Measurements
●● Units and basic quantities
Before a measurement can be made, a standard or unit must be chosen. The size of the quantity to be
measured is then found with an instrument having a scale marked in the unit.
Three basic quantities we measure in physics are length, mass and time. Units for other quantities are based on them. The SI (Système International d’Unités) system is a set of metric units now used in many countries. It is a decimal system in which units are divided or multiplied by 10 to give smaller or larger units.
Powers of ten shorthand
This is a neat way of writing numbers, especially if they are large or small. The example below shows how it works.
4000 = 4 × 10 × 10 × 10 = 4 × 103
400 = 4 × 10 × 10 = 4 × 102
40 = 4 × 10 = 4 × 101
4 = 4 × 1 = 4 × 100
0.4 = 4/10 = 4/101 = 4 × 10−1
0.04 = 4/100 = 4/102 = 4 × 10−2
0.004 = 4/1000 = 4/103 = 4 × 10−3
The small figures 1, 2, 3, etc., are called powers of ten. The power shows how many times the number has to be multiplied by 10 if the power is greater than 0 or divided by 10 if the power is less than 0. Note that 1 is written as 100.
This way of writing numbers is called standard notation.
●●Length
The unit of length is the metre (m) and is the distance travelled by light in a vacuum during a specific time interval. At one time it was the distance between two marks on a certain metal bar.
Submultiples are:
1 decimetre (dm) = 10−1 m
1 centimetre (cm) = 10−2 m
1 millimetre (mm) = 10−3 m
1 micrometre (μm) = 10−6 m
1 nanometre (nm) = 10−9 m
A multiple for large distances is
1 kilometre (km) = 103 m
Many length measurements are made with rulers; the correct way to read one is shown in Figure 1.2.
The reading is 76 mm or 7.6 cm. Your eye must be directly over the mark on the scale or the thickness of the ruler causes a parallax error.
To obtain an average value for a small distance, multiples can be measured. For example, in ripple
tank experiments (Chapter 25) measure the distance occupied by five waves, then divide by 5 to obtain the average wavelength.
●● Significant figures
Every measurement of a quantity is an attempt to find its true value and is subject to errors arising from limitations of the apparatus and the experimenter. The number of figures, called significant figures, given for a measurement indicates how accurate we think it is and more figures should not be given than is justified.
For example, a value of 4.5 for a measurement has two significant figures; 0.0385 has three significant figures, 3 being the most significant and 5 the least, i.e. it is the one we are least sure about since it might be 4 or it might be 6. Perhaps it had to be estimated by the experimenter because the reading was between two marks on a scale. When doing a calculation your answer should have the same number of significant figures as the measurements used in the calculation. For example, if your calculator gave an answer of 3.4185062, this would be written as 3.4 if the measurements had two significant figures. It would be written as 3.42 for three significant figures. Note that in deciding
the least significant figure you look at the next figure to the right. If it is less than 5 you leave the least significant figure as it is (hence 3.41 becomes 3.4) but if it equals or is greater than 5 you increase the least significant figure by 1 (hence 3.418 becomes 3.42).
If a number is expressed in standard notation, the number of significant figures is the number of digits before the power of ten. For example, 2.73 × 103 has three significant figures.
●●Area
The area of the square in Figure 1.3a with sides 1 cm long is 1 square centimetre (1 cm2). In Figure 1.3b the rectangle measures 4 cm by 3 cm and has an area of 4 × 3 = 12 cm2 since it has the same area as twelve squares each of area 1 cm2. The area of a square or rectangle is given by
area = length × breadth
Volume
Volume is the amount of space occupied. The unit of volume is the cubic metre (m3) but as this is rather large, for most purposes the cubic centimetre (cm3) is used. The volume of a cube with 1 cm edges is 1 cm3.
For a regularly shaped object such as a rectangular block, Figure 1.5 shows that
volume = length × breadth × height
The volume of a sphere of radius r is 4 3 πr3 and that of a cylinder of radius r and height h is πr2h. The volume of a liquid may be obtained by pouring it into a measuring cylinder, Figure 1.6a. A known volume can be run off accurately from a burette, Figure 1.6b. When making a reading both vessels must be upright and your eye must be level with the bottom of the curved liquid surface, i.e. the meniscus. The meniscus formed by mercury is curved oppositely to that of other liquids and the top is read.
Liquid volumes are also expressed in litres (l);
1 litre = 1000 cm3 = 1 dm3. One millilitre (1 ml) = 1 cm3.
●●Mass
The mass of an object is the measure of the amount of matter in it. The unit of mass is the kilogram (kg) and is the mass of a piece of platinum–iridium alloy at the Office of Weights and Measures in Paris. The gram (g) is one-thousandth of a kilogram.
1 g = 1000 kg = 10 –3kg = 0.001 kg
The term weight is often used when mass is really meant. In science the two ideas are distinct and have different units, as we shall see later. The confusion is not helped by the fact that mass is found on a balance by a process we unfortunately call ‘weighing’!
There are several kinds of balance. In the beam balance the unknown mass in one pan is balanced against known masses in the other pan. In the lever balance a system of levers acts against the mass when it is placed in the pan. A direct reading is obtained from the position on a scale of a pointer joined to the lever system. A digital top-pan balance is shown in Figure 1.7.
Figure 1.7 A digital top-pan balance
●●Time
The unit of time is the second (s) which used to be based on the length of a day, this being the time
for the Earth to revolve once on its axis. However, days are not all of exactly the same duration and
the second is now defined as the time interval for a certain number of energy changes to occur in the
caesium atom.
Time-measuring devices rely on some kind of constantly repeating oscillation. In traditional clocks
and watches a small wheel (the balance wheel) oscillates to and fro; in digital clocks and watches the
oscillations are produced by a tiny quartz crystal. A swinging pendulum controls a pendulum clock.
To measure an interval of time in an experiment, first choose a timer that is accurate enough for
the task. A stopwatch is adequate for finding the period in seconds of a pendulum, see Figure 1.8,
but to measure the speed of sound (Chapter 33), a clock that can time in milliseconds is needed. To
measure very short time intervals, a digital clock that can be triggered to start and stop by an electronic signal from a microphone, photogate or mechanical switch is useful. Tickertape timers or dataloggers are often used to record short time intervals in motion experiments (Chapter 2).
Accuracy can be improved by measuring longer time intervals. Several oscillations (rather than just one) are timed to find the period of a pendulum. ‘Tenticks’ (rather than ‘ticks’) are used in tickertape timers.
Practical work
Period of a simple pendulum
In this investigation you have to make time measurements using a stopwatch or clock. Attach a small metal ball (called a bob) to a piece of string, and suspend it as shown in Figure 1.8. Pull the bob a small distance to one side, and then release it so that it oscillates to and fro through a small angle.
Find the time for the bob to make several complete oscillations; one oscillation is from A to O to B to O to A (Figure 1.8). Repeat the timing a few times for the same number of oscillations and work out the average. The time for one oscillation is the period T. What is it for your system? The frequency f of the oscillations is the number of complete oscillations per second and equals 1/T. Calculate f.
How does the amplitude of the oscillations change with time? Investigate the effect on T of (i) a longer string, (ii) a heavier bob. A motion sensor connected to a data logger and computer (Chapter 2) could be used instead of a stopwatch for these investigations.
●● Systematic errors
Figure 1.9 shows a part of a rule used to measure the height of a point P above the bench. The rule chosen has a space before the zero of the scale. This is shown as the length x. The height of the point P is given by the scale reading added to the value of x. The equation for the height is height = scale reading + x
height = 5.9 + x
By itself the scale reading is not equal to the height. It is too small by the value of x. This type of error is known as a systematic error. The error is introduced by the system. A half-metre rule has the zero at the end of the rule and so can be used without introducing a systematic error. When using a rule to determine a height, the rule must be held so that it is vertical. If the rule is at an angle to the vertical, a systematic error is introduced.
●●Vernier scales and micrometers
Lengths can be measured with a ruler to an accuracy of about 1 mm. Some investigations may need a
more accurate measurement of length, which can be achieved by using vernier calipers (Figure 1.10) or a micrometer screw gauge.
Figure 1.10 Vernier calipers in use
a) Vernier scale
The calipers shown in Figure 1.10 use a vernier scale. The simplest type enables a length to be measured to 0.01 cm. It is a small sliding scale which is 9 mm long but divided into 10 equal divisions
One end of the length to be measured is made to coincide with the zero of the millimetre scale and
the other end with the zero of the vernier scale. The length of the object in Figure 1.11b is between
1.3 cm and 1.4 cm. The reading to the second place of decimals is obtained by finding the vernier mark which is exactly opposite (or nearest to) a mark on the millimetre scale. In this case it is the 6th mark and the length is 1.36 cm, since
Vernier scales are also used on barometers, travelling microscopes and spectrometers.
b) Micrometer screw gauge
This measures very small objects to 0.001 cm. One revolution of the drum opens the accurately flat,
parallel jaws by one division on the scale on the shaft of the gauge; this is usually 1 2 mm, i.e. 0.05 cm.
If the drum has a scale of 50 divisions round it, then rotation of the drum by one division opens the jaws by 0.05/50 = 0.001 cm (Figure 1.12). A friction clutch ensures that the jaws exert the same force
when the object is gripped.
The object shown in Figure 1.12 has a length of 2.5 mm on the shaft scale + 33 divisions on the drum scale
= 0.25 cm + 33(0.001) cm
= 0.283 cm
Before making a measurement, check to ensure that the reading is zero when the jaws are closed.
Otherwise the zero error must be allowed for when the reading is taken.
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