Simple harmonic motion
● VERBATIM — examined word-for-wordMotion in which the acceleration is directly proportional to the displacement from a fixed point and is always directed towards that point (opposite in direction to the displacement). Both clauses required — proportionality AND direction.
a = −ω²xWhat this actually means
Two clauses, both compulsory: acceleration proportional to displacement, AND always directed towards the fixed point. The equation a = −ω²x says both at once; the minus sign carries the direction clause.
Proportionality alone is not enough: a ball rolling off a dome also has acceleration growing with displacement, but pointing AWAY. The direction clause is what pulls the motion back and makes it oscillate.
When asked to 'show SHM': reduce the net force to F = −(constant)x, state both clauses in words, then read off ω² = constant/m.
Quoting a ∝ x without 'always directed towards the fixed point' — half the definition, half the marks.
Prove it — watch it be true
- Open the a–x tester on the true SHM candidate: the points fall on a straight line through the origin
- The gradient is negative, so a always points back at the fixed point — that is the second clause, and the line's straightness is the first
- Switch to the a ∝ x³ and a ∝ sign(x) candidates: both leave the straight line, so neither is SHM even though both still point inward