SHM, Springs, Pendulums, Energy · 5–8% of the AP Physics 1 exam
F = −kx
T_spring = 2π√(m/k)
T_pendulum = 2π√(L/g)
E = ½kA²
The basics
What's covered: Simple harmonic motion (SHM), period and frequency, sinusoidal representations, and energy of oscillators.
Exam weight: 5–8% — one of the lightest units.
The defining condition for SHM: a LINEAR restoring force, F = −kx. Pull farther, get pulled back harder.
The master idea: The period of SHM depends only on the system's properties — NOT on amplitude. Energy depends on amplitude (E = ½kA²), but period doesn't.
📐 Key equations
F = −kx
Hooke's Law. Force from an ideal spring. Negative because restoring. The defining force for SHM.
T = 2π√(m/k)
Period of a mass-spring system. Depends on m and k only. Doubled mass → period × √2. Doubled spring constant → period / √2.
T = 2π√(L/g)
Period of a simple pendulum (small angles). Length and gravity only. Mass-independent!
f = 1/T
Frequency. Number of oscillations per second. Units: Hz.
ω = 2π/T = 2πf
Angular frequency. Used in x(t) = A·cos(ωt). For a spring: ω = √(k/m).
x(t) = A·cos(ωt + φ)
Position vs. time in SHM. Sinusoidal. Amplitude A is the height; period T = 2π/ω.
v_max = Aω
Maximum speed. At equilibrium (x = 0). Also = A·√(k/m) for a spring.
a_max = Aω²
Maximum acceleration. At the turning points. Always: a = −ω²·x.
U = ½kx²
Potential energy in SHM. Stored in the spring. Max at the turning points.
E = ½kA² = ½mv_max²
Total mechanical energy. Conserved. Depends on amplitude SQUARED.
The 4 topics at a glance
7.1 Defining SHM
F = −kx. The linear restoring force that defines SHM. Springs obey Hooke's law exactly; pendulums obey approximately (small angles).
Position is sinusoidal: x(t) = A·cos(ωt + φ). Velocity peaks at equilibrium; acceleration peaks at extremes. v and a are 90° out of phase; x and a are 180° out of phase.
7.4 Energy
E = ½kA² = ½mv² + ½kx². Energy swaps between PE (max at extremes) and KE (max at equilibrium). Double A → quadruple E.
🧠 How to solve an SHM problem (4 steps)
1. Identify the oscillator. Is it a mass-spring, a pendulum, or something else? Pick the right period formula.
2. Find the period or angular frequency. T = 2π√(m/k) for spring; T = 2π√(L/g) for pendulum.
3. Use energy conservation for speeds/positions: ½kA² = ½kx² + ½mv². Solve for v at any x.
4. Use a = −ω²x for accelerations at any displacement.
⚠️ Common exam traps
"Will the period change if I…?" Most changes (amplitude, mass for pendulum, height of release) DO NOT affect period. Only changes to k, m (for spring), L (for pendulum), or g (for pendulum) do.
Pendulum period doesn't depend on mass. Same as how all objects fall at the same rate.
Amplitude affects energy (E ∝ A²) but NOT period. A common conceptual question.
v_max is at equilibrium; a_max is at turning points. Don't mix them up.
The phase relationships (x, v, a) are 90° apart. When x is max, v is zero. When v is max, x is zero AND a is zero.
Small-angle approximation for pendulum. The formula T = 2π√(L/g) only works for angles less than about 15°.