Simple harmonic motion
The idea
Many systems vibrate around a resting point — a mass bobbing on a spring, a swaying bridge, a plucked guitar string. Simple harmonic motion (SHM) is the cleanest version: it occurs whenever the restoring force is proportional to the displacement and points back toward equilibrium, as in Hooke's law F = −kx. The motion repeats with period T (seconds per cycle) and frequency f = 1/T (cycles per second), and its width is the amplitude A, the maximum distance from equilibrium.
The defining surprise of SHM is that the period does not depend on the amplitude. For a mass on a spring, T = 2π√(m/k): a stiffer spring means a quicker bounce, more mass means a slower one, and a bigger pull-back changes nothing about the timing. Energy sloshes back and forth — all spring potential energy (½kx²) at the turning points, all kinetic at the center — so the speed peaks at equilibrium and falls to zero at the extremes.
A frequent misconception is that the mass moves fastest where the force is biggest. It is exactly backwards: at maximum displacement the force and acceleration peak while the speed is momentarily zero, and at equilibrium the force vanishes while the speed is greatest. Force tells you how velocity is changing, not how large it is.
Worked example
A 0.50 kg block on a frictionless surface is attached to a spring of stiffness k = 200 N/m, pulled 0.10 m from equilibrium, and released. Find the period of the oscillation and the block's maximum speed.
- Identify the motion as SHM: the spring supplies a restoring force proportional to displacement, so the formulas for a mass-spring oscillator apply with m = 0.50 kg and k = 200 N/m.
- Compute the period: T = 2π√(m/k) = 2π√(0.50/200) = 2π√0.0025 = 2π × 0.050 ≈ 0.31 s, which is a frequency of about 3.2 Hz.
- For the maximum speed, use energy: at release all energy is spring potential, E = ½kA² = ½ × 200 × 0.10² = 1.0 J.
- At equilibrium all of that energy is kinetic: ½mv² = 1.0 J, so v = √(2 × 1.0/0.50) = √4.0 = 2.0 m/s.
- Sanity-check with the angular frequency route: ω = √(k/m) = √400 = 20 rad/s, and the maximum speed of SHM is ωA = 20 × 0.10 = 2.0 m/s — the two methods agree.
- Note what the amplitude did and did not control: pulling the block farther would raise the top speed but leave the 0.31 s period untouched.
Answer. The block oscillates with a period of about 0.31 s and reaches a maximum speed of 2.0 m/s as it passes through equilibrium.
Check your understanding
- Why does pulling the block back twice as far leave the period unchanged, even though it must now travel farther each cycle?
- Where in the cycle are the acceleration and the speed each at their maximum, and why are those two locations different?
- How does the energy picture of SHM explain why the block can never drift beyond its starting amplitude on a frictionless surface?
- What real-world effects make a child's swing or a car suspension deviate from ideal simple harmonic motion?
Build the foundations first
Simple harmonic motion builds on these concepts. If any feel shaky, start there.