# Datalogging S.H.M. of a mass on a spring

##### Demonstration and Class practical

To demonstrate S.H.M. of a mass on a spring and gather accurate data using a datalogger.

#### Apparatus and materials

Datalogger position sensor

Slotted 50 g masses and hanger

Expendable spring

Card

#### Health & Safety and Technical notes

Masses hanging from ceiling.

Be careful to set the mass moving only vertically, not swinging side to side. Also many position sensors do not work if the object gets too close - be careful to maintain at least the minimum working distance at all times.

#### Procedure

a Attach an expendable spring to the ceiling or a very high retort stand and hang a 50 g slotted mass hanger from it. Place a motion sensor underneath, pointing upwards. Displace the mass a small distance downwards. Position-time data can be recorded swiftly and easily.

b The experiment can be repeated using different numbers of masses, springs in series, and adding card to the bottom of the masses to increase the damping.

#### Teaching notes

1 If you have motion sensors this is much easier to set up than a pendulum attached to a rotary potentiometer.

2 This can work at every level. Initially data can be recorded over a few cycles and used to find the period and hence the frequency. The effect of changing the mass, springs and damping is quickly measured and compared directly to theory.

3 Position-time data can be analyzed to see if it is sinusoidal and if the period is constant as the amplitude damps down. Also, period being independent of initial amplitude can be checked.

4 Velocity and acceleration-time graphs can be plotted by the computer and shown to have the same shape and period but different phase to the position-time graph.

5 The damping can be investigated in various ways. Initially students can look and see if the presence or amount of damping affects the natural frequency. Secondly the amplitude can be extracted from each peak and a damping curve plotted. This can be tested to see if it is exponential.

6 As the data is on a computer it can be exported to a spreadsheet or mathematics package and fitted to a sinusoidal form (with exponential damping if appropriate).

How Science Works Extension: As well as illustrating the fundamental relationships of SHM (displacement, velocity, acceleration, time etc.), this experiment can be used as the basis of several open-ended investigations. Questions to tackle include:

• How does the period of oscillation T depend on the mass m, spring constant k and amplitudeA? (Note that, to double the spring constant, connect two springs side-by-side; to halve it, connect two end-to-end.)
• How does the amplitude of damped oscillations ‘decay’?
• Does damping affect the period of oscillation?

The mathematical relationship involved is T = 2 π√ m/k and this gives the opportunity to discuss how to choose appropriate axes to obtain a straight line graph. 2 is proportional to m and to 1/k, so you can introduce the idea of adding columns to results tables to calculate relevant quantities. You could also extend this to consider the quantities which can be deduced from the gradients of graphs.

A graph of 2 against 1/k has a gradient of 4π2 m; students can deduce m and compare it with the value of the mass they have used in the experiment. (Note that they may have to add the mass of the spring.)

You could point out that this is a method of determining mass which does not require gravity; fix a mass between two horizontal springs so that it oscillates from side to side, and deduce its mass from the period of oscillation. That would be useful in an orbiting spacecraft; it can also introduce the idea of inertial mass as opposed to gravitational mass.

The ‘decay’ of damped oscillations can be tested in two ways. Firstly, look for a ‘half-life’; does the decay of amplitude always halve over the same time interval? Next, plot a log-linear graph and show that it is a straight line. Points to make: a straight line graph makes use of all the data, rather than just selected points; and it will show up more clearly any errant data points.