# Relationships between acceleration, force and mass

This detailed experiment involves measurement of acceleration.

#### Apparatus and materials

*For each student groups*

Dynamics trolleys, up to 3

Rods for stacking trolleys

Elastic cords, 3

Runway

Ticker-tape

Ticker-timer with power supply unit

String

#### Health & Safety and Technical notes

Long runways or heavy shorter ones should be handled by two persons. In operation, ensure that a string is tied across the bottom of the runway, to prevent the trolley falling onto the floor (or someone's foot).

It might not be possible for every group to have three trolleys, and so groups may need to share.

To ensure the elastic cords (given to one group of students) all stretch by the same amount for the same force, set up a testing rig as shown in the diagram.

Oil the bearings on the trolley wheels. Do not use trolleys with bent axles (through dropping). Ensure the runway and the trolley wheels are clean.

#### Procedure

**The relationship between acceleration and force**

In this part, you will vary the force and measure different accelerations. Mass must stay the same.

**a** Set up the runway and compensate for friction, as in the experiment Compensating for friction.

**b** Set up the ticker-timer at the higher end of the runway.

**c** Accelerate a single trolley by a single strand of elastic cord. Use a ruler to help you to stretch the cord by a fixed amount, or extend the cords the full length of the trolley.

**d** Use the tape to find out the trolley's acceleration. (Use the method described in Finding average acceleration with a ticker-timer.)

**e** Repeat using two cords in parallel, stretched by the same amount as before. Measure and record the new acceleration.

**f** Repeat with three cords.

**g** Plot a graph of acceleration (*y* axis) against force (*x* axis). Simply use the number of cords, 1, 2 or 3, as a way of measuring force.

**The relationship between acceleration and mass**

**h** Repeat steps **a - g**, but this time apply the same force in all cases. Vary the mass by stacking up to three trolleys. (Two cords in parallel helps when pulling more trolleys.) For simplicity, you can use a trolley mass as a unit of mass (instead of mass in kilograms).

#### Teaching notes

**1** Students gain a great deal from feeling the effect of a constant force on increasing masses, and the 'sluggish' effect on their motion.

**2** The degree of necessary compensation varies with number of trolleys. Students will obtain best results if they readjust the slope of the runway when they increase the number of trolleys, so that they are still compensating for friction.

**3** The graphs each have three points. High precision of measurement is not possible. This can give rise to discussion. For example can a small number of measurements of modest precision yield valid conclusions? What is the nature of uncertainty and error here?

**4** The two investigations may take more than one lesson. You could save time by arranging for half the class to investigate the effect of force, *F*, and the other half to investigate the effect of mass, *m*. Then combine the results to arrive at *a = F/m*.

**5 How Science Works Extension:** This experiment is designed specifically to avoid a pitfall present in other experiments looking at *F* = *ma*. In some experiments (such as Investigating Newton's second law of motion) the force accelerating the mass is provided by hanging masses; their weight provides the force. However, this assumes that weight is proportional to mass, and so the relationship that the experiment is designed to show is already assumed in the design of the experiment.

You might discuss with your class how this experimental design overcomes this. The elastic cords are shown to be identical; if they are stretched the same amount, they provide the same force (although we don’t know what that force is in newtons). Similarly, the three trolleys are identical so their masses are equal.

In principle, we can only say from this experiment that *F* is proportional to *ma*. In the SI system of units, we define the newton so that *F* = *ma*.

For an example of some real ticker-tape charts, click here.

*This experiment was safety-checked in March 2005*