AP Physics C: Mechanics Unit 7 Cheatsheet: Oscillations

Simple Harmonic Motion (SHM) is periodic motion where restoring force is proportional to displacement from equilibrium.

Restoring Force Law:

ma_x = −kΔx

• Force always points toward equilibrium
• Magnitude increases with distance from equilibrium
• Equilibrium: net force = 0

Key distinction: SHM is a special case of periodic motion—not all periodic motion is SHM (e.g., uniform circular motion is periodic but not SHM).

Fundamental relationships:

T = 2π/ω = 1/f

Spring-mass system:

T_s = 2π√(m/k)

• Depends on mass and spring constant
• Independent of amplitude

Simple pendulum (small angle):

T_p = 2π√(ℓ/g)

• Depends on length and gravity
• Independent of mass and amplitude

Physical pendulum:

T_phys = 2π√(I/mgd)

• I = moment of inertia
• d = distance from pivot to center of mass

Position equation:

x = A cos(ωt + φ) or x = A sin(ωt + φ)

Derived from Newton's 2nd law:

d²x/dt² = −ω²x

Velocity:

v = −Aω sin(ωt + φ)

v_max = Aω (at equilibrium)

Acceleration:

a = −ω²x

a_max = Aω² (at maximum displacement)

Total mechanical energy:

E_total = U + K = constant

For spring-mass system:

E_total = ½kA²

• Depends only on amplitude and spring constant
• Independent of time

Energy exchange:

• At equilibrium: K is maximum, U is minimum (U = 0)
• At maximum displacement: U is maximum, K = 0
• Energy continuously converts between kinetic and potential