The use of graphs

In experimental physics, graphs have three main uses. The first is to determine the value of some quantity, usually the slope or the intercept of a straight line representing the relation between two variables. Although this use of graphs is often stressed in elementary teaching of practical physics, it is in fact a comparatively minor one. Whether we obtain the value of the slope of a straight line by the method of least squares or by taking the points in pairs (see chapter 4), we are of course not using the graph as such, but the original numbers. The only time we actually use the graph to determine the slope is when we judge or guess the best line through the points by eye. This is a crude method – though not to be despised on that account – and should only be used as a check on the result of a more sophisticated method, or when the value of the slope is not an important quantity in the final result.

The second use of graphs is by far the most important. They serve as visual aids. Suppose, for example, the rate of flow of water through a tube is measured as a function of the pressure gradient, with the object of determining when the flow ceases to be streamlined and becomes turbulent.

Introduction

In this chapter we shall consider some general principles for making measurements. These are principles which should be borne in mind, first in selecting a particular method and second in getting the most out of it. By the latter we mean making the method as precise or reproducible as possible, and – even more important – avoiding its inherent systematic errors.

We shall illustrate the various points by describing some specific examples of instruments and methods. Though chosen from several branches of physics, they are neither systematic nor exhaustive. The idea is that, having seen how the principles apply in these cases, you will be able to apply them yourselves in other situations. As always there is no substitute for laboratory experience. But experience without thought is a slow and painful way of learning. By concentrating your attention on certain aspects of measurement making we hope to make the experience more profitable.

Metre rule

We start with almost the simplest measuring device there is – a metre rule. Its advantages are that it is cheap to make and convenient to use. It can give results accurate to about ⅕ mm. However, to achieve this accuracy certain errors must be avoided.

Parallax error. If there is a gap between the object being measured and the scale, and the line of sight is not at right angles to the scale, the reading obtained is incorrect (Fig. 6.1a).

The system of units used throughout this book is known as SI, an abbreviation for Système International d'Unités. It is a comprehensive, logical system, designed for use in all branches of science and technology. It was formally approved in 1960 by the General Conference of Weights and Measures, the international organization responsible for maintaining standards of measurement. Apart from its intrinsic merits, it has the great advantage that one system covers all situations – theoretical and practical.

A full account of SI will be found in a publication of the National Physical Laboratory (Bell 1993). The following are the essential features of the system.

(1) SI is a metric system. There are seven base units (see next section), the metre and kilogram replacing the centimetre and gram of the old c.g.s. system.

(2) The derived units are directly related to the base units. For example, the unit of acceleration is 1 m s-2. The unit of force is the newton, which is the force required to give a body of mass 1 kg an acceleration of 1 m s-2. The unit of energy is the joule, which is the work done when a force of 1 N moves a body a distance of 1 m.

The use of auxiliary units is discouraged in SI. Thus the unit of pressure, the pascal, is 1 N m-2; the atmosphere and the torr are not used. Similarly the calorie is not used; all forms of energy are measured in joules.

In our experience, an understanding of the laws of physics is best acquired by applying them to practical problems. Frequently, however, the problems appearing in textbooks can be solved only through long, complex calculations, which tend to be mechanical and boring, and often drudgery for students. Sometimes, even the best of these students, the ones who possess all the necessary skills, may feel that such problems are not attractive enough to them, and the tedious calculations involved do not allow their ‘creativity’ (genius?) to shine through.

This little book aims to demonstrate that not all physics problems are like that, and we hope that you will be intrigued by questions such as:

• How is the length of the day related to the side of the road on which traffic travels?

• Why are Fosbury floppers more successful than Western rollers?

• How far below ground must the water cavity that feeds Old Faithful be?

• How high could the tallest mountain on Mars be?

• What is the shape of the water bell in an ornamental fountain?

• How does the way a pencil falls when stood on its point depend upon friction?

• Would a motionless string reaching into the sky be evidence for UFOs?

• How does a positron move when dropped in a Faraday cage?

• What would be the high-jump record on the Moon?

• Why are nocturnal insects fatally attracted to light sources?

• How much brighter is sunlight than moonlight?

• How quickly does a fire hose unroll?

• […]

The following chapter contains the problems. They do not follow each other in any particular thematic order, but more or less in order of difficulty, or in groups requiring similar methods of solution. In any case, some of the problems could not be unambiguously labelled as belonging to, say, mechanics or thermodynamics or electromagnetics. Nature's secrets are not revealed according to the titles of the sections in a text book, but rather draw on ideas from various areas and usually in a complex manner. It is part of our taskt o find out what type of problem we are facing. However, for information, the reader can find a list of topics, and the problems that more or less belong to these topics, on the following page. Some problems are listed under more than one heading. The symbols and numerical values of the principal physical constants are then given, together with astronomical data and some properties of material.

The majority of the problems are not easy; some of them are definitely difficult. You, the reader, are naturally encouraged to try to solve the problems on your own and, obviously, if you do, you will get the greatest pleasure. If you are unable to achieve this, you should not give up, but turn to the relevant page of the short hints chapter. In most cases this will help, though it will not give the complete solution, and the details still have to be worked out.

P1 Three small snails are each at a vertex of an equilateral triangle of side 60 cm. The first sets out towards the second, the second towards the third and the third towards the first, with a uniform speed of 5 cm min−1. During their motion each of them always heads towards its respective target snail. How much time has elapsed, and what distance do the snails cover, before they meet? What is the equation of their paths? If the snails are considered as point-masses, how many times does each circle their ultimate meeting point?

P2 A small object is at rest on the edge of a horizontal table. It is pushed in such a way that it falls off the other side of the table, which is 1 m wide, after 2 s. Does the object have wheels?

P3 A boat can travel at a speed of 3 m s−1 on still water. A boatman wants to cross a river whilst covering the shortest possible distance. In what direction should he row with respect to the bank if the speed of the water is (i) 2 m s−1, (ii) 4 m s−1? Assume that the speed of the water is the same everywhere.

P4 A long, thin, pliable carpet is laid on the floor. One end of the carpet is bent back and then pulled backwards with constant unit velocity, just above the part of the carpet which is still at rest on the floor.

Tools for applications

This chapter will introduce radionuclides and the emitted radiations as tools for applications. We shall begin with the well known α, β and γ radiations followed by a few characteristics of extranuclear electrons and X rays. Additional comments about neutrons (see Section 1.3.6), will follow in Section 5.4.4.

Properties of alpha particles

The nature and origin of alpha particles

Alpha particles are here called primary radiations because their emission is the first evidence of nuclear disintegrations which turn parent atoms into daughter atoms belonging to a different chemical element. Alpha particles are emitted either as a single branch, i.e. all with the same energy, or as several branches each with its own energy. A typical case is the α particle decay of americium-241. Close to 13% of the decays occur at an α particle energy of 5.44 MeV and close to 85% at 5.48 MeV, leaving three minor branches each with its own energy, overall exactly one α particle emitted per decay.

Properties of a particles were summarised in Table 1.2. With Z = 2 and A=4, α particles are physically identical to the nuclei of helium atoms. As calculated by Rutherford (Section 1.2) the diameter of α particles is about 10−14 m, or some 104 times smaller than the 10−10 m atomic diameters.