Key Equations

Simple Harmonic Motion — AQA A-Level Physics

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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.
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.
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.
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.
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.
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.
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).
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.
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.
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