Maximum velocity in SHM:
$$v_{max} = A\omega = A\sqrt{\frac{k}{m}}$$For P (spring $k_1$): $v_P = A_P\sqrt{k_1/m}$
For Q (spring $k_2$): $v_Q = A_Q\sqrt{k_2/m}$
Given $v_P = v_Q$ (same maximum velocity):
$$A_P\sqrt{\frac{k_1}{m}} = A_Q\sqrt{\frac{k_2}{m}}$$$$\frac{A_Q}{A_P} = \sqrt{\frac{k_1}{k_2}}$$SHM: $x = A\sin(\omega t + \phi)$. Restoring force: $F = -kx$. Acceleration: $a = -\omega^2 x$. Angular frequency: $\omega = \sqrt{k/m}$. Period: $T = 2\pi\sqrt{m/k}$. Energy: $E = \frac{1}{2}kA^2$ (constant). $KE = \frac{1}{2}m\omega^2(A^2-x^2)$. $PE = \frac{1}{2}kx^2$. Velocity: $v = \omega\sqrt{A^2-x^2}$. Maximum velocity at $x=0$: $v_{max} = \omega A = A\sqrt{k/m}$.
Horizontal spring: $T = 2\pi\sqrt{m/k}$. Vertical spring: $T = 2\pi\sqrt{m/k}$ (same! gravity only shifts equilibrium). Springs in series: $1/k_{eff} = 1/k_1 + 1/k_2$. Springs in parallel: $k_{eff} = k_1 + k_2$. If mass hung from two springs in series: equivalent to one spring of effective $k$. Springs connected to fixed wall in parallel: $k_{eff}$ adds. For spring cut into $n$ equal pieces: each piece has $k_{new} = nk$ (shorter spring is stiffer).
$T = 2\pi\sqrt{L/g}$. Valid for small oscillations ($\theta < 5°$). Independent of mass and amplitude (for small angles). $T \propto \sqrt{L}$: doubling length multiplies $T$ by $\sqrt{2}$. $T \propto 1/\sqrt{g}$: pendulum clock runs slower at higher altitudes (smaller $g$). Second pendulum: $T = 2$ s, $L \approx 1$ m. Seconds pendulum is used to measure $g$: $g = 4\pi^2 L/T^2$.
When driving frequency = natural frequency: resonance occurs. Amplitude becomes maximum (theoretically infinite without damping). Damping reduces amplitude at resonance. Quality factor $Q = \omega_0/(2\gamma)$ ($\gamma$ = damping coefficient). High $Q$: sharp resonance. Examples: pushing a swing at natural frequency. Breaking a wine glass with sound. Tacoma Narrows bridge collapse (wind resonance). Magnetic resonance imaging (MRI) uses nuclear spin resonance.
Total mechanical energy: $E = \frac{1}{2}kA^2 = \frac{1}{2}m\omega^2 A^2$ (constant, conserved). $KE = \frac{1}{2}mv^2 = \frac{1}{2}m\omega^2(A^2-x^2) = E\cos^2(\omega t)$. $PE = \frac{1}{2}kx^2 = \frac{1}{2}m\omega^2 x^2 = E\sin^2(\omega t)$. KE + PE = constant = $E$. At mean position: $KE = E$ (maximum), $PE = 0$. At extremes: $KE = 0$, $PE = E$ (maximum). Average KE = Average PE = $E/2$.
With resistive force $F_{damp} = -bv$: $x = Ae^{-bt/(2m)}\cos(\omega't + \phi)$. Damped frequency: $\omega' = \sqrt{\omega_0^2 - (b/2m)^2}$. Three cases: Underdamped ($b < 2m\omega_0$): oscillates with decreasing amplitude. Critically damped ($b = 2m\omega_0$): returns to equilibrium fastest without oscillating (desired in car suspension, galvanometer). Overdamped ($b > 2m\omega_0$): very slowly returns to equilibrium. Car shock absorbers: near critical damping.
Same frequency, same direction: resultant amplitude $A = \sqrt{A_1^2+A_2^2+2A_1A_2\cos\delta}$ where $\delta$ = phase difference. In phase ($\delta=0$): $A = A_1+A_2$. Antiphase ($\delta=\pi$): $A = |A_1-A_2|$. Perpendicular SHMs of same frequency: Lissajous figures (circles, ellipses, lines depending on phase). Perpendicular SHMs of different frequencies: complex Lissajous patterns (used to compare frequencies).
Progressive wave: each particle performs SHM with phase increasing with distance. $y = A\sin(\omega t - kx)$ for wave moving in $+x$. Wavelength $\lambda = 2\pi/k$. Speed $v = \omega/k = f\lambda$. Intensity $I \propto A^2$. Standing wave: $y = 2A\cos(kx)\sin(\omega t)$. Amplitude varies with position: nodes at $kx = n\pi$, antinodes at $kx = (n+1/2)\pi$. Standing waves in pipes and strings produce resonant frequencies discussed in wave theory.