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Simple Harmonic Motion

Simple Harmonic Motion
SHM

Any kind of periodic motion that can be modeled using a sinusoid. That is, motion that can be approximately or exactly described using a sine or cosine function. Examples include the swinging back and forth of a pendulum and the bobbing up and down of a mass hanging from a spring.

 

A pendulum diagram showing a fixed wall mount (labeled "pendulum") with a string hanging down ending in a circular bob.

 

See also

Period of periodic motion, frequency of periodic motion, periodic function

Key Formula

x(t)=Acos⁡(ωt+ϕ)x(t) = A \cos(\omega t + \phi)
Where:
  • x(t)x(t) = Position of the object at time t
  • AA = Amplitude — the maximum displacement from the equilibrium position
  • ω\omega = Angular frequency in radians per second, where ω = 2π/T
  • tt = Time
  • ϕ\phi = Phase shift — determines where in the cycle the motion begins at t = 0
  • TT = Period — the time for one complete oscillation

Worked Example

Problem: A mass on a spring oscillates in simple harmonic motion with an amplitude of 6 cm and a period of 2 seconds. If the mass starts at its maximum displacement (x = 6 cm) at time t = 0, find its position at t = 0.5 s and t = 1 s.
Step 1: Write the general equation. Since the mass starts at maximum displacement and we use cosine, the phase shift is 0.
x(t)=Acos⁡(ωt)x(t) = A\cos(\omega t)
Step 2: Calculate the angular frequency from the period T = 2 s.
ω=2πT=2π2=π rad/s\omega = \frac{2\pi}{T} = \frac{2\pi}{2} = \pi \text{ rad/s}
Step 3: Substitute A = 6 cm and ω = π to get the specific equation.
x(t)=6cos⁡(πt)x(t) = 6\cos(\pi t)
Step 4: Find the position at t = 0.5 s.
x(0.5)=6cos⁡(π⋅0.5)=6cos⁡ ⁣(π2)=6⋅0=0 cmx(0.5) = 6\cos(\pi \cdot 0.5) = 6\cos\!\left(\frac{\pi}{2}\right) = 6 \cdot 0 = 0 \text{ cm}
Step 5: Find the position at t = 1 s.
x(1)=6cos⁡(π⋅1)=6cos⁡(π)=6⋅(−1)=−6 cmx(1) = 6\cos(\pi \cdot 1) = 6\cos(\pi) = 6 \cdot (-1) = -6 \text{ cm}
Answer: At t = 0.5 s, the mass is at the equilibrium position (x = 0 cm). At t = 1 s, the mass is at x = −6 cm, its maximum displacement in the opposite direction. This makes sense: after half a period, the object has moved to the other extreme.

Frequently Asked Questions

What makes motion 'simple' harmonic instead of just harmonic?
The word 'simple' means the motion involves only a single sinusoidal component — one frequency, one amplitude. More complex periodic motions can be built by adding multiple sinusoids together (as in a Fourier series), and those are called harmonic but not simple harmonic. SHM is the purest, most basic form of oscillation.
Is a pendulum truly simple harmonic motion?
A pendulum is only approximately SHM. The approximation works well when the swing angle is small (roughly less than about 15°), because for small angles the restoring force is nearly proportional to displacement. For large swings, the relationship breaks down and the motion is periodic but no longer sinusoidal.

Simple Harmonic Motion vs. Periodic Motion

Periodic motion is any motion that repeats — it could follow any shape of repeating pattern. Simple harmonic motion is a specific type of periodic motion where the pattern is a pure sinusoid. For example, the orbit of a planet is periodic but not simple harmonic, while a mass bobbing on a spring is both periodic and simple harmonic.

Why It Matters

Simple harmonic motion is one of the most fundamental patterns in science and engineering. It describes the behavior of springs, pendulums, sound waves, alternating electrical circuits, and even the vibrations of atoms in a molecule. Understanding SHM gives you a foundation for analyzing any system where something oscillates around a stable point, making it central to physics, music, and signal processing.

Common Mistakes

Mistake: Confusing amplitude with period — thinking a larger amplitude means a longer period.
Correction: In true SHM, the period does not depend on the amplitude. A mass on a spring takes the same time to complete one cycle whether it is displaced 1 cm or 10 cm. Amplitude controls how far the object moves, while period controls how fast it oscillates.
Mistake: Forgetting to convert between angular frequency (ω) and ordinary frequency (f) or period (T).
Correction: Remember the relationships: ω = 2πf and ω = 2π/T. Angular frequency is measured in radians per second, while ordinary frequency f is in cycles per second (hertz). Mixing these up leads to answers that are off by a factor of 2π.

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