AP Physics 1 · Unit 7: Oscillations ·  Lesson 7.1

Deep Dive: Defining Simple Harmonic Motion

🔬 Deep Dive
This is your textbook for this topic. Take your time. Read it more than once.
Lesson 7.1: Defining Simple Harmonic Motion
7.1.A.1Concept

SHM Is a Special Case of Periodic Motion

Periodic motion just means motion that repeats itself in a fixed time interval. A satellite orbiting Earth is periodic. A ball bouncing down a hallway is periodic. A metronome ticking is periodic. All periodic — but not all of them are simple harmonic motion.

💡SHM is periodic motion that satisfies one extra rule about the force driving it. "Repeats" gets you into the category of periodic motion. A specific relationship between force and displacement is what gets you into the much smaller category of SHM.

That extra rule is what the rest of this lesson is about.

7.1.A.2.iConcept

The Restoring Force

A restoring force is a force exerted in the direction opposite an object's displacement from its equilibrium position. Push a spring's mass to the right, and the spring pulls it left. Pull a pendulum bob to one side, and gravity pulls it back toward center. In both cases, the force's whole job is to undo the displacement.

🔑The word "opposite" is doing real work here — it's exactly why a minus sign shows up in every version of the SHM force equation you'll see. The restoring force isn't just related to displacement; it's always pointed against it.
7.1.A.2 – 7.1.A.2.iiMath

The SHM Condition & Equilibrium Position

An equilibrium position is the location at which the net force on an object or system is zero. It's the spot the restoring force is always trying to bring the system back to.

Simple harmonic motion results when the magnitude of the restoring force is proportional to the object's displacement from that equilibrium position:

F_restore ∝ −x

This is the general SHM condition — it doesn't commit to any particular system. The most familiar system that satisfies it exactly is an ideal spring, where Hooke's law makes the proportionality explicit:

F = −kx

Here k (the spring constant) is just the proportionality constant — a bigger k means a stiffer spring and a stronger restoring force at any given displacement.

Drag the displacement and watch the restoring force respond. Toggle between the spring and the small-angle pendulum — same proportionality, different variable names.

equilibriumrestoring
Displacement x (m)0.20
Spring constant k (N/m)20.00
Displacement x
0.20
Restoring Force F
-4.00
F = −(20)(0.20) = -4.00 N
displacement →

The line is always straight through the origin — that's what "proportional" looks like.

ExampleWorked Example — Reading the SHM Condition from Data

A student pulls a spring-mass system to different displacements and measures the restoring force at each: x = 0.10 m, F = 2.0 N; x = 0.20 m, F = 4.0 N; x = 0.30 m, F = 6.0 N. Show that this system satisfies the SHM condition and find the spring constant k.

7.1.A.2.iiiConcept

The Small-Angle Pendulum Model

A pendulum doesn't obviously look like a spring — there's no x, no k, nothing labeled "Hooke's law." But swing it through a small angle and the same SHM condition reappears, just written in terms of rotation: the restoring torque is proportional to the angular displacement.

⚠️This only works for small angular displacements — the model breaks down as the swing angle gets large, because the actual restoring torque depends on sin θ, and sin θ only stays proportional to θ itself when θ is small (roughly under 15°–20°). A pendulum swung through 60° is still periodic — it's just no longer well-modeled as SHM.
ExampleGuided Example — Is It SHM?

Four systems: (1) a block on a spring following Hooke's law F = −kx, (2) a basketball bouncing on a hard floor, (3) a pendulum swinging through a 5° angle, and (4) a child on a swing pushed to a large 60° angle. Which of these exhibit simple harmonic motion?

Step 1 — Apply the SHM test
For each system, ask the same question: is the magnitude of the restoring force (or torque) proportional to the displacement from equilibrium?
← Back to Lesson 7.1Next: Lesson 7.2 →Frequency and Period of SHM — how long does one cycle take?