Which one of the following four particlcs, whose displacement .t and acceleration ax are related as follows, is executing simple harmonic motion ?

Simple Harmonic Motion (SHM) is defined by a specific relationship where the acceleration of a particle is directly proportional to its displacement from the equilibrium position and is always directed towards that position. Mathematically, this is expressed as a = -ω²x, where 'a' is acceleration, 'x' is displacement, and ω² is a positive constant representing the square of the angular frequency. The negative sign indicates that the acceleration (and the restoring force) acts in the opposite direction to the displacement [1]. Among the given options, only 'ax = -ix' (where 'i' represents a positive constant) satisfies this linear restoring condition. Other options involving positive displacement (ax = +3x) or time-dependent variables (ax = 3 - t²) do not represent the characteristic restoring force required for SHM.

Sources

1. [1] https://kparker.bg-research.cc.ic.ac.uk/homepage/Mechanics%20of%20the%20Circulation/Chap_08/_Chapter_08.htm