7.1 Defining Simple Harmonic Motion (SHM)

Before the Fall 2024 course update, oscillations were Unit 6. They're now Unit 7, so older videos and practice may call this unit "Unit 6."

Periodic motion is any motion that repeats itself in equal time intervals: a swing, a heartbeat, a planet's orbit. Simple harmonic motion is a special case. In SHM, a graph of position against time is a smooth sine or cosine curve.

Not every repeating motion is SHM. A ball bouncing on the floor repeats, but its position–time graph is a series of parabolas, not a sine curve.

The equilibrium position is where the net force on the object is zero. If the object sits there at rest, it stays there.

A restoring force is a force that points back toward equilibrium whenever the object is displaced. Pull a mass on a spring to the right and the spring pulls left. Push it to the left and the spring pushes right.

Why doesn't the object just stop at equilibrium? At the turning points, the restoring force is biggest and the object is momentarily at rest. As it heads back, it speeds up. When it reaches equilibrium, the net force is zero, but it's moving at top speed, so its inertia carries it past. On the other side, the restoring force slows it down until it stops and turns around again.

You get SHM when the restoring force is proportional to the displacement from equilibrium. With x measured from equilibrium, $$F = -kx$$ The minus sign says the force points opposite the displacement. This is Hooke's law for a spring, where k is the spring constant in N/m.

So a quick test for SHM is to write the net force (or the acceleration) as a function of displacement. If it has the form $a = -(\text{positive constant})\,x$, the motion is SHM.

An energy test works too. A force $F = -kx$ comes from a potential energy $U = \tfrac{1}{2}kx^2$. So if U(x) is a parabola around its minimum, the motion is SHM. Forces like $-cx^3$ are restoring forces, but they aren't proportional to x, so the motion repeats without being SHM.

Hang a mass m on a vertical spring and it stretches until the spring force balances gravity: $k\,\Delta y_0 = mg$. That's the new equilibrium position.

Measure displacement y from this new equilibrium. Then the net force is just −ky: gravity's pull is canceled by the extra stretch at equilibrium. So a vertical spring–mass system does SHM about its stretched equilibrium point, with the same k and the same period as it would have horizontally.