Simple Harmonic Motion

SHM occurs when:

• The acceleration is always directed towards a fixed equilibrium point
• The acceleration is proportional to the displacement from equilibrium

$\boxed{a = -\omega^2 x}$

The negative sign means the acceleration is always in the opposite direction to the displacement (restoring force).

Displacement

$x = A\cos(\omega t) \quad \text{or} \quad x = A\sin(\omega t)$

(Use cosine if timing starts from maximum displacement; sine if from equilibrium.)

Velocity

$v = -A\omega\sin(\omega t) \quad \text{or} \quad v = A\omega\cos(\omega t)$

Maximum velocity: $v_{\text{max}} = A\omega$ (at equilibrium, $x = 0$)

Alternative form: $v = \pm\omega\sqrt{A^2 - x^2}$

Acceleration

$a = -A\omega^2\cos(\omega t) = -\omega^2 x$

Maximum acceleration: $a_{\text{max}} = A\omega^2$ (at maximum displacement, $x = \pm A$)

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

Mass-Spring System

$\boxed{T = 2\pi\sqrt{\frac{m}{k}}}$

$\omega = \sqrt{k/m}$

• Period depends on mass and spring constant
• Period does NOT depend on amplitude

Simple Pendulum

$\boxed{T = 2\pi\sqrt{\frac{L}{g}}}$

$\omega = \sqrt{g/L}$

• Period depends on length and $g$
• Period does NOT depend on mass or amplitude (for small angles, $\theta < 10°$)

$KE = \frac{1}{2}m\omega^2(A^2 - x^2)$ $PE = \frac{1}{2}m\omega^2 x^2$ $E_{\text{total}} = \frac{1}{2}m\omega^2 A^2 = \text{constant}$

Key Relationships

• At $x = 0$ (equilibrium): KE is maximum, PE is minimum (zero)
• At $x = \pm A$ (extremes): KE is zero, PE is maximum
• Total energy is constant (for undamped SHM)
• KE and PE oscillate between 0 and $E_{\text{total}}$
• Energy graphs show KE and PE as sinusoidal, with KE + PE = constant

Damped oscillations lose energy over time (to friction, air resistance).

Critical damping is used in car shock absorbers and galvanometers.

Forced oscillation: An external periodic force drives the system.

Resonance occurs when the driving frequency equals the natural frequency of the system. At resonance:

• Amplitude is maximum
• Energy transfer from driver to oscillator is most efficient

Effects of damping on resonance:

• Lower peak amplitude
• Broader resonance peak
• Resonant frequency shifts slightly lower

Examples of resonance:

• Tuning a radio (electrical resonance)
• Tacoma Narrows Bridge collapse (wind-driven resonance)
• Microwave ovens (resonance of water molecules)
• Musical instruments (air column/string resonance)

• SHM: $a = -\omega^2 x$ (acceleration proportional to displacement, towards equilibrium)
• $x = A\cos(\omega t)$, $v = \pm\omega\sqrt{A^2 - x^2}$, $v_{\max} = A\omega$, $a_{\max} = A\omega^2$
• Mass-spring: $T = 2\pi\sqrt{m/k}$; Pendulum: $T = 2\pi\sqrt{L/g}$
• Energy: $E_{\text{total}} = \frac{1}{2}m\omega^2A^2$ = constant (undamped)
• Resonance: driving frequency = natural frequency → max amplitude
• Damping reduces amplitude; critical damping = fastest return without oscillation