Everything you need to master Unit 7 — simple harmonic motion, period and frequency, sinusoidal motion, and energy of oscillators. Mass-spring systems and pendulums are the canonical examples.
5–8% of the AP exam
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Unit 7 is about oscillations — anything that moves back and forth in a regular, repeating pattern. Springs bobbing up and down, pendulums swinging, a kid on a swing set — they all fit the same mathematical template called Simple Harmonic Motion (SHM). SHM happens whenever a restoring force is proportional to displacement: pull farther, get pushed back harder.
The College Board breaks Unit 7 into 4 topics: (7.1) Defining SHM, (7.2) Frequency and Period of SHM, (7.3) Representing and Analyzing SHM, and (7.4) Energy of Simple Harmonic Oscillators. The two formulas you'll use most are T = 2π√(m/k) for a mass-spring system and T = 2π√(L/g) for a pendulum.
Unit 7 is 5–8% of the AP exam — relatively light. Once you have the equations, most problems are plug-and-chug, but watch for conceptual traps like "does amplitude affect the period?" (Answer: no.)
Key terms preview
A taste of what you'll find in The Essentials and Flashcards.
Simple Harmonic Motion
Motion driven by a linear restoring force (F = −kx). The position follows a sinusoidal curve in time.
Period (T)
Time for one complete oscillation. Units: seconds. Frequency f = 1/T (Hz).
Amplitude (A)
Maximum displacement from equilibrium. Doesn't affect the period in SHM (a key result).
Mass-Spring Period
T = 2π√(m/k). Depends on mass and spring stiffness — NOT on amplitude or gravity.
Pendulum Period
T = 2π√(L/g) (for small angles). Depends on length and gravity — NOT on the mass of the bob.
Energy of an Oscillator
E_total = ½kA². Energy swaps between PE (½kx²) and KE (½mv²) as the system oscillates.
1. SHM happens whenever there's a linear restoring force
If the force is always directed back toward equilibrium AND proportional to displacement (F = −kx), the motion is sinusoidal. Mass-spring systems and small-angle pendulums both qualify. The bigger you pull it, the harder it pulls back — that's the defining feature.
2. Period depends on the system, not on amplitude
The period of SHM depends only on the system's properties — for a mass-spring, that's m and k; for a pendulum, it's L and g. Surprisingly, amplitude does NOT affect period. A spring with a 1 cm pull and a spring with a 10 cm pull have the same period. This is a hallmark feature of SHM that distinguishes it from other periodic motion.
3. Energy oscillates between PE and KE
Total mechanical energy is conserved: E = ½kA². The energy swaps between potential (½kx², maximum at the turning points) and kinetic (½mv², maximum at equilibrium). The oscillator is fastest in the middle and momentarily at rest at the extremes.