Wait, What? You can increase the hanging mass and accidentally test two variables at once.

That is one of the classic traps in a Newton’s second law trolley experiment. If you simply add more mass to the hanger, you increase the driving force—but you also increase the total mass of the accelerating system. The resulting graph may still look sensible while no longer testing the clean relationship you thought you were testing.

The practical job is therefore not “show that F = ma.” It is to design a system in which force, mass and acceleration are measured or controlled in a way that lets one relationship be tested at a time, while accounting for friction, pulley effects and the finite response of the measuring system.

The scientific model

For a system of constant mass:

F_net = ma

Therefore, if m is held constant, acceleration a should be proportional to net force. If net force is held constant, acceleration should be inversely related to mass.

The Institute of Physics uses a trolley, pulley, hanging load and light gate to investigate this relationship. Crucially, it recommends transferring masses from the trolley to the hanger so the total accelerating mass stays constant while the driving force changes. See the IOPSpark Newton’s second law practical.

What counts as the system?

The trolley does not accelerate alone. The hanging mass also accelerates, and the string and pulley can contribute to the dynamics. In the simplest school model, we treat trolley + hanger + added masses as one accelerating system.

If the hanging mass is m_h, its weight m_hg provides the nominal driving force. But the true net force is smaller because friction, pulley resistance and other losses oppose motion.

This makes the practical a strong lesson in model boundaries: the force you calculate from hanging weight is not automatically the net force acting on the whole system.

Why moving masses matters

Suppose the trolley system initially has 500 g on the trolley and 50 g on the hanger. If you move 50 g from the trolley to the hanger, the total system mass remains the same, but the hanging weight increases.

That isolates the effect of changing driving force far better than simply adding more hanging mass from outside the system.

Friction can write the intercept

If friction is approximately constant, the simple model becomes:

m_hg − F_friction ≈ ma

Rearranging:

a ≈ (g/m)m_h − F_friction/m

A graph of acceleration against nominal driving force may therefore be straight but not pass through the origin. The intercept can contain information about friction or systematic offsets.

Do not “fix” the graph by forcing the best-fit line through zero unless the method and evidence justify that constraint.

Runway alignment matters

A tilted runway adds or subtracts a component of the trolley’s weight along the track. A tiny slope can therefore imitate an extra driving force or extra resistance.

Level the track carefully. One useful diagnostic is to see whether the trolley tends to roll systematically when nominally unloaded. If it does, the runway may not be horizontal or the wheels may not be mechanically balanced.

How light gates infer acceleration

A light gate does not “measure acceleration” directly in the same way a balance measures mass. It detects when a card interrupts the beam. Software can infer speeds from card width and interruption time, then acceleration from changes in speed over time or between segments.

This creates a measurement chain:

beam interruption → time interval → speed → acceleration

If the card width is entered incorrectly, every inferred speed can be biased. A digital readout is only as trustworthy as the geometry supplied to the software.

Alternative: motion sensor or video

A motion sensor can provide position or velocity against time. Video analysis can track the trolley frame by frame. These methods produce richer datasets but add new assumptions: sampling rate, tracking point, scale calibration and perspective.

The best instrument is not automatically the most digital one. Choose the method whose measurement chain you can explain and control.

Pulley inertia is real

A real pulley rotates, so some of the driving energy accelerates the pulley’s rotational motion. This means the simple translational model can underestimate the system’s effective inertia.

At school level, pulley inertia is often neglected. At higher resolution, it helps explain why measured acceleration can be smaller than a naive model predicts even after obvious trolley friction is reduced.

String mass and stretch

An ideal string is massless and inextensible. A real string has mass and can stretch slightly. If string mass is not negligible relative to the moving masses, or if it stretches dynamically, tension need not be represented perfectly by the simplest model.

These are usually second-order effects in a school setup, but they define the evidence boundary.

Quantitative window: one force-series design

Suppose total accelerating mass is kept at 1.00 kg. Hanging masses are 0.05, 0.10, 0.15 and 0.20 kg. Using g ≈ 9.81 m s⁻², nominal driving forces are approximately:

• 0.49 N
• 0.98 N
• 1.47 N
• 1.96 N

If friction were negligible, expected accelerations would be roughly the same numerical values in m s⁻² because total mass is 1.00 kg.

If measured accelerations are 0.35, 0.84, 1.33 and 1.81 m s⁻², the approximately constant shortfall suggests an opposing force of roughly 0.14–0.15 N.

Graph reasoning

Plot acceleration a against nominal force F. In the ideal constant-mass model:

a = F/m

so gradient = 1/m.

If the measured gradient is 0.98 kg⁻¹, that supports an effective mass near 1.02 kg. A negative intercept can indicate friction. This is more informative than calculating F/a separately for each point and ignoring systematic structure.

Second experiment: vary mass at fixed force

To test a ∝ 1/m, keep the driving force as constant as possible and add mass to the trolley. Plot a against 1/m. A linear relationship supports the model.

Do not plot a directly against m and expect a straight line; the theoretical relationship is inverse.

Observation versus inference

Observation: “When nominal driving force increased from 0.49 N to 1.96 N, measured acceleration increased from 0.35 to 1.81 m s⁻².”

Inference: “Acceleration increased approximately linearly with applied driving force for the constant-mass system.”

Stronger inference: “The data support F_net = ma over the tested range, with an approximately constant opposing force accounting for the non-zero intercept.”

Failure modes that cap standards

• Adding hanging mass without removing mass from trolley: force and total mass change together.
• Track not level: gravity contributes along the runway.
• Large wheel friction: nominal force differs substantially from net force.
• Card width entered wrongly: light-gate speed is systematically wrong.
• Hanger hits floor too soon: the driving force suddenly disappears.
• String slack: force transmission is not continuous at release.
• Few widely spaced data points: curvature or intercept behaviour can be missed.

Unfamiliar transfer: air-track glider

An air track reduces contact friction dramatically, but the same logic survives: define the system, hold the right variable constant, measure acceleration, and ask whether the nominal applied force is really the net force. Lower friction improves one assumption; it does not remove the need for system thinking.

Secondary → JC → deeper Physics

Secondary: vary force at constant mass, measure acceleration, plot a against F and recognise friction as an opposing force.

JC: define the system carefully, analyse intercepts and gradients, test inverse-mass relationships, and discuss pulley inertia, timing resolution and uncertainty.

Deeper Physics: dynamics experiments extend to force sensors, rotational inertia, system identification, friction models, uncertainty propagation and numerical fitting.

Checkpoint 1: the double-change trap

A student increases the hanging load from 50 g to 100 g by adding 50 g from the bench. What two physical quantities changed?

Checkpoint 2: the non-zero intercept

An acceleration-versus-force graph is straight but crosses the force axis at +0.12 N. What does that suggest?

Answer key and WHY reasoning

Checkpoint 1: driving force increased and total accelerating mass increased. The experiment no longer isolates the force–acceleration relationship cleanly.

Checkpoint 2: about 0.12 N of applied force may be needed to overcome an approximately constant opposing force before acceleration extrapolates to zero. Friction is a plausible explanation, though other systematic offsets should also be checked.

How to study this practical

Before touching the apparatus, draw a boundary around the accelerating system. Then list every external force on that system. Finally write what must remain constant for the graph you plan to make. If you cannot define the system, you are not yet ready to interpret the graph.

Evidence boundaries

The trolley experiment tests Newtonian dynamics over modest speeds and accelerations where relativistic effects are irrelevant and apparatus friction can be approximated. It does not prove that friction is exactly constant, the pulley is massless or the string ideal. Those are model assumptions whose adequacy is judged from the data.

Authoritative next steps

• Institute of Physics: Investigating Newton’s second law of motion
• Institute of Physics: supporting A-Level practical requirements
• SEAB O-Level syllabus directory
• SEAB A-Level syllabus directory

Teaching Guide

Ask students to design two versions of the experiment: one that changes force while keeping total mass constant, and one that accidentally changes both. Let them predict the graphs before running either. The deepest learning comes when they see that a graph can be numerically smooth yet answer the wrong scientific question.