5.4 Investigating Hooke's law — Learn

Prerequisites: the whole of this chapter, and reading a graph.

Objectives: describe how to measure the extension of a spring as the force on it changes; find the spring constant from the gradient of the graph; say what the graph shows beyond the limit of proportionality; state the uncertainty in a reading.

By the end of this section, you can carry out the investigation and read the spring constant off your own graph.

This section is the practical part of the chapter, and it is also preparation for the Physics In Practice Investigation, which is 40% of your Leaving Certificate mark. The skills are the same: take the readings, plot them, find what the graph says, and be honest about what you could not control.

The apparatus and the method

You need:

a retort stand, a boss and a clamp;
a spring;
a metre stick, fixed vertically beside the spring;
a set of known masses and a hanger;
a pointer attached to the bottom of the spring, so you read the same point each time.

Method.

Record the position of the pointer with no masses on the hanger. This gives the natural length.
Add one mass at a time. Record the new pointer position each time.
Work out the extension each time: the new position minus the natural position. Do not use the total length.
For each mass, work out the force. The weight is $W = mg$ , with $g = 9.8\,\text{N/kg}$ .
Plot force on the vertical axis against extension on the horizontal axis.
Draw the straight line of best fit through the points that lie on a straight line, and find its gradient.

figure hooke-law

Predict, then test. Before you plot anything, write down what you expect the graph to look like if the spring obeys Hooke's law. Then look at your points. Explain any point that does not lie on the line — a single stray point usually means a misreading, but a run of points curving away at large extensions means the spring has passed its limit of proportionality.

Reading the spring constant off the graph

The gradient of the straight line is the spring constant.

\text{gradient} = \frac{\text{change in force}}{\text{change in extension}} = \frac{\Delta F}{\Delta x} = k

Example 1. Two readings from one experiment: a force of 2.0 N gives an extension of 0.05 m, and a force of 6.0 N gives 0.15 m. Calculate the spring constant.

k = \frac{\Delta F}{\Delta x}

k = \frac{6.0 - 2.0}{0.15 - 0.05}

k = \frac{4.0}{0.10}

k = 40\,\text{N/m}

Use the two points as far apart as possible on the straight line. That reduces the effect of any small reading error on the gradient.

Why a graph rather than one pair of readings. A single pair of readings gives a spring constant, but it cannot tell you whether the spring was behaving. A straight line through every point shows that it was; a line that bends shows where it stopped. The gradient also uses all the points, so one poor reading has less effect.

The uncertainty in a reading

A metre stick read to the nearest millimetre has a reading uncertainty of $\pm 0.001\,\text{m}$ . It is worth giving that as a percentage of the value, because that says how much the uncertainty matters.

\text{percentage uncertainty} = \frac{\text{uncertainty}}{\text{reading}} \times 100\%

Example 2. An extension is measured as 0.050 m. Calculate the percentage uncertainty.

\text{percentage uncertainty} = \frac{0.001}{0.050} \times 100\%

\text{percentage uncertainty} = 2\%

Watch out. A small extension measured on a metre stick carries a large percentage uncertainty. The same $\pm 0.001\,\text{m}$ on an extension of 0.010 m is 10%. That is why the method asks you to stretch the spring enough to get readable extensions, and why the smallest readings on your graph are usually the ones that scatter most.

Sources of error, and what to do about them

Reading the pointer from an angle. Parallax makes the reading depend on where your eye is. Fix your eye level with the pointer, or use a set square.
Reading the total length instead of the extension. The single most common mistake. Always subtract the natural length.
Using the mass instead of the weight. A 100 g mass has a weight of $0.100 \times 9.8 = 0.98\,\text{N}$ , not 100 N. Convert.
Overloading the spring. Past the limit of proportionality the graph bends and the gradient no longer gives $k$ . Stop before the last few points curve.
The spring swinging. Let it settle before you read.

Progress check

This is where the chapter problem is answered, and it is worth writing down properly.

You have a graph with a straight part and a bend. State the spring constant, with its unit, from the gradient of the straight part. Then write down the extension at which your graph stopped being straight. Finally, explain in two sentences what the spring is doing past that point, using the words limit of proportionality and elastic limit — and say which one comes first.

Keep this. Chapter 22 is about writing up an investigation, and this is the shape of it.

You should now be able to

describe how to measure force and extension for a spring, and name the apparatus;
find the spring constant from the gradient of a force-extension graph, using two points far apart;
say why a graph is a better way to find $k$ than a single pair of readings;
calculate the percentage uncertainty in a reading;
describe the main sources of error and what to do about each.