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Simple Harmonic Systems

Physics: AQA A Level, topic 6.1.5

SHM Experiment (Mass-Spring System)

A mass suspended on the end of a spring is also an excellent example of SHM.

Equipment

Equipment

  • Suspend a mass, m, from a spring as shown in the diagram.
Method

Method

  • The formula for the period, T, is T=2πmkT=2\pi \sqrt{\frac{m}{k}} where k is the spring constant of the spring.
  • Find the length of time for 10 complete oscillations, using the pointer against the vertical scale.
Method 2

Method 2

  • Repeat this and find the average time for 10 oscillations. Divide that time by 10 to find the period, T.
  • The mass should be varied and the period remeasured for each value of m.
  • Goggles/safety specs should be worn because there is a spring under tension.
Getting quality data

Getting quality data

  • The fiducial mark should be horizontal. Some form of pointer could be attached to the suspended mass.
  • You can use a small section of string between the clamp and the mass-hanger to reduce the effect of any side to side motion given to the mass in releasing it.
Obtaining quality data 2

Obtaining quality data 2

  • You should check the mass, m, with a balance.
  • The extension should not be so large that the elastic limit of the spring is reached. The spring should not be compressed so much that it no longer provides a restoring force.
  • Keep the pointer close to but not touching the scale.
  • Observe the pointer from eye level. Do this to reduce parallax errors.
Analysis

Analysis

  • You should plot a graph of T² against m to verify the equation.
  • You would expect to plot a straight line through the origin with a gradient =4π2k=\frac{4\pi^2}{k}.
  • The straight line might miss the origin because the effective mass of the spring has not been taken into account.
  • Plotted or given error bars would help to provide an estimate of the uncertainty in g by finding the worst acceptable line of fit.

SHM Experiment (Simple Pendulum System)

The simple pendulum is a classic experiment to demonstrate SHM.

Equipment

Equipment

  • Suspend a pendulum bob by a thread from a clamp.
  • The thread should be squashed between two small blocks of wood or a bung/cork split in two. This makes the suspension point easy to define.
Method

Method

  • Time for 10 oscillations. Repeat and get an average.
  • Divide this average time by 10 to find the period.
  • Repeat for different lengths of pendulum thread.
Obtaining quality data

Obtaining quality data

  • Use a ruler to measure the length of the string from the suspension point to the centre of the bob.
  • A micrometer or calipers might be useful to measure the diameter of the bob.
  • Pull the bob to one side (position A) so that the thread makes a small angle with the vertical (use a protractor to measure this angle to keep it consistent and small. The maximum angle should be 10 degrees).
Obtaining quality data 2

Obtaining quality data 2

  • Start and then later stop the timer when the bob passes in front of a fiducial mark placed behind the thread (i.e. position B).
  • This can be a card with a vertical line drawn on, or the clamp stand.
  • This mark needs to be at the equilibrium position. This is where the bob is travelling fastest, so means there is the smallest error in starting and stopping the timer.
Analysis

Analysis

  • The equation for this is supposed to be T=2πlgT = 2\pi \sqrt{\frac{l}{g}}. So you should plot a graph of T² against l to verify the equation.
  • You would expect to see a straight line through the origin with gradient =4π2g= \frac{4\pi^2}{g}.
  • You can find a value of g from the gradient (g=4π2gradientg = \frac{4\pi^2}{gradient}).
  • Plotted or given error bars would help to provide an estimate of the uncertainty in g by finding the worst acceptable line of fit.

Test yourself on Simple Harmonic Systems

  • What is the correct equation for the period T of a mass-spring system?

  • Finding spring constant experiment

  • The fiducial mark for the timer should be set when the bob of the pendulum is moving fastest, where is this?

  • Which statements about a pendulum SHM system are true?

  • What is the gradient of a graph of T2 against l in simple harmonic motion?

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