Section 12

Oscillations and periodic motion

f = \frac{1}{T}, T = \frac{1}{f}

ω = 2 π f = \frac{2 π}{T}

x (t) = A cos (ω t + ϕ)

a = - ω^{2} x

For a mass–spring system

a = - \frac{k}{m} x

v = - A ω sin (ω t + ϕ)

Also

v^{2} = ω^{2} (A^{2} - x^{2})

ω = \frac{k}{m}

T = 2 π \frac{m}{k}

Equivalent form

f = \frac{1}{2 π} \frac{k}{m}

E = \frac{1}{2} m v^{2} + \frac{1}{2} k x^{2} = \frac{1}{2} k A^{2}

v_{m a x} = A ω

a_{m a x} = A ω^{2}

ω = \frac{g}{L}

T = 2 π \frac{L}{g}

Equivalent form

f = \frac{1}{2 π} \frac{g}{L}

ω = \frac{m g d}{I}

Equivalent form

T = 2 π \frac{I}{m g d}

τ = - κ θ

Equivalent form

ω = \frac{κ}{I}

Equivalent form

T = 2 π \frac{I}{κ}

Note: $κ$ is used for the torsional constant, to avoid confusing it with the constant $k$ of a linear spring.