Some Basic Ideas
Activity 1 | Week 1 | Resources
What to expect
Scientific measurement begins with a question. In Activity 1, you will measure the period of a pendulum and investigate:
The experiment uses a pendulum, but the reasoning involved is useful far beyond this activity and this course. For instance, similar questions arise when:
- measuring a length with a ruler or microscope
- estimating a concentration from an image
- comparing the brightness of two samples
- measuring the time between biological events
- deciding whether an observed change is larger than ordinary measurement variation
In fact, our broader question is:
The pendulum provides a simple, concrete example through which to explore this question. In this chapter, I will use it to introduce statistical analysis and the propagation of uncertainty. You may have encountered some of these ideas elsewhere, but this activity gives you an opportunity to use them together in a complete scientific investigation.
But first, let me briefly introduce the simple pendulum and some important ideas about experimental design.
1 The simple pendulum
A simple pendulum consists of a mass, called the bob, suspended from a fixed pivot by a light string.
When the pendulum is at rest, the bob hangs directly below the pivot. This is its equilibrium position. If the bob is displaced and released, gravity pulls it back towards equilibrium. The bob then moves repeatedly from side to side: it oscillates.
The length, \(L\), of the pendulum is measured from the pivot to the centre of the bob.
The period, \(T\), is the time taken for one complete oscillation. For example, we could begin timing when the bob passes a chosen point in one direction and stop when it next passes the same point in the same direction.
The amplitude is the maximum angle between the string and the downward vertical. In this activity, we set the initial amplitude by choosing the angle from which the bob is released.
A simple analysis, which we will explore next week, predicts that the mass of the bob does not affect the period. You may already be familiar with the following approximate relationship for a simple pendulum:
\[ T \approx 2\pi\sqrt{\dfrac{L}{g}}, \tag{1}\]
where:
- \(T\) is the period
- \(L\) is the pendulum length
- \(g\) is the acceleration due to gravity, approximately \(9.81\ \mathrm{m\,s^{-2}}\)
This relationship is known as the small-angle approximation because it becomes less accurate as the amplitude increases.
2 Designing the measurement
The activity question may look simple, but it raises several important practical questions:
- What exactly counts as one complete oscillation?
- What should one trial involve: timing one oscillation or several?
- How many trials should be carried out?
- How do we reliably determine the angles?
- Could the instrument or procedure introduce uncertainty or bias?
- How can we quantify and analyse measurement uncertainty?
More generally, a useful measurement process involves:
- clearly defining the quantity to be measured
- identifying important assumptions and possible sources of uncertainty
- designing a repeatable procedure
- collecting and describing the data
- estimating the measurement uncertainty
- deciding what claims the evidence supports
- comparing the result with a prediction or model, when possible
A systematic approach gives your results credibility. It also allows others to understand, reproduce, and evaluate your experiment.