What makes it simple harmonic

Simple harmonic motion is the back-and-forth motion of a mass on a spring, a pendulum at small angles, and many other systems. What ties them together is not that they repeat, but a single defining rule: the acceleration is proportional to the displacement from equilibrium and directed opposite to it.

In symbols this is the second-order relationship $a=\dfrac{d^2x}{dt^2}=-\omega^2 x$, equivalently a restoring force $F=-kx$ with $\omega^2=k/m$. This is the first second-order differential equation in the course, so it is worth saying in words: the second derivative of position is proportional to the negative of position.

Three habits carry the topic: keep the minus sign, so the acceleration restores rather than repels; remember the acceleration is not constant but largest at the turning points and zero at the center; and require the force to be linear in $x$, since a restoring force alone does not make SHM.

The defining equation $a=-\omega^2 x$ packs in two requirements. First, the acceleration is proportional to the displacement: the farther the mass sits from equilibrium, the harder it is pulled back. Second, the minus sign makes the acceleration point opposite to the displacement, always back toward equilibrium.

That minus sign is the whole difference between oscillation and runaway. With $a=-\omega^2 x$ the mass is always pulled back, so it overshoots, returns, and repeats. With $a=+\omega^2 x$ the mass is pushed away from equilibrium and its displacement grows without bound. On a graph of acceleration against position, SHM is a straight line through the origin with negative slope $-\omega^2$.

The defining relation is $a=-\omega^2 x$: acceleration proportional to displacement and opposite in direction. The minus sign is what makes the motion oscillate.

Because $a=-\omega^2 x$ depends on position, the acceleration changes throughout the motion. At the turning points, where $|x|=A$, the acceleration reaches its largest magnitude $\omega^2 A$ and points back toward the center. At equilibrium, where $x=0$, the acceleration is exactly zero, even though the speed is greatest there.

This is why the constant-acceleration kinematics equations do not apply. A formula such as $v^2=v_0^2+2a\,\Delta x$ assumes a single fixed $a$, but in SHM there is no such value. Treating an oscillator like a uniformly accelerated body, or reaching for $x=\tfrac{1}{2}at^2$, is a common and costly error; the speed at equilibrium comes from energy instead.

The acceleration varies with position: largest in magnitude, $\omega^2 A$, at the turning points and zero at equilibrium. Constant-acceleration formulas do not apply.

A force that points back toward equilibrium is restoring, but that alone does not make simple harmonic motion. The force must also be linear in the displacement, $F=-kx$. Only a linear restoring force produces $a=-\omega^2 x$ and the smooth sinusoidal motion that defines a simple harmonic oscillator.

Many motions repeat without being simple harmonic. A ball bouncing on the floor feels a constant weight while in flight, not a force proportional to its height. A pendulum is simple harmonic only at small angles, where the restoring torque, proportional to $\sin\theta$, is nearly linear in $\theta$; at large angles that nonlinearity breaks it. Periodic is not the same as simple harmonic.

SHM requires a force linear in displacement, $F=-kx$. A restoring force that is constant or nonlinear gives periodic motion that is not simple harmonic.

To decide whether a motion is simple harmonic, apply one test: is the acceleration proportional to the displacement and opposite to it? If yes, the motion is SHM and obeys $a=-\omega^2 x$. If not, it may still be periodic, but it is not simple harmonic.

The three traps line up against the three parts of the definition. Dropping the minus sign fails the direction requirement. Treating the acceleration as constant ignores that it must vary with position. Accepting any restoring force ignores the linearity requirement. Get all three right and the motion is genuinely simple harmonic.

One test defines SHM: the acceleration is proportional to and opposite the displacement, $a=-\omega^2 x$. Repetition by itself is not enough.

Deciding whether a motion is truly simple harmonic tends to fail in three specific ways.

• Dropped signWriting the relation without its minus sign, so the acceleration points the same way as the displacement. Fix: $a=-\omega^2 x$, restoring and opposite to $x$.
• Constant accelerationTreating the acceleration as fixed and reaching for constant-acceleration kinematics. Fix: $a=-\omega^2 x$ varies with position, zero at the center and largest at the ends.
• Any restoring forceAssuming any back-toward-center force is SHM. Fix: the force must be linear, $F=-kx$; constant or nonlinear restoring forces are not simple harmonic.

Keep the minus sign, remember the acceleration is not constant, and require a linear restoring force. The whole topic comes down to those three habits.