V11 - physical pendulum

For the simple pendulum a point mass hangs from a massless string — an idealisation. A physical pendulum is the real case: an extended body, free to rotate about an axis Athat does not pass through its centre of mass. In the experiment it is an irregular body on a knife edge.

Two points determine everything: the axis of rotation A and the centroid S. Their distance is called s. At rest, S hangs vertically below A — any other position would turn there by itself.

Figure: the body deflected by α = 25°. The centre of mass shown is the true area centroid of the drawn outline; the lever arm of the weight is the horizontal distance s·sin α. The seven holes are the plate's suspension points in the laboratory — here it hangs from hole 1.

• A — axis of rotation (the knife edge)
• S — centroid of the body
• s — distance between the axis of rotation and the centre of mass
• α — deflection angle from the rest position
• M·g — weight, imagined as acting at the centre of mass

When deflected, the weight pulls the body back — but it does not pull at the pivot, it pulls beside it. This sideways offset is the lever arm, and force times lever arm is the torque, which drives the oscillation.

The torque is the decisive quantity of this experiment: it decides how strongly the body is turned back — and thus the period you measure.

Notation

Two meanings for “m” — a trap

In the evaluation, the Increase of the regression line is traditionally called m. That is why this module uses the capital M throughout for the Mass of the pendulum body. Anyone who confuses the two will reliably calculate wrongly later on.

The oscillation arises from an interplay: the weight turns the body back towards the rest position, its inertia makes it overshoot. How fast this happens follows from the angular momentum theorem.

What the formula contains

Three measurable quantities and one that is sought

M you weigh, s you measure, T you time — and g sits in the middle of it. Only J is still unknown: it depends on where where the axis lies. That is exactly what the next chapter is about.

The moment of inertia is not a fixed value of a body — it always belongs to a particular axis. For every axis parallel to the centroidal axis, the Parallel Axis Theoremapplies: to the body's own part J_S an addition is made that grows with the square of the distance between the centre of mass and the pivot.

Figure: the same body, again deflected by α . J_S belongs to the axis through the centre of mass, J to the axis of rotation at the distance s.

This equation is known as the Parallel Axis Theorem denotes.

For the irregular body of this experiment, J_S not cannot be calculated from the geometry — its shape yields no formula. This is not a weakness of the experiment but its appeal: J_S and g can be obtained together from one series of measurements in which you vary s .

If you suspend the body at a different point, s and J change — and with them the period. That dependence is exactly what you measure.

The approximation sin α ≈ α from chapter 2 is the price of the simple formula. It costs something — but little, as long as the deflection stays small. How little is shown by the drawing: it plots sin α and α itself against the angle.

Figure: sin α and α in radians over the angle. Up to about 20° both curves coincide; beyond that they diverge. The dashed marker is the limit below which measurements are taken.

What the approximation leaves out reappears in the period: at large deflection the real pendulum swings more slowly than the formula from chapter 2 promises — and all the more clearly, the further sin α lies below the straight line. The table gives the deviation from T₀, the period for very small angles:

deflection α	Deviation from T0
5°	+0,05 %
10°	+0,19 %
20°	+0,76 %
25°	+1,19 %

The drawing in chapter 1 shows 25° — clearly visible, but already at the edge of what is allowed. Measurements are made at smaller angles.

Check on the test bench

See for yourself

The digital test bench calculates without the approximation, with the full sin α — just like the real pendulum. Set the same distance s there and measure once with 5° and once with 20° deflection. The difference in the period is small, but it is there — and it is the reason why measurements are taken with small angles.

Measuring practice

Why at least 30 oscillations

Starting and stopping the watch introduces an error of a few tenths of a second — for a single period of about one second that would be deviations in the double-digit percentage range. If instead you stop 30 oscillations and divide by 30, the same error is spread over all periods and shrinks to below one percent.

The error does not vanish — it is divided. That is the cheapest gain in accuracy this experiment has to offer.

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 Axle section the straight line.

The formula from chapter 2 is not linear in s — and J itself depends on s . Both together are the reason why a raw plot of T on s yields nothing. The test bench's example series shows it:

Figure: period against the distance s. The curve first falls, passes through a minimum and rises again — no straight line can be fitted anywhere here. Only the right transformation turns this curve into a straight line; which one that is, is the first guiding question.

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

A word on naming: what the supplementary questions and the lab instructions call y-intercept is called here and in the evaluation tool c — both mean the value of the line at the position x = 0.

In this experiment both parameters carry something: one holds the gravitational acceleration, the other the moment of inertia of the body. Which parameter carries which quantity is the second key question.

On the digital pendulum, set the distance s between the axis of rotation and the centre of mass, deflect the body and measure the period — for several distances. In the lab you time the 30 oscillations from chapter 4; the digital test bench makes do with ten to save time.

A button then takes you straight on to station 9, the guiding questions.

Drawing

The pendulum, once more

You know the drawings from chapters 1 and 3. Now the labels are missing: drag them to the right place — with the mouse, with your finger, or by tapping first a label and then a field.

Your answers

Questions on the evaluation

1. Which quantities do you have to plot against each other so that a linear regression can be carried out? (This is — besides recording the measurements conscientiously — the decisive task of this experiment.)

2. How can the value of the acceleration due to gravity be determined from the parameters of the linear regression (slope, intercept)?

3. What error estimate Δg follows from the uncertainties of slope and/or intercept for the value of the acceleration due to gravity?

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: for each quantity you are after, first state the relationship symbolically and then evaluate it with your own regression parameters.

Your apparatus: Masse des Pendelkörpers M = steht am Prüfstand

1. Acceleration due to gravity g

   Relationship, symbolic Numerical value from your measurements
   m/s2

2. Intrinsic moment of inertia J_S of the pendulum 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.