Oscillations & Waves — Advanced Multi-Step Practice (Hard)

Quick Recap — Advanced Oscillations & Waves (Multi-step)

• Timing within SHM ($x=A\sin\omega t$): mean$\to\dfrac{A}{2}$ takes $\dfrac{T}{12}$; mean$\to\dfrac{A}{\sqrt2}$ takes $\dfrac{T}{8}$; mean$\to A$ takes $\dfrac{T}{4}$.
• Combining SHMs (same frequency, phase difference $\phi$): resultant amplitude $\sqrt{A_1^2+A_2^2+2A_1A_2\cos\phi}$; perpendicular equal amplitudes at $90^\circ$ give a circle.
• Energy split: $KE=PE$ at $x=\dfrac{A}{\sqrt2}$; at $x=\dfrac{A}{2}$, $KE=\dfrac34 E$.
• From $v_{max}$ and $a_{max}$: $\omega=\dfrac{a_{max}}{v_{max}}$, $A=\dfrac{v_{max}^2}{a_{max}}$.
• Wave power on a string: $P=\tfrac12\mu\omega^2 A^2 v$. Seconds pendulum ($T=2$ s) has length $\approx1$ m.

Worked mini-example. Two SHMs of amplitude 3 cm and 4 cm along the same line, differing in phase by $90^\circ$, combine to amplitude $\sqrt{3^2+4^2}=5$ cm.