Oscillations & Waves — SHM Applied (Medium)

Quick Recap — Simple Harmonic Motion

• Kinematics: $x=A\sin(\omega t+\phi)$, $v=\omega\sqrt{A^2-x^2}$, $a=-\omega^2 x$; $v_{max}=\omega A$, $a_{max}=\omega^2 A$. Velocity leads displacement by $\dfrac{\pi}{2}$.
• Period: $T=\dfrac{2\pi}{\omega}$; spring–block $T=2\pi\sqrt{\dfrac{m}{k}}$; simple pendulum $T=2\pi\sqrt{\dfrac{L}{g}}$ (independent of mass and amplitude).
• Springs: parallel $k_{eff}=k_1+k_2$; series $\dfrac{1}{k_{eff}}=\dfrac{1}{k_1}+\dfrac{1}{k_2}$; cutting a spring in half doubles $k$.
• Energy: total $E=\tfrac12 kA^2=\tfrac12 m\omega^2 A^2$; $KE=\tfrac12 m\omega^2(A^2-x^2)$, $PE=\tfrac12 m\omega^2 x^2$. At $x=\dfrac{A}{2}$, $KE:PE=3:1$.

Worked mini-example. A body in SHM has $A=5$ cm and $\omega=10$ rad/s. Then $v_{max}=\omega A=0.5$ m/s, $a_{max}=\omega^2 A=5$ m/s$^2$, and $T=\dfrac{2\pi}{\omega}=0.63$ s.