Simple Harmonic Motion

Question

A particle moves such that its acceleration is always directed toward a fixed point and is proportional to its distance from that point. Which characteristic best identifies this motion as simple harmonic?

RankGuru · Accepted Answer

Simple harmonic motion is defined by the dynamical condition $a = -\omega^2 x$, meaning the acceleration is directly proportional to the displacement from the mean position and always points back toward that position. The negative sign encodes the restoring nature of the force, which is what makes the motion oscillatory rather than runaway. This single relation guarantees a sinusoidal solution $x = A\sin(\omega t + \phi)$ for the position with time. The option claiming constant acceleration describes uniformly accelerated motion, not oscillation, since SHM acceleration clearly varies with position. The option stating velocity is maximum at the extremes is wrong because speed is actually zero at the extremes and maximum at the mean position where displacement vanishes. The option with acceleration proportional to the square of displacement violates the strict linear proportionality that defines SHM and would not yield sinusoidal motion. This matches the NCERT formulation of SHM as motion under a linear restoring force. As a consistency check, dimensional analysis confirms $\omega^2$ has units of $	ext{s}^{-2}$, so $\omega^2 x$ correctly carries the dimensions of acceleration.

Simple Harmonic Motion

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