# Investigating a mass-on-spring oscillator

##### Demonstration

A mass suspended on a spring will oscillate after being displaced. The period of oscillation is affected by the amount of mass and the stiffness of the spring. This experiment allows the period, displacement, velocity and acceleration to be investigated by datalogging the output from a motion sensor. It is an example of simple harmonic motion.

#### Apparatus and materials

Motion sensor, interface and computer

Slotted masses on holder, 100 g-400 g

Clamp and stand

String

Springs, 3

Card

#### Health & Safety and Technical notes

Unless the stand is very heavy, use a G-clamp to anchor it to the bench.

Suspend the spring from a clamp and attach a mass to the free end. Adjust the height of the clamp so that the mass is about 30 cm above the motion sensor, which faces upwards.

The clarity of measurements depends upon the choice of spring stiffness and mass. Good results can be obtained with three springs linked in series, and masses in the range 100 - 400 g. With this choice, it is necessary to place the sensor on the floor and allow the mass and spring to overhang the edge of the bench.

When the mass is displaced and released, its vertical motion is monitored by a motion sensor connected via an interface to a computer.
In general, the magnitude of the initial displacement should not exceed the extension of the spring. It is best to lift the mass to displace it, rather than pull it down.

The mass may acquire a pendulum type of motion from side to side. Eliminate this by suspending the spring from a piece of string up to 30 cm.

For the collection and analysis of the data, data-logging software is required to run on the computer. Configure the program to measure the distance of the mass from the sensor, and to present the results as a graph of distance against time. Scale the vertical axis of the graph to match the amplitude of oscillation.

#### Procedure

Data collection

a Lift and release a 400 g mass to start the oscillation. Start the data-logging software and observe the graph for about 10 seconds.

b Before the oscillation dies away, restart the data-logging software and collect another set of data, which can be overlaid on the first set.

c Repeat the experiment with 300 g, 200 g and 100 g masses.

Analysis
Measurement of period
d The period of the sinusoidal graph may be measured using a time-interval analysis tool in the software. Measure the period from peak to peak.

e Take measurements at several different places on the time axis, and observe that the period does not vary with elapsed time.

f Take similar measurements on the set of results with a smaller amplitude, and observe that the period appears to be independent of amplitude.

Effect of mass
g Measure the period for each of the other graphs resulting from using different masses. Plot a new graph of period against mass. (Y axis: period; X axis: mass.)

h Use a curve-matching tool to identify the algebraic form of the relationship. This is usually of the form 'period is proportional to the square root of mass'.

i Use the program to calculate a new column of data representing the square of the period. Plot this against mass on a new graph. A straight line is the usual result, showing that the period squared is proportional to the mass.

Velocity and acceleration
j On the 'distance vs. time' graph, the gradient at any point represents the velocity of the oscillating mass. Choose the clearest set of data and use the program to calculate the gradient at every point on the graph.

A plot of the resulting data shows a 'velocity vs. time' graph. Note that the new graph is also sinusoidal. However, compared with the 'distance vs. time' graph, there is a phase difference - the velocity is a maximum when the displacement is zero, and vice versa.

k A similar gradient calculation based on the 'velocity vs. time' graph yields an 'acceleration vs. time' graph. Comparing this with the original 'distance vs. time' graph shows a phase difference of 180°. This indicates that the acceleration is always opposite in direction to the displacement.

#### Teaching notes

1 This experiment illustrates the value of rapid collection and display of data in assisting thinking about the phenomenon under investigation. Data is collected within a few seconds and the graph is presented simultaneously. Students can observe connections between features on the graph and the actual motion of the mass. For example, the crests and troughs on the graph represent the mass at the extremes of its displacement.

The parameters suggested here usually produce displacements of a few centimetres. The motion sensors can detect these with suitable precision. Small amplitude oscillations produce rather noisy data. Starting with the largest mass, shows the clearest results first.

Software tools for taking readings from the graph are employed: measuring gradients and time intervals. The detail available in the data allows the idea that the periodic time remains constant for a given mass to be tested.

2 A particularly useful software function is that which calculates the velocity for all points on the graph and plots these as a new graph. A notable feature of the velocity graph is the phase difference from the distance data. This can provoke useful discussion about the change in magnitude and direction of velocity during each cycle of oscillation. The 'noisiness' of the measurements begins to show more markedly on the velocity graph. The process by which the program calculates the velocity (usually by taking differences between distance readings) should be questioned.

3 The further derivation of acceleration from the gradients of the velocity graph usually shows even more measurement noise. Nevertheless, the form of the graph convincingly shows the antiphase relation with the distance graph. This is useful for prompting discussion about the conditions for simple harmonic motion (SHM). This can be reinforced by plotting a further graph of acceleration against displacement. The negative gradient straight line supports the basic condition for SHM: acceleration is proportional to displacement, but in the opposite direction.