AP Physics C: Mechanics · Unit 7 · Topic 1 of 5 · 6 min lesson

SHM Equations

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

The big idea

Any object pulled back by a force proportional to its displacement, F = −kx, obeys d²x/dt² = −ω²x. The solution is x = A cos(ωt + φ): A is the amplitude, ω = √(k/m) is the angular frequency, and the phase φ is set by where the motion starts.

1. From force to motion

ma = −kxNewton’s second law

d²x/dt² = −ω²xwith ω² = k/m

x = A cos(ωt + φ)general solution

v = −Aω sin(ωt + φ)differentiate once

a = −Aω² cos(ωt + φ) = −ω²xdifferentiate twice

The phase φ shifts the curve sideways. Here φ = π/3 makes it start at A/2 heading toward −x.

2. Reading the equations

Whenever you can write a = −(something)x, that something is ω², and the motion is SHM with T = 2π/ω.

Maximum speed v_max = Aω at x = 0; maximum acceleration a_max = Aω² at x = ±A. So ω = a_max/v_max.

Finding φ: plug in t = 0. x(0) = A cos φ and v(0) = −Aω sin φ. Starting at +A gives φ = 0; at x = 0 moving −x, φ = π/2; at x = 0 moving +x, φ = −π/2.

Forces like F = −kx³ aren’t SHM: the restoring force must be linear in x.

3. Example, then try it

Example

An object moves with x(t) = 0.20 cos(5t) (SI units). Find its angular frequency, period, maximum speed and maximum acceleration.

ω = 5 rad/s, so T = 2π/5 ≈ 1.26 s.

v_max = Aω = 1.0 m/s; a_max = Aω² = 5.0 m/s².

Try it · work it on paper first

An object obeys d²x/dt² = −25x (SI units). At t = 0 it is at x = 0 moving at 0.50 m/s in the −x direction. Find ω, the period, the amplitude, and φ, and write x(t).

ω: ω² = 25, so ω = 5.0 rad/s and T = 2π/5.0 ≈ 1.26 s.

Amplitude: it passes x = 0 at top speed, so A = v_max/ω = 0.50 ÷ 5.0 = 0.10 m.

Phase: x(0) = A cos φ = 0 → φ = ±π/2. v(0) = −Aω sin φ = −0.50 → sin φ = +1, so φ = π/2.

Result: x(t) = 0.10 cos(5t + π/2), which is the same as −0.10 sin(5t).

Answer: ω = 5.0 rad/s, T ≈ 1.26 s, A = 0.10 m, φ = π/2.

4. Common mistakes

Mixing up ω and f.ω = 2πf, in rad/s; f is in Hz.

Forgetting radians.Your calculator must be in radian mode for cos(ωt).

Picking the wrong sign of φ.Check both x(0) and the direction of v(0).

Check yourself

An oscillator has a maximum speed of 2.0 m/s and a maximum acceleration of 8.0 m/s². What is its angular frequency?

a_max/v_max = Aω²/(Aω) = ω = 8.0 ÷ 2.0 = 4.0 rad/s.

Ready to practice?

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

Quiz this topic All of Unit 7