Volatile Yard: Simple Harmonic Motion

Basics

Simple harmonic motion (SHM) results from a body that moves in such a way that its acceleration \(a \propto x\) is directed towards a fixed point (equilibrium position) in its path and is directly proportional to its displacement \(x\) from that point
SHM, with amplitude \(A\) and period \(T\), as a projection of uniform circular motion, with angular velocity \(\omega\) and radius \(A\)

Displacement of the body in SHM from its equilibrium position \(x = A \cos{\omega t}\) is the \(x\)-component of the position of body in uniform circular motion
Velocity \(v = \frac{dx}{dt} = - \omega A \sin{\omega t} = \pm \omega \sqrt{A^2 - x^2}\)
Acceleration \(a = \frac{d^2x}{dt^2} = - \omega^{2} A \cos{\omega t} = - \omega^{2} x\)
Period \(T = \frac{2\pi}{\omega}\)

Modeled as SHM with \(\omega = \sqrt{\frac{k}{m}}\), where \(m\) is the mass of the body and \(k\) is the spring constant

Derivation: Linear elastic restoring force is given by Hooke's Law \(F = -kx\) which results in acceleration \(a = \frac{F}{m} = - \frac{k}{m} x\). Compare with \(a = -\omega^{2} x\), we have \(\omega = \sqrt{\frac{k}{m}}\)

Modeled as SHM with \(\omega = \sqrt{\frac{g}{l}}\), where \(l\) is the length of the inextensible string with negligible weight and \(g\) is the gravitational acceleration

Derivation (a): Consider bob of mass \(m\) with angular displacement \(\theta\). Restoring force \(F = -mg\sin{\theta} = ma\) at displacement \(x \approx l \sin{\theta}\), thus \(a \approx -\frac{g}{l}x\). Compare with \(a = -\omega^{2}x\), we have \(\omega \approx \sqrt{\frac{g}{l}}\).
Derivation (b): Consider bob of mass \(m\) with angular displacement \(\theta\). Restoring force \(F = - mg\sin{\theta} = m\frac{dv}{dt} = m\frac{d}{dt}(l\frac{d\theta}{dt}) = ml\frac{d^2\theta}{dt^2}\), therefore \(\frac{d^2\theta}{dt^2} = - \frac{g}{l} \sin{\theta} \approx - \frac{g}{l} \theta\) with small \(\theta\) (\(< 5^{\circ}\)). Compare with \(\frac{d^{2}x}{dt^2} = - \omega^{2} x\), we have \(\omega \approx \sqrt{\frac{g}{l}}\).

Simply look at SHM as a projection of uniform circular motion
Model spring with \(\omega = \sqrt{\frac{k}{m}}\) and pendulum with \(\omega = \sqrt{\frac{g}{l}}\)
Others are easy