The SHM Differential Equation

You’ll be able to

Recognize simple harmonic motion from the equation d²x/dt² = −ω²x
Verify that x(t) = A cos(ωt + φ) satisfies the SHM equation
Identify the amplitude, angular frequency, period, and phase of an oscillation

What makes motion simple harmonic

Simple harmonic motion (SHM) arises whenever the restoring force is proportional to the displacement and points back toward equilibrium: F = −kx. Newton's second law then reads m d²x/dt² = −kx, or d²x/dt² = −ω²x with ω² = k/m. This differential equation — acceleration proportional to the negative of position — is the definition of SHM. Any system that reduces to it oscillates sinusoidally.

The SHM equation and its solution

d²x/dt² = −ω²x x(t) = A cos(ωt + φ)

A is the amplitude, ω the angular frequency, φ the phase constant set by initial conditions. The period is T = 2π/ω.

Verifying the sinusoidal solution

Take x(t) = A cos(ωt + φ) and differentiate: the velocity is v = −Aω sin(ωt + φ), and the acceleration is a = −Aω² cos(ωt + φ) = −ω²x. The acceleration comes back proportional to −x, exactly satisfying the SHM equation — proof that a cosine (or sine) is the motion. The period T = 2π/ω is the time for one full cycle, and the frequency f = 1/T = ω/2π counts cycles per second.

Period, frequency, and angular frequency

T = 2π/ω f = 1/T ω = 2πf

ω is in rad/s, f in hertz, T in seconds. Together they describe how fast the oscillation repeats, independent of amplitude.

Worked example

Verify that x(t) = A cos(ωt) satisfies d²x/dt² = −ω²x, and state the period. For a system with ω = 10 rad/s, give the numerical period.

1.Differentiate once: v(t) = dx/dt = −Aω sin(ωt).
2.Differentiate again: a(t) = d²x/dt² = −Aω² cos(ωt).
3.Recognize A cos(ωt) = x, so a = −ω²x — the SHM equation is satisfied.
4.The period is T = 2π/ω = 2π/10 ≈ 0.63 s.

Answer: a = −Aω² cos(ωt) = −ω²x, confirming SHM; with ω = 10 rad/s, T = 2π/ω ≈ 0.63 s

Checkpoint

Which equation of motion defines simple harmonic motion?

Checkpoint

A system oscillates with x(t) = A cos(ωt). What is its acceleration as a function of time?

Tip

To confirm a motion is SHM, differentiate the position twice and check that the acceleration comes back as −ω² times the position. The coefficient of x in the equation of motion is ω², so you can read the angular frequency straight off the differential equation.

Answer the 2 checkpoints as you read.