Chapter 1: Working scientifically and measurements
The chapter problem: the pendulum in the church tower
A church clock is driven by a pendulum that hangs down the inside of the tower. The pendulum is a metal bob on a thin wire. A technician measures the wire from its pivot to the centre of the bob with a metre rule, and times the swing with a stopwatch.
These are the facts to use all the way through the chapter.
- The length from the pivot to the centre of the bob is 0.800 m.
- The metre rule has a smallest division of 1 mm.
- The time for 20 complete oscillations is measured three times: 35.90 s, 35.85 s and 35.95 s.
- The stopwatch reads to 0.01 s.
- The accepted value of the acceleration due to gravity is 9.81 m s⁻².
By the end of the chapter you will work out the period of the pendulum, use it to find the acceleration due to gravity, and state the uncertainty in your answer. You will then decide whether the experiment agrees with the accepted value. At the end you will find the length a pendulum needs if the clock is to keep time, and explain why the 0.800 m pendulum cannot do the job.
First contact: no working yet. Estimate. A pendulum 1.0 m long has a period of about 2.0 s. This pendulum is 0.800 m long. Is its period more or less than 2.0 s? Roughly how long will 20 oscillations take? Write down both estimates.
If you are not sure where to start
A shorter pendulum has a shorter period. The period does not fall in proportion to the length: it depends on the square root of the length. So shortening the pendulum from 1.0 m to 0.800 m shortens the period by a smaller factor than 0.800. Estimate the period first, then multiply it by 20.
By the end of this chapter you will be able to
- write a quantity in SI base units, and use prefixes correctly;
- check that an equation is homogeneous by comparing base units;
- estimate a physical quantity to the nearest order of magnitude;
- explain the limits of a measurement: resolution, accuracy, precision, systematic error and random error;
- combine repeated readings into a mean, and state its uncertainty;
- find percentage uncertainties, and propagate them through a product, a quotient and a power;
- quote a result with its uncertainty to a sensible number of significant figures;
- decide whether a measurement agrees with an accepted value.
Where it is used
Every experiment in physics rests on these ideas. An engineer trusts a strain reading only when the uncertainty is known. A doctor acts on a blood test only when the measurement is reliable. Astronomers quote the distance to a star with an uncertainty, because the number alone would overstate what they know.
The same ideas decide whether a result is worth reporting. A value quoted without its uncertainty, or to more figures than the instrument can support, is not a measurement. It is a guess with extra digits. These skills also carry the practical work in every later chapter of this book.
Your turn to connect it
Think of a measurement you rely on in a normal week. It could be a thermometer, a speedometer, the weight printed on a food packet, or a stopwatch in sport. Write two examples. For one of them, say what could make the reading wrong.