V07 - parallel-axis theorem

A turntable sits on a vertical shaft. Between the shaft and the frame a spiral spring is clamped. Twist the table by the angle φ and let go, and the spring turns it back; it overshoots the rest position — the pendulum oscillates, but it oscillates about its own axis.

The factor C* means torsional constant of the spring and is measured in N·m. It says the same about the torsion spring as the spring constant says about a tension spring: which torque belongs to a given angle. The minus sign is essential — the torque always points back towards the rest position.

Older texts still add “/rad” to the torsional constant. That notation is outdated: an angle in radians is a pure number, and the unit remains N·m.

Figure: top view of the turntable. The added body sits at the distance a from the axis of rotation; the pointer at the rim shows the deflection φ relative to the rest position.

Principles

The analogy between linear motion and rotation

Between linear motion and rotation there is an analogy: every quantity of one has a counterpart in the other, and the laws keep their form. The translation rules of the analogy are:

straight-line	becomes, for rotation,
Distance s	Angle φ
Velocity v	Angle speed ω
Acceleration a	Angular acceleration φ̈
Force F	Torque M
Mass m	Moment of inertia J
spring constant C	torsional constant C*

Anyone who is fluent in one of the two columns and knows these translation rules can also argue with the quantities of the other column.

The difference lies in the moment of inertia: the mass of a body is a number. Its moment of inertia depends on which axis it rotates about — and that is exactly what this experiment is about.

Notation

Two meanings for “m” — a trap

In this module m stands for the mass of the added body.

In the evaluation, however, the slope of the regression line is also traditionally called m — that is how the tool labels it. Take care which of the two quantities is meant.

And M is always a torque in this module, never a mass.

Seven mounting points

The turntable has a radial slot with seven holes. The added body fits into each of them; its distance from the axis of rotation ranges from 0,040 m to 0,130 m. There is no hole over the axis itself — the hub sits there.

Apparatus

Two numbers you will need later

Each of the two additional bodies has the mass m = 0.86478 kg. The homogeneous cylinder has the diameter d = 0.068015 m. Chapter 4 calculates with these two values.

The equation of motion takes two lines — and from the first one it is linear. Click through it step by step; the drawing highlights what is being discussed.

Remarkable

The period does not depend on the amplitude

The formula contains no φ. Whether you twist the table by 5° or by 40°: the period stays the same. For a string pendulum this holds only approximately and only for small angles, because there the restoring torque goes with sin φ ; here it holds exactly.

This is not grey theory but testable in the experiment: measure the same mounting point once with a small and once with a large deflection. If the times differ, it is your timing — not the physics.

The period formula from chapter 2 needs the moment of inertia J_ges of the entire rotating system. It has two parts: the additional body and the turntable. Only the additional body has anything to do with the Parallel Axis Theorem — so the derivation starts with it.

Overview

Four moments of inertia, one experiment

Symbol	Moment of inertia …	Known?
JS	of the additional body about its centre of mass (intrinsic moment of inertia)	homogeneous cylinder: calculable · inhomogeneous body: sought
JK = JS + m·a²	of the additional body about the axis of rotation — this is where the parallel axis theorem acts	follows from JS and the plug position a
JT	of the turntable about the axis of rotation	unknown, a property of the rotary apparatus
Jges = JT + JK	of the whole rotating system — this is what appears in the period formula	is contained in the measured period

Principles

Where the term m·a² comes from

The moment of inertia is the sum of all mass elements times the square of their distance from the axis of rotation. If the distances are measured from the centre of mass of the body and that centre is displaced by a, the sum splits into three parts:

The middle term vanishes because the distances are counted from the centroid — there the moments of the mass elements cancel by definition. What remains is the Parallel Axis Theorem:

Spelled out, it states: the moment of inertia of a body about any axis is its moment of inertia about the parallel axis through the centre of mass, increased by the product of mass and the square of the distance between the axes.

It holds only for a displaced, parallel axis — and only if J_S refers to the centre-of-mass axis.

The experiment comes with two added bodies of the same mass, but with quite different mass distributions.

Figure: on the left the homogeneous solid cylinder — centre of mass at the middle, mass distribution known. On the right the inhomogeneous body: one side is heavier, the centre of mass lies off centre.

For the homogeneous solid cylinder the intrinsic moment of inertia can be found in any formula collection. With the mass and diameter from chapter 1 it is therefore a known number:

For the inhomogeneous body this does not work: you cannot tell from the outside how the mass is distributed inside. Its J_S is exactly what you are meant to determine in the end.

Principles

Which contribution is known?

Of the three contributions from chapter 3, the distance a only ever yields the Steiner term m·a² — for both bodies, because m is known. For the homogeneous cylinder, J_S is added; for the inhomogeneous body it remains unknown, and J_T is known in neither case.

The test bench therefore offers these contributions as axis options on both axes: the Steiner term m·a², the intrinsic moment of inertia J_S of the homogeneous cylinder and the moment of inertia of the additional body about the axis of rotation, J_K = J_S + m·a². Both options involving J_S use the known value of the cylinder — for the inhomogeneous body it is only a fixed offset, not its own intrinsic moment of inertia. J_ges and J_T are not on offer: they are the unknowns that only follow from the regression. Which choice yields a straight line for which body remains your decision.

The thread

First determine the system parameters, then measure

This dictates the course of the experiment. The formula from Chapter 3 contains three quantities that nobody can measure directly: the torsional constant C*, the moment of inertia of the turntable J_T and — for the inhomogeneous body — its J_S.

1. System parameters with the homogeneous cylinder. Here J_S is known. A series of measurements across the mounting points yields C* and J_T — both properties of the rotary apparatus.
2. Measuring the inhomogeneous body. The same rotary apparatus, hence the same C* and the same J_T. A second series of measurements then yields the sought J_S.

The values from part 1 are therefore not incidental in part 2 but a prerequisite. Note them down before you reset the table for the second series.

Each measurement scatters. One Straight line can be determined robustly and transparently from scattered data points, whereas fitting a curved relationship is more difficult. A nonlinear relationship is therefore linearised by choosing suitable variables for the axes. The quantity being determined is then contained in the Increase or in y-intercept the straight line.

In this form the formula from Chapter 3 is not linear: the period sits under a square root. The test bench's example series show it for both bodies:

Figure: period against the distance a for the homogeneous cylinder and the inhomogeneous body, calculated with the example values of the apparatus — your test bench draws its own for each session. Both curves bend upwards, neither is a straight line; the inhomogeneous one lies higher because of the larger J_S .

Before plotting, the formula must therefore be rearranged. Which quantities then belong on which axis is the first guiding question — and it is not the same for the two bodies, because more is known for the homogeneous cylinder than for the inhomogeneous one.

Linear regression determines that straight line y = m·x + cthat best fits your measurement points. Slope m and y-intercept c only carry a physical meaningif you derive the axes from the basic equation of the experiment. That choice is yours to make.

What the supplementary sheet calls y-intercept is called in the evaluation tool c — it is the value of the line at the position x = 0.

In this experiment both regression parameters carry a result: the slope gives a property of the spring, the y-intercept together with the slope gives a property of the rotating body. A y-intercept close to zero would therefore not be a good sign here but a hint at an error.

You measure twice with the same rotary apparatus — first with the homogeneous cylinder, then with the inhomogeneous body. The test bench counts the oscillations; stop each measurement after ten, which reduces the stopwatch error. Proceed as follows:

1. Select the body “Homogeneous” and stop the period for all seven plug positions; add each measurement to the table with “Add”.
2. Evaluate and note down slope and intercept — they yield C* and J_T, which you need in the second series.
3. Switch to the second measurement series: the green button in the evaluation window now reads "Next: Inhomogeneous body" and changes the test bench over. Doing it by hand works just as well — select the "Inhomogeneous" body. Either way the first measurement series stays saved.
4. Measure the same seven mounting points again.
5. Evaluate again — both evaluations appear further down on this page and go into the protocol.

After each evaluation a button takes you to the next measurement series; once all of them are evaluated it leads straight on to station 9, the guiding questions.

Drawing

The torsion pendulum, once more

You know the drawings from Chapters 1 to 4. Now the labels are missing: drag them to the right place — with the mouse, by tapping, or by first tapping a building block and then a field.

Your answers

Questions on the homogeneous metal cylinder

1. Which quantities do you have to plot against each other in a diagram in order to perform a linear regression?

2. How can the system parameters be determined from the regression parameters, that is, slope and y-intercept — the moment of inertia J_T of the turntable and the torsional constant C* of the torsion spring?

Questions on the inhomogeneous metal cylinder

1. Which quantities do you have to plot against each other here in order to perform a linear regression?

2. How can the system parameters J_T and C* from the evaluation of the homogeneous cylinder, together with the new regression parameters, give the moment of inertia J_S of the inhomogeneous body?

Your answers are stored locally and are transferred into your Preparation report taken over. There is no automatic evaluation — discuss your thoughts in the lab and in the colloquium.

Your result

Calculating the quantities you are after

Finally, complete the evaluation: the first two quantities follow from the measurement series with the homogeneous cylinder, the third from the second series. State each relationship symbolically first, then evaluate it with your own regression parameters.

1. Torsional constant C* of the torsion spring

   Relationship, symbolic Numerical value from your measurements
   N·m

2. Moment of inertia J_T of the turntable

   Relationship, symbolic Numerical value from your measurements
   kg·m²

3. Intrinsic moment of inertia J_S of the inhomogeneous body

   Relationship, symbolic Numerical value from your measurements
   kg·m²

There is no automatic assessment here either. Both entries go into your preparation report — so that in the end it contains not only the approach but also the result.

Note

Two measurement series, two evaluations

The protocol contains both measurement series — for each body the measurement table, the chart and the regression parameters. If an evaluation is missing, the protocol says so; therefore export only once both series have been evaluated.