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A particle moves with simple harmonic motion if its acceleration is proportional to and opposite its displacement from some origin. \[ \begin{align*} F &=-kx\\ a &=-\frac{k}{m}x \end{align*} \]
simple harmonic motion, position function of the particle. $C$ is the mean value or average value of the function $x(t).$ (cf. Plane Trigonometry, Rice, Strange, p. 196) \[ \begin{align*} x &=A\cos{\left(\omega t-\phi\right)}+C\\ &=A\cos{\omega\left(t-\frac{\phi}{\omega}\right)}+C\\ &=A\cos{\omega\left(t-t_0\right)}+C \end{align*} \]
phase \[ \omega t-\phi \]
phase constant (phase angle) \[ \phi=\tan^{-1}{\left(-\frac{v_i}{\omega x_i}\right)} \]
phase shift, the value of $t$ for which the argument of the sinusoidal function is 0. Thus, the phase shift is $\frac{\phi}{\omega}.$ This value represents some initial time for the motion. (cf. Plane Trigonometry, Rice, Strange, p. 206.) \[ t_0=\frac{\phi}{\omega} \]
amplitude (max, peak). If $\left|A\right|\gt 1$ then the amplitude compared with $\sin{t}$ is increased, if $\left|A\right|\lt 1$ then it is decreased. The amplitude is also the maximum or peak of the sinusoid. \[ \left|A\right|=\sqrt{x_i^2+\left(\frac{v_i}{\omega}\right)^2} \]
angular frequency (angular speed) \[ \omega=\sqrt{\frac{k}{m}}=\frac{2\pi}{T}=2\pi f \]
frequency \[ f=\frac{1}{T}=\frac{\omega}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{k}{m}} \]
period \[ T=\frac{2\pi}{\omega}=2\pi\sqrt{\frac{m}{k}} \]
speed \[ v=\frac{dx}{dt}=-\omega A\sin{\left(\omega t+\phi\right)} \]
speed as a function of position \[ v=\pm\omega\sqrt{A^2-x^2} \]
acceleration \[ a=\frac{dv}{dt}=-\omega^2A\cos{\left(\omega t+\phi\right)}=-\omega^2x \]
initial position \[ x_i=A\cos{\theta} \]
initial velocity \[ v_i=-\omega A\sin{\phi} \]
max velocity \[ v_{\mathrm{max}}=\omega A \]
max acceleration \[ a_{\mathrm{max}}=\omega^2A \]
kinetic energy \[ K=\frac{1}{2}m\omega^2A^2\sin^2{\left(\omega t+\phi\right)} \]
potential energy \[ U=\frac{1}{2}kA^2\cos^2{\left(\omega t+\phi\right)} \]
total mechanical energy \[ E=K+U=\frac{1}{2}kA^2 \]
simple pendulum angular acceleration \[ \ddot{\theta}=-\frac{g}{L}\theta \]
simple pendulum position \[ \theta=A\cos{\left(\omega t+\phi\right)} \]
simple pendulum angular frequency \[ \omega=\sqrt{\frac{g}{L}} \]
simple pendulum period \[ T=2\pi\sqrt{\frac{L}{g}} \]