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Mechanics · Quick Reference
Simple Harmonic Motion Cheat Sheet
In one line: simple harmonic motion (SHM) is the most fundamental type of oscillation, where the restoring force is directly proportional to displacement — producing smooth, predictable sine-wave motion.
Key Ideas
1Defining Condition for SHM. The restoring force (and acceleration) is directly proportional to displacement from equilibrium, and always points opposite to that displacement: F = -k*x.
2Mass on a Spring. A classic SHM example, governed by Hooke's Law F=-kx, where k is the spring constant; the period is T = 2*pi*sqrt(m/k).
3Simple Pendulum (Small Angles). For small swing angles, a pendulum approximates SHM with period T = 2*pi*sqrt(L/g), where L is the pendulum's length.
4Position, Velocity, and Acceleration in SHM. Position varies as a sine or cosine wave over time; velocity and acceleration are also sinusoidal, but shifted in phase relative to position.
5Energy in SHM. Total mechanical energy stays constant, continuously converting between kinetic energy (maximum at equilibrium) and potential energy (maximum at the extremes of motion).
Worked Examples
A 2 kg mass is attached to a spring with spring constant k=8 N/m. Find the period of oscillation.
T = pi ≈ 3.14 seconds
Find the period of a simple pendulum with length 1m (g=9.8 m/s^2).
T ≈ 2.0 seconds
In SHM, where is kinetic energy at its maximum, and where is potential energy at its maximum?
KE is maximum at equilibrium; PE is maximum at the extremes of motion
Formulas
F = -k*x (Hooke's Law)
Period of mass-spring system: T = 2*pi*sqrt(m/k)
Period of simple pendulum: T = 2*pi*sqrt(L/g)
Practice Yourself
A 1 kg mass on a spring with k=4 N/m. Find the period.
T=2*pi*sqrt(1/4)=2*pi*0.5=pi≈3.14s
Find the period of a pendulum with length 4m (g=9.8).
T=2*pi*sqrt(4/9.8)≈4.01s
What is the defining force equation for SHM?
F = -k*x