3.6.1.2
Displacement follows a cosine (or sine) function
Simple Harmonic Motion — AQA A-Level Physics
$$x = A\cos(\omega t)$$
- $x$: displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). (m)
- $A$: amplitudeThe maximum displacement of a point on a wave from its equilibrium (rest) position. Measured in metres (m). (m)
- $\omega$: angular frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz). (rad \(s^{-1}\))
- $t$: time (s)
- Use cosine when the oscillation starts from maximum displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). ($x = A$ at $t = 0)$.
- Use $x = A \sin(\omega t)$ when the oscillation starts 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 ($x = 0$ at $t = 0)$.
- The displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). varies between +A and -A.
- $omega = 2 \pi / T = 2 \pi f.$
Worked Example
A 55 g mass on a spring is pulled down 4.3 cm and released at t = 0. The periodThe time taken for one complete oscillation or wave cycle. Measured in seconds (s). is 0.8 s. Calculate the displacement at t = 0.3 s.
Show Solution
1
Choose cosine (released from max displacement)
$$x = A\cos(\omega t)$$
2
Calculate angular frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz).
$$\omega = \frac{2\pi}{T} = \frac{2\pi}{0.8} = 7.85 \text{ rad s}^{-1}$$
3
Substitute values (calculator in radians)
$$x = 4.3 \cos(7.85 \times 0.3) = -3.0 \text{ cm (2 s.f.)}$$
Answer
$x = -3.0$ cm (the mass is 3.0 cm above 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..)
Examiner Tips and Tricks
- Displacement is a vector.
- Keep the minus sign in your answer -- it tells you which side of 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.. the object is on.
- Losing the sign can lose you a mark.