CXC CSEC Physics

Galileo and the method of physics

Before Galileo Galilei (1564–1642), most accounts of motion and the heavens rested on the authority of ancient writers, above all Aristotle. Galileo's contribution was a way of working rather than a single discovery: he observed carefully, described what he saw in mathematics, and tested ideas by experiment, accepting the result even when it contradicted received opinion.

Several of his studies show the method. Rolling balls down gently sloping planes slowed the motion enough to time it, and showed that the distance covered grows with the square of the time, so that a falling body gains speed steadily. He argued, and tested with bodies of different weight, that in the absence of air resistance heavy and light objects fall together, which contradicted Aristotle's claim that heavier bodies fall faster. He timed the swing of a pendulum and found that the period hardly changes as the swing dies away. With a telescope he saw moons circling Jupiter, the phases of Venus and mountains on the Moon, and by patient daily observation of sunspots he showed that they lie on the surface of the Sun and move with its rotation. Each of these was a measurement that other people could repeat.

The steps of the scientific method as Galileo used it, and as physicists still use it, are these.

1. Observe something and ask a definite question about it.
2. Form a hypothesis, a proposed explanation that could turn out to be wrong.
3. Predict what should happen if the hypothesis is true.
4. Design an experiment that tests the prediction, changing one quantity at a time.
5. Record the measurements, analyse them (often by a graph) and compare them with the prediction.
6. Accept, change or reject the hypothesis, and publish the method and results so that others can repeat them.

A hypothesis is useful only if an experiment could show it to be false. Repeating the experiment, and having other people repeat it, guards against a result that came from a mistake or a chance.

Variables and a fair test

The quantity the experimenter deliberately changes is the independent variable. The quantity that is measured to see the effect is the dependent variable. Every other quantity that could affect the result is a control variable and is kept constant, otherwise it is not clear which change caused the effect. A test in which only one variable is changed at a time is a fair test.

Writing up an investigation

An experimental report is written so that someone else could repeat the work from it alone. It normally has these parts, in order: the aim; the hypothesis or prediction; the apparatus, usually with a labelled diagram; the method, as numbered steps in the past tense, stating which variable is changed, which is measured and which are controlled; a table of results with headings and units; the analysis, including graphs and any calculation; the conclusion, which answers the aim and says whether the hypothesis is supported; and the sources of error, with the precautions taken and improvements that could be made.

The simple pendulum

A simple pendulum is a small heavy bob on a light string fixed at the top, swinging through a small angle. One oscillation (one complete swing) is the motion from one extreme, across to the other side and back again. The period $T$ is the time for one oscillation.

Three factors might affect the period: the length of the string, the mass of the bob and the angle through which it is pulled aside before release (the angle of displacement).

Method

1. Fix the string firmly between two wooden blocks or in a clamp, so that the length does not change as the pendulum swings.
2. Measure the length $L$ from the point of suspension to the centre of the bob.
3. Pull the bob through a small angle (no more than about $10^\circ$), release it, and start the stopwatch when it passes through the middle of its swing.
4. Time 20 oscillations, and divide by 20 to find $T$. Repeat and average.
5. Change one factor only: vary $L$ with the same bob and angle; then vary the mass of the bob with the same length and angle; then vary the angle with the same length and mass.

Timing 20 swings rather than one reduces the effect of the experimenter's reaction time, which is about the same at the start and the end but is now shared among 20 periods. Starting the timing at the centre of the swing, where the bob moves fastest and so passes a fixed marker (a pin behind it) in the shortest time, makes it easier to judge the instant.

Results

The period depends on the length of the string and not, for small angles, on the mass of the bob or the size of the swing. Increasing the length increases the period, but not in proportion: doubling $L$ does not double $T$. A plot of $T$ against $L$ is a curve through the origin. A plot of $T^2$ against $L$ is a straight line through the origin, so $T^2$ is directly proportional to $L$. The relationship, which can be derived theoretically, is

where $g$ is the gravitational field strength, $\mathrm{N\,kg^{-1}}$.

The gradient of the $T^2$ against $L$ graph is $4\pi^2/g$, which allows $g$ to be found:

Worked example: a class measures the time for 20 oscillations at four lengths.

The points lie close to a straight line through the origin. Taking the points $(0.25, 1.00)$ and $(1.00, 4.04)$ on the line, the gradient is

and so $g = 4\pi^2/4.05 = 39.5/4.05 \approx 9.75\ \mathrm{m\,s^{-2}}$. This agrees with the accepted value of about $9.8\ \mathrm{m\,s^{-2}}$ to within the precision of the timing.

The period does not depend on the mass because a heavier bob is pulled harder by gravity but also has more inertia, and the two effects cancel. The simple relationship above holds only for small angles; for large swings the period is slightly longer.