Why Is There a Minus Sign in the SHM Equation?

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This condition comes from a restoring force. For a mass attached to a spring, Hooke’s law gives $F=-kx$. Combining this with Newton’s second law, $F=ma$, gives $ma=-kx$, so $a=-(k/m)x$. Comparing this with $a=-\omega^2x$ shows that $\omega^2=k/m$. The minus sign in both equations has the same physical meaning: the force and acceleration oppose displacement. This is the mechanism that produces oscillation rather than motion away from equilibrium.

The factor $\omega^2$ is always positive, so the minus sign determines the direction of acceleration:

For example, if $x=+0.040\,\text{m}$ and $\omega=5.0\,\text{rad s}^{-1}$, then

The negative acceleration points back toward the equilibrium position. If the displacement were $-0.040\,\text{m}$, the acceleration would be $+1.0\,\text{m s}^{-2}$, again pointing toward equilibrium.

This relationship also means that the magnitude of acceleration is proportional to the magnitude of displacement: $|a|=\omega^2|x|$. Acceleration is greatest at the extreme positions and zero at equilibrium. On an acceleration-displacement graph, the gradient is negative and equal to $-\omega^2$.

A common misconception is that the minus sign means the object is always slowing down. This is incorrect. The sign indicates direction, not whether speed is decreasing. An oscillator speeds up while moving toward equilibrium and slows down while moving away from it.

Exam technique: For an IB Explain question, state both required ideas: acceleration is proportional to displacement, and acceleration is opposite to displacement, so it is directed toward equilibrium. Do not describe the minus sign merely as “negative acceleration.”