V05 - spring-mass oscillator (dynamic)

Pull on a helical spring and it pulls back — the further it is stretched, the more strongly. In 1676 Robert Hooke put this into a sentence that still holds today: The force is proportional to the deflection.

The proportionality factor C means spring constant and is measured in N/m. Determining it is the goal of this part of the experiment — here by way of time instead of distance.

Figure: when a mass hangs on the spring, a new Restoration is established. Everything that follows is measured from it.

That is the decisive step: in the rest position, weight and spring force already balance each other. Displace the mass from there and all that remains is the additional spring force — and that always points back towards the rest position. Gravity only shifts the rest position; it does not drive the oscillation.

Notation

Three masses, two letters — a trap

In this module M is the attached mass and m_F is the mass of the spring.

The lower-case m is reserved for the slope of the regression line — that is its traditional name and the one the tool uses. The measurement column of the attached mass is nevertheless labelled “m / kg” at the test rig, for historical reasons.

Pull the mass out of its rest position and let go, and it oscillates. Click through it step by step; the drawing highlights what is being discussed.

Remarkable

Gravitational acceleration does not appear

The formula contains no g. A spring pendulum oscillates on the Moon just as fast as on Earth — the rest position simply lies elsewhere.

The derivation in Chapter 2 pretends the spring is massless. It is not — and its mass moves along. But not equally everywhere: the coils close to the suspension stay almost still, the lowest ones travel the full path of the mass.

Figure: the arrows show how far the coils move — from zero at the top to the full excursion at the bottom.

If you add up the kinetic energy of all the coils, the spring behaves as if a third of its mass were attached at its end as well. The period is therefore:

For heavy loads this hardly matters, for light ones it very much does. And it has a nice consequence: anyone who records a series of measurements over several masses can obtain from it not only the spring constant but also the mass of the spring — without ever putting it on a balance. How, that is the last key question of this part of the experiment.

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.

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

Figure: period against the attached mass M, calculated with example values for spring and spring mass — your test bench draws its own for each session. The curve rises like a square root, steeply at first and then ever more gently; no straight line fits anywhere.

Before plotting, the formula must therefore be rearranged — the supplementary questions call it “juggling the formulas a little”.

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.

Watch the intercept: it contains the co-oscillating spring mass from Chapter 3. Here it is not a nuisance term but a second measurement result — an intercept close to zero would not be a good sign but a hint at an error.

Attach the intended masses one after another, deflect the spring and measure the period — over many oscillations so that the stopwatch error stays small.

A button then takes you straight on to station 8, the guiding questions.

Drawing

The oscillating pendulum, once more

You know the drawing from Chapters 2 and 3. 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 dynamic determination

1. Which quantities do you have to plot against each other in a diagram in order to perform a linear regression? (Careful: here you have to juggle the formulas a little.)

2. What is the relation between the spring constant you are looking for and the regression parameters you determined, that is, slope and intercept?

3. One of the two regression parameters is not needed for the spring constant at all — but it can be used to determine the mass of the spring. How does the spring mass influence the period, and how do you get at it through the regression?

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.

1. Spring constant C

   Relationship, symbolic Numerical value from your measurements
   N/m

2. Spring mass m_F

   Relationship, symbolic Numerical value from your measurements
   kg

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.