Key Equations

Oscillations - OCR A-Level Physics

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SHM defining equation
$$a = -\omega^2 x$$
  • Where:
    • $a$ = m \(s^{-2}\)
    • $\omega$ = rad \(s^{-1}\)
    • $x$ = m
The defining equation for SHM. The negative sign means acceleration opposes displacement.
SHM displacement
$$x = A\cos(\omega t)$$
  • Where:
    • $x$ = m
    • $A$ = m
    • $\omega$ = rad \(s^{-1}\)
    • $t$ = s
Assumes object starts at x = +A at t = 0. Calculator must be in radian mode.
SHM velocity
$$v = -A\omega\sin(\omega t)$$
  • Where:
    • $v$ = m \(s^{-1}\)
    • $A$ = m
    • $\omega$ = rad \(s^{-1}\)
    • $t$ = s
Derivative of x = A\cos(\omega t). Velocity leads displacement by \pi/2.
Period of mass-spring system
$$T = 2\pi\sqrt{\frac{m}{k}}$$
  • Where:
    • $T$ = s
    • $m$ = kg
    • $k$ = N \(m^{-1}\)
Derived from \omega = \sqrt{k/m}. Period independent of amplitude.
Period of simple pendulum
$$T = 2\pi\sqrt{\frac{l}{g}}$$
  • Where:
    • $T$ = s
    • $l$ = m
    • $g$ = m \(s^{-2}\)
Valid for small angles only (\theta < 10\degree). Period independent of mass and amplitude.
Maximum velocity in SHM
$$v_{max} = A\omega$$
  • Where:
    • $v_{max}$ = m \(s^{-1}\)
    • $A$ = m
    • $\omega$ = rad \(s^{-1}\)
Must memorise. Occurs at x = 0 (equilibrium position).
Kinetic energy in SHM
$$KE = \frac{1}{2}m\omega^2(A^2 - x^2)$$
  • Where:
    • $KE$ = J
    • $m$ = kg
    • $\omega$ = rad \(s^{-1}\)
    • $A$ = m
    • $x$ = m
Must memorise. Derived from KE = \frac{1}{2}mv^2 with v = \omega\sqrt{\(A^{2}\) - \(x^{2}\)}.
Total energy in SHM
$$E = \frac{1}{2}m\omega^2 A^2$$
  • Where:
    • $E$ = J
    • $m$ = kg
    • $\omega$ = rad \(s^{-1}\)
    • $A$ = m
Must memorise. Total energy is constant for undamped SHM and proportional to \(A^{2}\).
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