Mathematics 11–12 · Year 12

Mass on a spring and a conical pendulum: period against mass, radius and angle

Applications of calculus to mechanics: Simple harmonic motion; Forces and further motion in a straight line (Mathematics Extension 2, Year 12)

Practical, model not builtLow risk

This site has no interactive model of its own. Where a step or a material names a Concept Studio model, simulation or tool, it has not been built; an external simulation a step names (for example PhET) is not part of this site.

The idea

A restoring force proportional to displacement gives simple harmonic motion whose period depends on the mass and the stiffness and not on the amplitude, and resolving the tension in a whirled line gives a conical pendulum whose period depends on the line length and the cone angle and not on the mass.

Safety card

Low riskAn adult supervises

Hazards

slotted masses sliding off the carrier onto hands or feet
a spring loaded past its elastic limit springing back
a whirled bung striking a person, a window or glassware

Controls

G-clamp the stand to the bench and stand the loaded spring over a foam tray
keep the load under the spring's stated limit and check the spring returns to its unloaded length between sets
wear safety glasses while loading the spring and while the bung is moving
clear a 2 m circle, use a rubber bung on braided cord, and tape the knot at the bung

Note

No chemicals and no heat are used, so the NSW Department of Education Chemical Safety in Schools package does not apply; record the loaded spring and the whirled mass in the school's risk assessment system, such as RiskAssess, before the lesson.

What you need

A helical steel spring with a spring constant near 25 N/m, hung from a boss and clamp on a retort stand that is G-clamped to the bench
A slotted mass carrier with 50 g slotted masses to 500 g, and a balance reading to 1 g (the spring is weighed as well)
A metre rule clamped vertically beside the spring and a card pointer taped to the mass carrier
A stopwatch reading to 0.01 s, or a phone running the phyphox Spring experiment taped to the carrier
A foam tray or a folded towel on the bench below the masses
For the conical pendulum: 1.20 m of thin braided cord, a 50 g rubber bung drilled for the cord, a second retort stand with a clamp as the fixed suspension point, a protractor and a tape measure
A phone camera and Tracker on a laptop for filming the circle from above and from the side

How to do it

Hang the carrier alone and record the pointer reading against the metre rule. Add 50 g at a time to 500 g, recording the pointer each time, and plot extension in metres against load in newtons. Take the gradient and invert it to get the spring constant k, using 9.80 m/s^2 for g.
Load 200 g, pull the carrier down 30 mm, release it and time 20 complete oscillations. Repeat the timing three times and divide the mean by 20 for the period. Repeat for 300 g, 400 g and 500 g.
Plot the square of the period against the load in kilograms. Take the gradient and compare it with 4 pi^2 / k.
Weigh the spring and repeat the comparison with one third of the spring's mass added to every load, then state which version fits the measured periods better.
Repeat step 2 at an amplitude of 15 mm instead of 30 mm and record whether the period changes by more than the timing uncertainty.
Hang the bung on a 0.600 m cord from the clamp. Start it moving in a horizontal circle by hand, let the motion settle, then film it from the side and from above.
Measure the circle radius from the overhead frames and time 20 complete circuits three times. Compute the cone angle from sin(theta) = r / L.
Predict the period 2 pi sqrt(L cos(theta) / g) and the speed 2 pi r / T, compare both with the measured values, and repeat at a larger radius.
Resolve the tension for the measured angle: the vertical component carries the weight and the horizontal component supplies m v^2 / r. Check the two expressions for the horizontal force against each other.

What you should see

A spring that extends 39.2 mm under a 100 g load has k = 0.98 / 0.0392 = 25.0 N/m. With 200 g the period is 2 pi sqrt(0.200 / 25.0) = 0.562 s, so 20 oscillations take 11.2 s; the periods for 300 g, 400 g and 500 g are 0.688 s, 0.795 s and 0.889 s. The plot of the squared period against load is a straight line of gradient 4 pi^2 / k = 1.579 s^2/kg. A spring of mass 20 g adds 6.7 g to every load and lifts the 200 g period from 0.562 s to 0.571 s, a 1.7 per cent change that shows as 0.19 s over 20 oscillations. Halving the amplitude does not change the period, because n = sqrt(k / m) = 11.18 rad/s carries no amplitude; the 30 mm amplitude gives a maximum speed n A = 0.335 m/s and a maximum acceleration n^2 A = 3.75 m/s^2. On a 0.600 m cord at a 30 degree cone angle the radius is 0.300 m, the predicted period is 1.447 s (20 circuits in 28.9 s) and the predicted speed is 1.303 m/s; the tension on a 50 g bung is 0.566 N, whose horizontal component 0.283 N equals m v^2 / r. Opening the cone to 45 degrees shortens the period to 1.307 s and raises the speed to 2.04 m/s, and as the angle closes towards zero the period approaches the simple pendulum value 2 pi sqrt(L / g) = 1.555 s. The learner knows it worked when the squared-period line is straight with a gradient within 5 per cent of 4 pi^2 / k, when the measured conical period sits within 5 per cent of the value predicted from the measured radius and cord length, and when the two expressions for the horizontal force agree.

What changes

What you change: suspended mass on the spring, and the radius of the circle for the conical pendulum
What you measure: period of oscillation, and the time for one circuit of the circle
What you keep the same: the same spring throughout
amplitude held constant within each set
cord length of 0.600 m
g taken as 9.80 m/s^2
timing over 20 complete cycles

Common misconceptions

Each of these ideas is wrong, and the activity is a chance to test it.

A heavier load oscillates faster because gravity pulls it harder; the period grows as the square root of the mass, and the added weight only shifts the centre of the oscillation.
The restoring force is measured from the spring's unloaded length; in simple harmonic motion it is measured from the loaded equilibrium, where the spring force already balances the weight.
A larger amplitude gives a longer period; for a linear spring the period is independent of the amplitude.
A heavier bung takes longer to go round; the mass cancels out of the conical pendulum period, as it does for a simple pendulum.
An object moving at a steady speed in a circle has no acceleration; its direction changes, so it accelerates towards the centre at v^2 / r.
The cord can be whirled until it is horizontal; the vertical component of the tension has to carry the weight, so the cord can never reach the horizontal.

Curriculum references

The NSW syllabus outcomes and Australian Curriculum v9 codes this activity supports. They are references, not a verified or complete curriculum alignment.

Mathematics Extension 2 11–12 Syllabus (2024), Year 12 focus area Applications of calculus to mechanics; Extension 2 is a Year 12 course taught from Term 4 2026 with the first HSC examination in 2027, and it replaces the Mathematics Extension 2 Stage 6 Syllabus (2017) taught until then; page read 2026-09-23ME2-12-05
Australian Curriculum v9No Australian Curriculum v9 code is listed.

Sources

The pages the author read to write this activity.

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