Oscillations

AP Physics C

Simple Harmonic Motion Equation

A mathematical representation of oscillatory motion, defined as x(t) = A cos(ωt + φ), where A is the amplitude, ω is the angular frequency, and φ is the phase constant.

angular frequency equation

is given by ( ω = 2π/T or ω = 2πf) where (f) is the frequency of the motion.

amplitude equation

A=1/2 (xmax - xmin)

Kinetics of simple harmonic motion

KE →/← u

E = K + u = constant : KE is kinetic energy and u is potential energy in the system.

u_{s ⁼} ½ KA²cos²ω∝ and KE = ½ mω²A²sin²ωt. In this context, kinetic energy is maximized when the displacement is zero, while potential energy is maximized at the amplitude.

Ideal spring equation for angular frequency

ω = √(k/m)

Ideal spring equation for period

T = 2π√(m/k)

v(max), a(max), and E for ideal spring

v(max) = Aω

a(max) = Aω²

E = ½ kA²

vertical spring xeq equation

x(eq) = mg/k

The equilibrium position equation for a vertical spring accounts for gravitational forces (Fg=Fs)

Simple pendulum period

T = 2π√(L/g)

The time it takes for a pendulum to complete one full oscillation

Physical pendulum period

T = 2π√(I/mgx) where I is the moment of inertia and x is the distance from the pivot to the center of mass.

*Use Icm+mx² to find I.