AP Physics 1 · Unit 7 · Topic 1 of 5 · 5 min lesson

SHM Characteristics

A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.

Quiz this topic · 5 questions Watch the review video ↗

The big idea

Simple harmonic motion (SHM) happens when a restoring force pulls an object back toward equilibrium, and the force is proportional to the displacement: F = −kx. Twice as far out means twice the pull back. Masses on springs and small-angle pendulums both do this.

1. Where things are biggest

At the ends, the object stops for an instant but the force is largest. At the middle, the force is zero but the speed is largest.

F = −kxrestoring force; the minus means “back toward equilibrium”

a = F ÷ mso a is also proportional to x

At x = ±Av = 0; |F| and |a| largest

At x = 0F = 0 and a = 0; speed largest

2. What it looks like

The object speeds up as it heads toward equilibrium (force and velocity point the same way) and slows down as it moves away (they point opposite ways).

It overshoots equilibrium because it’s moving fastest there. Nothing stops it at the middle: the force there is zero.

A position–time graph of SHM is a sine or cosine curve. Free fall (constant force) and a car braking (constant force) aren’t SHM: their forces don’t grow with distance from an equilibrium point.

3. Example, then try it

Example

A spring (k = 40 N/m) holds a 0.50 kg block. The block is 0.050 m from equilibrium. Find the restoring force and the acceleration. What if it’s twice as far out?

F = kx = (40)(0.050) = 2.0 N toward equilibrium; a = F ÷ m = 2.0 ÷ 0.50 = 4.0 m/s².

At 0.10 m, the force and acceleration double: 4.0 N and 8.0 m/s².

Try it · work it on paper first

A 0.20 kg block on a spring (k = 50 N/m) oscillates with amplitude 0.10 m. Find the largest force and acceleration and where they occur. What are the force and acceleration when the block is 0.04 m from equilibrium?

Largest values are at the ends, x = ±0.10 m: F = kA = (50)(0.10) = 5.0 N.

Largest acceleration: a = 5.0 ÷ 0.20 = 25 m/s², pointing toward equilibrium.

At 0.04 m: F = (50)(0.04) = 2.0 N, a = 2.0 ÷ 0.20 = 10 m/s².

At equilibrium: F = 0 and a = 0, but the speed is largest.

Answer: 5.0 N and 25 m/s² at the ends; 2.0 N and 10 m/s² at 0.04 m.

4. Common mistakes

“Speed is largest at the ends.”At the ends the object stops for an instant. Speed is largest at equilibrium.

“At rest means no acceleration.”At the ends v = 0, but the force, and so the acceleration, is largest.

“The force points the way it’s moving.”The restoring force always points toward equilibrium, even while the object moves away.

Check yourself

A mass on a spring is at its maximum displacement. Which describes its velocity and acceleration at that instant?

At the turning point the mass stops for an instant, so v = 0. It is as far from equilibrium as it gets, so the restoring force and acceleration are largest there.

Ready to practice?

5 questions on SHM Characteristics, each with a hint and an explanation for every wrong answer.

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