SUMMARY

This unit's core results, gathered for quick revision: Oscillatory motion is back-and-forth motion about a reference point; SHM is the special case where the restoring force/acceleration is proportional to displacement and directed towards the mean position, $F_x=-kx$. Displacement $y=A\sin\omega t$; velocity $v=A\omega\cos\omega t=\omega\sqrt{A^2-y^2}$; acceleration $a=d^2y/dt^2=-\omega^2y$. Time period $T=2\pi/\omega$; frequency $f=1/T$. Angular SHM has frequency $f=\dfrac{1}{2\pi}\sqrt{\kappa/I}$. Springs in series: $\dfrac{1}{k_s}=\sum\dfrac{1}{k_i}$; in parallel: $k_p=\sum k_i$. Simple pendulum: $T=2\pi\sqrt{l/g}$, independent of mass and (for small angles) of amplitude. U-tube liquid column: $T=2\pi\sqrt{l/2g}$. Since $F=-dU/dx$, potential energy in SHM is $U(x)=\tfrac12 m\omega^2x^2$; kinetic energy $KE=\tfrac12 mv^2=\tfrac12 m\omega^2(A^2-x^2)$; total energy $E=\tfrac12 m\omega^2A^2=$ constant. The four types of oscillation are free …