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4.3 Experiments to investigate Newton's 2nd law of motion (F = ma)

The experiments described below can be used to investigate how force and mass affect acceleration.

These experiments use light gates to measure speed - so they need to be explained!

A light gate directs a beam of light from one side to a detector on the other side (no details shown, just the labels!).

When the light beam is cut by something passing through the light gate (e.g. the a card on a trolley) the light gate measures the time the light beam is not detected.

To make an acceleration measurement the moving object must break the light beam twice and there are two ways of doing this - versions 1 and 2 experiments are both described below. Yu need two speeds and the difference between them to calculate an acceleration.

The light gate is connected to a computer and the rest is explained via the diagram below.

In version 1, with one light gate, you use an inverted Π shaped card (rectangle with a middle section cut out).

The lengths of the upright sections that cut the light beam are d1 and d2 (which you can make the same).

As the trolley and card pass through the light gate, the first section of card of length d1, cuts light beam for time t1 (for speed 1).

When the second section of the card of length d2 passes through the light gate, the beam is cut for time t2 (for speed 2).

At this point the combined light gate also records the time difference t3 between the two speed measurements (for acceleration a). So for speed s ...

s1 = d1/t1, s2 = d2/t2, acceleration = change in speed / time taken = a = (s2 - s1) / t3

However, all you do is input d1 (= d2) into the computer and the software does the acceleration calculation for you!

Obviously, the computer software system must be set up to suit the way you have set up the experiment.

 

In version 2, with two light gates, you use a rectangular shaped card [   ] of length d.

You set up two light gates a suitable distance apart.

When the trolley and card pass through the first light gate, the beam is cut for t1.

After passing through the 2nd light gate, time t2 is recorded.

At this point the combined light gates also record the time difference t3 between the two speed measurements.

Its now a similar data situation to version 1 of the experiment, so for speed s ....

s1 = d/t1, s2 = d/t2, acceleration = change in speed / time taken = a = (s2 - s1) / t3

Again, the computer software system must be set up to suit the way you have set up the experiment and will do the acceleration calculation for you.

Example of calculation for experiment version 2

Suppose the length of the trolley is 10 cm (0.10 m) and interrupts the 1st light gate for 0.25 s.

The velocity at the 1st light gate is 0.10 / 0.25 = 0.40 m/s.

The trolley interrupts the 2nd light gate for 0.10 s.

The velocity at the 2nd light gate is 0.10 / 0.10 = 1.0 m/s.

The time taken for the trolley to travel between the light gates was 0.80 s.

Therefore the acceleration = (v - u ) / ∆t = (1.0 - 0.40) / 0.80 =  0.75 m/s²

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Investigation version 1 using one light gate

Error correction NOT shown in the diagram above

Strictly speaking you should do the experiment on a slightly tilted running board rather than the horizontal laboratory bench shown in the diagrams above.

You tilt the running board ramp down to the right until the trolley (without hook attached) just begins to move on its own.

You then prop up the left-hand end at the same height and then perform the experiments as described.

After doing this, the weight of the trolley down the slope will compensate for the friction acting between the wheels of the trolley and the running board ramp.

It is also possible to do these investigations with a horizontal air track on which a 'trolley' hovers on jets of air - this is an alternative investigation design to minimise the effects of friction in the experiments.

The experiment is conducted on a smooth bench on which a trolley can freely run.

You need to mark on a start line to keep things consistent.

The total mass of the trolley and extra weights (m) is measured and can be altered by adding extra weights.

On the trolley is fixed a 'rectangular' Џ shaped card, whose two arms will block the light beam twice when the trolley passes through the light gate when it measures the time taken twice.

The trolley is connected by wire or string to the weights via the pulley wheel, that on falling, provide the downward force to accelerate the trolley along the 'runway'

The time measured by the light gate system is recorded electronically and passed to the connected data logger or computer.

As the Џ card passes through the light gate, the light beam is cut off twice, so two velocities are measured and the time interval between them.

The principles of the F = ma calculations for version 1.

The accelerating force F is given by the falling weight: mass in kg x 9.8 N/kg

The mass being accelerated = total mass of trolley plus weights (m)

The acceleration is given by the light gate velocities divided by the time interval between them: a = v₂ - v₁ / Δt

You can do a series of experiments with constant trolley mass m, and varying the accelerating force F by varying the falling weights on the end of the hook.

Theoretically, a graph of force F versus acceleration a should be linear - or look something like it!

The gradient is given by the trolley mass m (the constant).

You should definitely be able to show that increasing the applied force a greater acceleration is produced, ideally your results are a linear graph.

Newton's 2nd law equation: F = ma, F a, and the constant gradient should be the total mass of the trolley.

 

Extending the investigation

You can then investigate the effect of varying the weights on the trolley, to vary its mass and applying a constant force of acceleration using the same falling weight.

You measure the acceleration in the same way as described.

You should be able to show the greater the total mass of the trolley m (the greater its inertia), the smaller the acceleration produced by a constant accelerating force (F).

A graph of mass versus acceleration should be a downward curve.

F = ma, as m gets bigger, a gets smaller for constant F.

m = F/a   and  a = F / m

You can do a linear graph analysis of your results e.g.

A graph of m versus 1/a should be linear with a gradient F.

A graph of acceleration a versus 1/m should be linear with a gradient F.

 

Sources of error

Friction in the wheel axles and friction between the wheels and the running board will have a small retarding effect. This will reduce the acceleration a little bit.

One way to compensate for this is to tilt the running board slightly until the trolley just moves, but don't overdo it or you will record accelerations greater than you should.

You can do more accurate experiments using an air track on which the 'trolley' is made to hover on jets of air.

Using the light gates and a computer is more accurate than trying to make direct observations with a stopwatch.

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Investigation version 2 using two light gates

Error correction NOT shown in the diagram above

Strictly speaking you should do the experiment on a slightly tilted running board rather than the horizontal laboratory bench shown in the diagrams above.

You tilt the running board ramp down to the right until the trolley (without hook attached) just begins to move on its own.

You then prop up the left-hand end at the same height and then perform the experiments as described.

After doing this, the weight of the trolley down the slope will compensate for the friction acting between the wheels of the trolley and the running board ramp.

It is also possible to do these investigations with a horizontal air track on which a 'trolley' hovers on jets of air - this is an alternative investigation design to minimise the effects of friction in the experiments.

As the [    ] card goes past the light gates, the two time intervals for blocking the light give you an initial and final velocity, and the time interval between these two events can be used to calculate the acceleration of the trolley.

The investigations, calculations and graphs you can do are described above in method 1.

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Version 3 of acceleration experiments.

You can run the trolley down an inclined ramp onto the level running board and then past two light gates (2 and 3) to measure the final velocity.

You can set the first light gate (1) near the start line of the inclined ramp - the initial velocity should be close to zero.

You can vary the angle of the ramp - the steeper angle should give a greater acceleration.

You can vary the length of the inclined ramp to see if a longer running downhill distance increases acceleration.

You can vary the total mass of the trolley to if this has any effect on the acceleration.

 

INDEX for physics notes on Newton's Laws of Motion

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Keywords, phrases and learning objectives for experiments to show the validity of Newton's 2nd law of motion

Be able, with the aid of diagrams, describe and explain experiments to investigate Newton's 2nd law of motion using light gates and an accelerating wheeled object of some description.

Be able to the methods and procedures to gather data and calculate the results and produce graphs that prove Newton's second law of motion.

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