Simple Harmonic Motion

Simple harmonic motion is an oscillatory motion in which an object's position over time can be modelled as a smooth sinusoidal function.

$$x(t) = A\cos\left( \frac{2\pi}{T} t \right)$$

where $A$ (amplitude) is the maximum displacement and $T$ is the period of a single oscillation.

Its velocity can then be described as the rate of change of position over time:

$$v(t) = \frac{dx}{dt} = -\frac{2\pi}{T}A\sin(\frac{2\pi}{T}t)$$

$$v_{x}(t) = \frac{dx}{dt} = -\frac{2\pi A}{T}\sin( \frac{2\pi t}{T}) = -2\pi fA\sin(2\pi ft) = -\omega A\sin(\omega t)$$

$$v_{max} = \frac{2\pi A}{T} = 2\pi fA = \omega A$$

Phase Shift

$$x(t) = A\cos(\omega t + \phi_{0})$$ $$v_{x}(t) = -\omega A\sin(\omega t + \phi_{0}) = -v_{max}\sin(\omega t + \phi_{0})$$

Energy in SHM

$$U_{spring} = \frac{1}{2}k\Delta x^2$$$$ E = K + U \therefore$$

$$\omega = \sqrt{ \frac{k}{m} } \quad \quad f = \frac{1}{2\pi} \sqrt{ \frac{k}{m} } \quad\quad T = 2\pi \sqrt{ \frac{m}{k} } \quad \text{(15.24)}$$ $$ E = \frac{1}{2}mv^2 + \frac{1}{2}k\Delta x^2 = \frac{1}{2}kA^2 = \frac{1}{2}m(v_{max})^2 \quad \text{(15.25)} $$

Damping

$$