3.6.1.3
The mass-spring system: $T = 2 pi sqrt(m/k)$
Simple Harmonic Motion — AQA A-Level Physics
- The restoring force in a mass-spring system is given by Hooke's lawThe extension of a spring is directly proportional to the applied force, provided the limit of proportionality is not exceeded.: $F = -kx$.
- This is directly proportional to displacementThe distance moved in a particular direction from a starting point. A vector quantity. Measured in metres (m). and in the opposite direction, so the motion is SHM.
$$T = 2\pi\sqrt{\frac{m}{k}}$$
- $T$: time periodThe time taken for one complete oscillation or wave cycle. Measured in seconds (s). (s)
- $m$: mass (kg)
- $k$: spring constantThe force per unit extension of a spring. A measure of the stiffness of the spring. Measured in N m⁻¹. (N \(m^{-1}\))
- Applies to both horizontal and vertical mass-spring systems.
- The time periodThe time taken for one complete oscillation or wave cycle. Measured in seconds (s). does not depend on gravity. Oscillations have the same periodThe time taken for one complete oscillation or wave cycle. Measured in seconds (s). on Earth and the Moon.
- Stiffer spring (larger k) means shorter time period.
- Heavier mass means longer time period.
Worked Example
A 2.0 kg mass is attached to a spring of spring constantThe force per unit extension of a spring. A measure of the stiffness of the spring. Measured in N m⁻¹. 0.9 N \(m^{-1}\). Calculate the frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz)..
Show Solution
1
Calculate time period
$$T = 2\pi\sqrt{\frac{m}{k}} = 2\pi\sqrt{\frac{2.0}{0.9}} = 9.37 \text{ s}$$
2
Calculate frequencyThe number of complete oscillations passing a point per unit time. Measured in hertz (Hz).
$$f = \frac{1}{T} = \frac{1}{9.37} = 0.11 \text{ Hz}$$
Answer
$f = 0.11$ Hz
Examiner Tips and Tricks
- Hooke's lawThe extension of a spring is directly proportional to the applied force, provided the limit of proportionality is not exceeded. often gets combined with SHM questions.
- The spring constantThe force per unit extension of a spring. A measure of the stiffness of the spring. Measured in N m⁻¹. k appears in both $F = kx$ and $T = 2 \pi \sqrt{m/k}$.
- Make sure you are comfortable moving between these.