Oscillations

Simple harmonic motion, damping, resonance, and energy in oscillating systems.

Spec Points Covered
  • I can define simple harmonic motionOscillatory motion in which the accelerationThe rate of change of velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.. A vector quantity. Measured in m s⁻². is directly proportional to the displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). from the equilibriumAn object is in equilibrium when the resultant force on it is zero. The object is either stationary or moving at constant velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.. position and always directed towards it. as motion where accelerationThe rate of change of velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.. A vector quantity. Measured in m s⁻². is proportional to displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). and directed towards the equilibriumAn object is in equilibrium when the resultant force on it is zero. The object is either stationary or moving at constant velocity. position.
  • I can use a = -$\omega$^2 x to identify and analyse SHM.
  • I can use x = $A\cos$($\omega$t) and v = -$A\omega\sin$($\omega$t) to describe displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). and velocity in SHM.
  • I can calculate maximum velocity using $v_{max} =$ $A\omega.$
  • I can derive and use the periodThe time taken for one complete oscillation or wave cycle. Measured in seconds (s). equations T = $2\pi\sqrt{m/k}$ for a mass-spring system and T = $2\pi\sqrt{l/g}$ for a simple pendulum.
  • I can describe energyThe capacity to do work. Measured in joules (J). interchange between kinetic and potential energyThe capacity to do work. Measured in joules (J). during SHM and sketch energyThe capacity to do work. Measured in joules (J).-displacement graphs.
  • I can distinguish between free oscillations, forced oscillations, and resonanceThe condition where the driving frequency matches the natural frequency of a system, causing maximum amplitude of oscillation and maximum energy transfer..
  • I can describe and compare light, heavy, and critical dampingThe reduction in amplitude (and energy) of an oscillation over time due to resistive forces such as friction or air resistance..
  • I can explain the effect of dampingThe reduction in amplitude (and energy) of an oscillation over time due to resistive forces such as friction or air resistance. on the resonanceThe condition where the driving frequency matches the natural frequency of a system, causing maximum amplitude of oscillation and maximum energy transfer. curve (lower peak, broader curve, shifted peak).
  • I can describe the phase relationships between displacement, velocity, and accelerationThe rate of change of velocity. A vector quantity. Measured in m s⁻². in SHM.
  • I can apply SHM analysis to real situations including springs, pendulums, and vibrating systems.
Notes
01 Simple harmonic motion (SHM) Simple harmonic motion (SHM) → 02 The cosine form assumes the object starts at $x = +A$ when $t = 0 (released$ from maximum $x = A\cos(\omega t)$ → 03 Displacement x = $A\cos$($\omega$t) → 04 In SHM → 05 Derived by combining F = -kx (Hooke's law) with F = ma and the SHM condition a = -$\omega$^2 x $T = 2\pi\sqrt{\frac{m}{k}}$ → 06 Valid only for small angles of oscillation (less than about $10\degree$) $where \sin\theta$ $T = 2\pi\sqrt{\frac{l}{g}}$ → 07 Free oscillation Free oscillation → 08 Resonance Resonance → 09 Combining x = $A\cos$($\omega$t) and v = -$A\omega\sin$($\omega$t) with the $identity \sin$^2 →
Σ Key Equations Full Reference →
On Data Sheet
Not on Data Sheet
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 \(A^{2}\).