Simple Harmonic Motion

The defining equation, displacement-velocity-acceleration relationships, mass-spring and pendulum systems, energy in SHM, and the required practical.

Spec Points Covered
  • State the two conditions for SHM and apply the defining equation $a = -\omega^2 x$.
  • Use $x = A \cos(\omega t)$ to calculate displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). at any time.
  • Apply $v = +/- \omega \sqrt{A^2 - x^2}$ and derive $v_{\max} = \omega A$.
  • Apply $a_{\max} = \omega^2 A$ and relate 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. to displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). graphically.
  • Use $T = 2 \pi \sqrt{m/k}$ for a mass-spring system and $T = 2 \pi \sqrt{L/g}$ for a simple pendulum.
  • Describe the energyThe capacity to do work. Measured in joules (J). interchange between kinetic and potential energyThe capacity to do work. Measured in joules (J). in SHM.
  • Describe and analyse the required practical for investigating SHM.
Notes
01 SHM requires acceleration proportional to and opposite to displacement Simple harmonic motion 3.6.1.2 → 02 The defining equation links acceleration, angular frequency, and displacement $a = -\omega^2 x$ 3.6.1.2 → 03 Displacement follows a cosine (or sine) function $x = A\cos(\omega t)$ 3.6.1.2 → 04 Speed depends on position: fastest at equilibrium, zero at amplitude $v = \pm\omega\sqrt{A^2 - x^2}$ 3.6.1.2 → 05 Maximum speed and maximum acceleration have simple expressions $v_{\max} = \omega A$ 3.6.1.2 → 06 The displacement, velocity, and acceleration graphs are 90 degrees out of phase 3.6.1.2 → 07 The mass-spring system: T = 2 pi sqrt(m/k) $T = 2\pi\sqrt{\frac{m}{k}}$ 3.6.1.3 → 08 The simple pendulum: T = 2 pi sqrt(L/g) $T = 2\pi\sqrt{\frac{L}{g}}$ 3.6.1.3 → 09 Energy in SHM: kinetic and potential swap but total stays constant 3.6.1.2 → 10 Required practical: investigating SHM with a mass-spring system $k = \frac{4\pi^2}{\text{gradient}}$ 3.6.1.3 → 11 Required practical: investigating SHM with a simple pendulum $g = \frac{4\pi^2}{\text{gradient}}$ 3.6.1.3 → 12 Reducing errors in SHM practicals 3.6.1.3 →
Σ Key Equations Full Reference →
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Defining equation of SHM
$$a = -\omega^2 x$$
  • Where:
    • $a$ = acceleration (m \(s^{-2}\))
    • $\omega$ = angular frequency (rad \(s^{-1}\))
    • $x$ = displacement (m)
The minus sign is essential. It shows a and x are in opposite directions.
Maximum speed
$$v_{\max} = \omega A$$
  • Where:
    • $v_{\max}$ = maximum speed (m \(s^{-1}\))
    • $\omega$ = angular frequency (rad \(s^{-1}\))
    • $A$ = amplitude (m)
Obtained by setting x = 0 in the speed equation.
SHM displacement (from amplitude)
$$x = A\cos(\omega t)$$
  • Where:
    • $x$ = displacement (m)
    • $A$ = amplitude (m)
    • $\omega$ = angular frequency (rad \(s^{-1}\))
    • $t$ = time (s)
Use cosine when starting from maximum displacement. Use sine when starting from equilibrium.
Maximum acceleration
$$a_{\max} = \omega^2 A$$
  • Where:
    • $a_{\max}$ = maximum acceleration (m \(s^{-2}\))
    • $\omega$ = angular frequency (rad \(s^{-1}\))
    • $A$ = amplitude (m)
Obtained by setting x = A in a = -omega^2 x (drop the minus for magnitude).
SHM speed at displacement x
$$v = \pm\omega\sqrt{A^2 - x^2}$$
  • Where:
    • $v$ = speed (m \(s^{-1}\))
    • $\omega$ = angular frequency (rad \(s^{-1}\))
    • $A$ = amplitude (m)
    • $x$ = displacement (m)
Derived by differentiating the displacement equation. Maximum speed omega A at x = 0.
Spring constant from \(T^{2}\) vs m graph
$$k = \frac{4\pi^2}{\text{gradient}}$$
  • Where:
    • $k$ = spring constant (N \(m^{-1}\))
    • $gradient$ = gradient of \(T^{2}\) vs m graph (\(s^{2}\) kg^{-1})
From squaring T = 2 pi sqrt(m/k). Required practical analysis.
Period of mass-spring system
$$T = 2\pi\sqrt{\frac{m}{k}}$$
  • Where:
    • $T$ = time period (s)
    • $m$ = mass (kg)
    • $k$ = spring constant (N \(m^{-1}\))
Independent of g. Same period on Earth and Moon.
g from \(T^{2}\) vs L graph
$$g = \frac{4\pi^2}{\text{gradient}}$$
  • Where:
    • $g$ = gravitational field strength (m \(s^{-2}\))
    • $gradient$ = gradient of \(T^{2}\) vs L graph (\(s^{2}\) \(m^{-1}\))
From squaring T = 2 pi sqrt(L/g). Required practical analysis.
Period of simple pendulum
$$T = 2\pi\sqrt{\frac{L}{g}}$$
  • Where:
    • $T$ = time period (s)
    • $L$ = length from pivot to centre of bob (m)
    • $g$ = gravitational field strength (m \(s^{-2}\))
Valid for small angles only (theta < 10 degrees). Depends on g, so different on other planets.
Q Retrieval Practice All 15 Questions →
Q1. State the two conditions required for 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..
(1) The accelerationThe rate of change of velocity. A vector quantity. Measured in m s⁻². is proportional to the displacement. (2) The acceleration is always 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 (opposite to displacement).
Q2. Write the defining equation of SHM.
a = -omega^2 x, where a is acceleration, omega is angular frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz)., and x is displacement from equilibriumAn object is in equilibrium when the resultant force on it is zero. The object is either stationary or moving at constant velocity..
Q3. Write the SHM displacement equation and state when to use cosine vs sine.
x = A cos(omega t) when starting from maximum displacement. x = A sin(omega t) when starting from the equilibrium position.
Q4. State the equation for the speed of an oscillator at displacement x.
v = +/- omega sqrt(\(A^{2}\) - \(x^{2}\)).
Q5. State the maximum speed and maximum acceleration of an SHM oscillator.
v_max = omega A (at x = 0). a_max = omega^2 A (at x = +/- A).