proportionality constant must be positive in magnitude with a negative sign in the equation.

Final Answer:

Acceleration is proportional to the displacement and directed towards the reference (restoring).