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°.