B. Dynamics · Dynamics: Forces and Motion in a Plane

Investigating Circular Motion: The Whirling Stopper, Video Analysis and Graphs

B2.7 A1.1 A1.2 A1.5 A1.6 A1.8 A1.10 A1.11 A1.13About 75 minutes

Learning goals

I can plan and carry out an inquiry into the relationships among centripetal force, mass, radius, period, frequency and speed.
I can linearize data (for example, graph F against f²) and use the slope to test a theory.
I can measure a period and a radius from a video of a ride and calculate its centripetal acceleration.
I can estimate uncertainties, identify sources of error and write a conclusion supported by evidence.

Words to know

linearize
To choose what to graph (for example F against f² instead of F against f) so that a theoretical relationship gives a straight line.

slope
The rise over run of a best-fit line. Its value, with units, can be compared with a theoretical prediction.

random error
Unpredictable scatter in measurements (such as reaction time on a stopwatch). It is reduced by repeating trials and timing many cycles.

systematic error
An error that pushes every measurement the same way, such as friction in the tube or a string that is not horizontal.

percent uncertainty
The uncertainty of a measurement divided by the measurement, times 100%.

Testing Fc = 4π²mrf²

You derived the equations for uniform circular motion. Now a good scientist asks: does nature agree? In this lesson you will test the relationships with the classic whirling stopper experiment, analyse a video of an amusement-park ride, and learn how to turn a curved relationship into a straight-line graph whose slope tells you whether the theory works.

The curriculum suggests investigations such as video analysis of a ride and measuring the forces and period of a tetherball. The same skills apply to all of them.

The whirling stopper

A small rubber stopper (mass m) is tied to a string that passes through a smooth tube held in your hand. The other end of the string holds a hanging mass M. You whirl the stopper in a horizontal circle above your head while keeping the hanging mass still. Because M is at rest, the tension equals Mg, and that tension supplies the stopper's centripetal force:

Mg = 4π²mrf²

A paper clip on the string just below the tube marks the radius: if the clip stays just below the tube, r is constant. You change one variable at a time:

Vary M (the force) with r and m fixed: theory says F ∝ f².
Vary r with M and m fixed: theory says T² ∝ r.
Vary m (stack two stoppers) with M and r fixed: theory says T² ∝ m.

The whirling-stopper apparatus. The hanging mass sets the tension; the paper clip keeps the radius constant; the frequency is found by timing 10 revolutions.

Worked example

Example 1: Analysing whirling-stopper data

A 0.015 kg stopper is whirled at r = 0.60 m. The table gives the time for 10 revolutions for four hanging masses: M = 0.050 kg: 8.7 s; 0.100 kg: 6.1 s; 0.150 kg: 5.0 s; 0.200 kg: 4.3 s. Test the theory F = 4π²mrf².

Force F = Mg: 0.49, 0.98, 1.47, 1.96 N.
Frequency f = 10 ÷ time: 1.149, 1.639, 2.000, 2.326 Hz. Square them: f² = 1.321, 2.687, 4.000, 5.408 Hz².
A graph of F against f is curved. Theory says F = (4π²mr) f², so a graph of F against f² should be a straight line through the origin with slope 4π²mr.
The best-fit line has slope ≈ 0.36 N/Hz² (that is, kg·m) and an intercept of about 0.01 N, close to zero.
Predicted slope: 4π²(0.015)(0.60) = 0.355 kg·m. Percent difference = |0.361 − 0.355| ÷ 0.355 × 100% ≈ 2%.

The F–f² graph is linear and passes very close to the origin, and the slope agrees with the prediction to about 2%. The evidence supports F = 4π²mrf².

Force against frequency squared for the data in Example 1. The points lie close to a straight line through (nearly) the origin, as the theory predicts.

Worked example

Example 2: Video analysis of a carousel

A student films a carousel from directly above at 30 frames per second. A horse on the outer ring returns to the same position after 540 frames. Using a 1.0 m railing in the video as a scale, she finds the outer ring's radius is 7.5 m. Find the period and the centripetal acceleration of a rider on the outer ring.

T = 540 frames ÷ 30 frames/s = 18 s.
a = 4π²r ÷ T² = 4π²(7.5) ÷ 18² = 296 ÷ 324 = 0.91 m/s².

T = 18 s, a ≈ 0.91 m/s² (about 0.09g). Measuring several revolutions, and checking the scale against a second known length, would improve the result.

Errors, uncertainty and conclusions

Time many cycles. Reaction time adds an uncertainty of about ±0.2 s to each stopwatch reading. Over one revolution of 0.8 s that is 25%; over 10 revolutions (8 s) it is only 2.5%.
Uncertainties grow when you square. If T has a 2.5% uncertainty, then f² (or T²) has about a 5% uncertainty.
Systematic errors: friction where the string rubs the tube; a string that droops below the horizontal (so the circle's radius is smaller than the string length, and only the horizontal part of the tension is centripetal; if you use Mg and the full string length L, these two effects cancel exactly, but the circle is harder to keep steady); the hanging mass bobbing up and down.
A good conclusion answers the question, states whether the evidence supports the hypothesis, quotes numbers (slope, percent difference), names the main sources of error and suggests a specific improvement.
Report precision honestly (A1.13): your slope is 0.36, not 0.36092.

Just for fun

Why did the rubber stopper skip the party?

It was tired of going around in circles with the same old crowd, and it always got strung along.

Hands-on activity

Build and test a whirling stopper

You need: an empty ballpoint-pen barrel (ink tube removed) or a short piece of plastic tubing, about 1.5 m of strong thread or fishing line, a rubber stopper or a large eraser, several identical metal washers or coins in a small zip bag, a paper clip, tape, a metre stick, a stopwatch and safety glasses. Work outdoors or in a large room, away from people, lights and windows.

Tie the stopper firmly to one end of the thread. Pass the thread through the tube and tie the bag of washers to the other end.
Measure 0.50 m of thread from the stopper and attach the paper clip there, just below the tube.
Whirl the stopper in a horizontal circle over your head, adjusting the speed until the paper clip hovers just below the tube without touching it.
Have a helper time 10 revolutions. Repeat 3 times.
Repeat with 2, 3, 4 and 5 washers. Weigh one washer on a kitchen scale (or look up its mass) and weigh the stopper.
Graph F = Mg against f². Find the slope and compare it with 4π²mr. Write a full conclusion with sources of error.
Extension: keep the washers fixed and vary r (0.30 m to 0.80 m). Graph T² against r.

Check the knots before every trial.

Your turn

A stopper makes 10 revolutions in 7.5 s. Find its period and frequency.

Which graph should be a straight line if F = 4π²mrf²: F against f, or F against f²? Why?

Predict the slope of an F–f² graph for a 0.020 kg stopper at r = 0.50 m.

If the hanging mass is doubled (same r and m), by what factor does the frequency change?

Why should you time 10 revolutions instead of 1?

In the experiment that varies the hanging mass, name the independent, dependent and controlled variables.

More practice: warm-up

What supplies the stopper's centripetal force?

What does the paper clip do?

At 30 frames per second, how long is 90 frames?

Is reaction time a random or a systematic error?

If a measurement has a 3% uncertainty, about what is the uncertainty of its square?

More practice: on your own

With M fixed (F = 1.96 N), a student measures: r = 0.40 m, T = 0.50 s; r = 0.60 m, T = 0.61 s; r = 0.80 m, T = 0.71 s. Show that T² ∝ r, and estimate the stopper's mass.

In a video at 30 frames/s, a Ferris wheel turns a quarter of a revolution in 150 frames. Its radius is 12 m. Find the period and a rider's centripetal acceleration.

A student's period measurement has an uncertainty of ±3%. Estimate the uncertainty in her calculated centripetal force, F = 4π²mr ÷ T² (m and r are precise).

The stopper's string droops below the horizontal during a trial. Explain two ways this affects the experiment.

A tetherball on a 1.6 m rope swings in a horizontal circle with the rope at 40° to the pole. Predict its period using T = 2π√(L cos θ ÷ g) (from Centripetal Force). A student measures 2.3 s. Find the percent difference.

Challenge

Use the slope from Example 1 (0.361 kg·m) to calculate the stopper's mass, and compare it with the true mass of 0.015 kg.

The best-fit line in Example 1 has a small positive intercept (about 0.01 N) instead of passing exactly through the origin. Suggest a physical cause and say whether it matters.

Plan an inquiry to test whether centripetal acceleration is proportional to the radius for objects on a turntable spinning at a fixed rate. Give the hypothesis, variables, method and the graph you would draw.

Check your understanding

Quiz

In the whirling-stopper experiment, the centripetal force on the stopper is supplied by:

To get a straight-line graph from F = 4π²mrf², you should plot F against:

The slope of an F–f² graph equals:

Timing 20 revolutions instead of 2 mainly reduces:

A ride is filmed at 30 frames per second and completes one revolution in 450 frames. Its period is:

Which is a systematic error in the whirling-stopper experiment?

For parents and teachers

Big idea: Students test the circular-motion equations by experiment (B2.7) and practise the full inquiry cycle: question and hypothesis, variables, safe procedure, data tables, linearized graphs, uncertainty and evidence-based conclusions (A1.1, A1.2, A1.5, A1.6, A1.8, A1.10, A1.11, A1.13).

How to help: Be the timer! Then ask your teen why the graph uses f² instead of f, and what the slope should be. If the slope matches the prediction, the theory survives the test.

Common difficulties:

Plotting F against f and trying to draw a straight line through curved data.
Using the string length rather than the circle's radius.
Writing a conclusion with no numbers, or blaming "human error" without saying which error and how it affected the result.
Quoting slopes with far too many digits.

Safety: safety glasses, strong knots, and plenty of clear space for the whirling stopper.