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 $T = \frac{2\pi}{\omega}$

Body on a String

• 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 $\omega = \sqrt{\frac{k}{m}}$
• Total mechanical energy $E = \frac{1}{2} kA^2 = \frac{1}{2} m\omega^{2}A^2$

Pendulum Motion

• 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 $\omega \approx \sqrt{\frac{g}{l}}$.

Remarks / Tips

• 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