Oscillations

考纲要求

• 17.1 Simple harmonic oscillations
  • Use displacement, amplitude, period, frequency, angular frequency, phase difference
  • SHM: $a \propto -x$ (acceleration proportional to displacement, opposite direction)
  • Use $a = -\omega^2 x$ and $x = x_0 \sin \omega t$
  • Use $v = v_0 \cos \omega t$ and $v = \pm \omega \sqrt{x_0^2 - x^2}$
  • Analyse graphs of $x$, $v$, $a$ for SHM
• 17.2 Energy in SHM
  • Describe KE-PE interchange during SHM
  • Use $E = \frac{1}{2} m \omega^2 x_0^2$ for total energy
• 17.3 Damped and forced oscillations, resonance
  • Resistive force causes damping
  • Light, critical, heavy damping — sketch $x$-$t$ graphs
  • Resonance: max amplitude when driving frequency = natural frequency

常见题型

• 定义题：State what is meant by simple harmonic motion
• 计算题：求频率、最大速度、最大加速度、能量等
• 作图题：Sketch $x$-$t$, $v$-$t$, $a$-$t$ 图或 $v$-$x$ 图
• 解释题：Explain damping / resonance / energy interchange

核心公式

$a = -\omega^2 x$

$x = x_0 \sin \omega t$

$v = v_0 \cos \omega t = \pm \omega \sqrt{x_0^2 - x^2}$

$v_0 = \omega x_0$

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

$E = \frac{1}{2} m\omega^2 x_0^2$

常见错误

• 位移最大时速度也为零（不是最大）
• 加速度始终指向平衡位置（$a = -\omega^2 x$）
• 混淆 $\omega$ 和 $f$：$\omega = 2\pi f$
• 共振条件：驱动力频率 = 系统自然频率