Units and Measurement: Class 11 Physics · Chapter 1

1.1 Introduction

A tailor in Kolkata notes that a sleeve should be 58 cm long. A pharmacist in Pune weighs out 250 mg of a powder. Both are doing the same thing: holding a quantity up against an agreed standard and counting how many times the standard fits into it. That standard is a unit (मात्रक). All of physics rests on such comparisons, so we begin with units, and with how far a measured number can be trusted.

A measured result therefore has two parts: a pure number, its numerical measure, and a unit. “The corridor is 40” tells you nothing; “the corridor is 40 m” is a measurement. If you switch to a smaller unit, the number grows in proportion, so that the product of the two stays the same (Fig. 1.1):

For example, $2.5\mathrm{m}$ is also $250\mathrm{c}\mathrm{m}$: the unit became 100 times smaller and the number 100 times larger.

Speed is distance divided by time. Density is mass divided by volume. Force is mass times acceleration. Because physical quantities are linked like this, a handful of independent standards is enough for all of them. We pick a few base quantities (also called fundamental quantities), such as length, mass and time, and agree on a base unit for each. Every other unit is then built by multiplying and dividing base units; these are derived units (व्युत्पन्न मात्रक), such as the metre per second. A chosen set of base units, with all the derived units that follow from them, is called a system of units (Fig. 1.2).

1.2 The international system of units

Until the middle of the twentieth century, different countries, and even different branches of science, measured in different systems. Three were especially common. They differed in the base units chosen for length, mass and time (Table 1.1).

Today almost the whole world uses one system, the Système International d'Unités (French for International System of Units), or SI. It was adopted in 1960 at a meeting of the General Conference on Weights and Measures (CGPM), an intergovernmental body served by the BIPM, the International Bureau of Weights and Measures, near Paris. The SI reached its present seven base units in 1971, when the mole was added. In November 2018 the CGPM redefined all seven in terms of fixed values of constants of nature, and the new definitions came into force on 20 May 2019. SI multiples go in powers of ten, so converting between millimetres, metres and kilometres only moves the decimal point. From here on, every quantity in this book is in SI units unless we say otherwise.

How can a constant define a unit? Take the metre. Light in vacuum always travels at the same speed, and the SI simply declares that speed to be exactly 299 792 458 metres per second. A metre is then the distance light covers in $1\mathrm{/}299792458$ of a second. Any well-equipped laboratory can realise it, with no master rod to consult. The other units are fixed in the same spirit: the second by a frequency of the caesium atom, the kilogram through the Planck constant, and so on.

One caution about the mole: always name the entities you are counting. A mole of hydrogen atoms and a mole of hydrogen molecules both contain about $6.022\times {10}^{23}$ particles, but the second has twice the mass of the first.

Every other unit is derived from the seven. Several derived units have special names, such as the newton (force), joule (energy), watt (power), pascal (pressure) and hertz (frequency). A few units outside the SI are so convenient that they stay in use beside it (Table 1.3): farmland is sold by the hectare, grain prices are quoted per quintal, and diamonds are weighed in carats.

Prefixes scale a unit by powers of ten (Table 1.4). Symbols follow a few fixed rules, which keep a written value unambiguous everywhere in the world:

• Unit symbols are printed upright and never take a plural s or a full stop: 5 kg, not 5 kgs.
• Leave a space between the number and the symbol, as in 3 m or 25 °C; the angle signs are the only exception, as in 30°.
• Symbols of units named after people are capitals (N, Pa, W), but the names themselves are written in lower case (newton, pascal, watt).
• Use one prefix at a time: write 1 ng, not 1 mµg; a prefix and its unit form a single symbol, so km² means (km)².

Plane angle and solid angle

Two more SI units measure angles. A plane angle is the opening between two lines that meet at a point. Draw a circle of radius $r$ about that point; the lines cut off an arc of length $s$. The angle is defined as the ratio

Its unit is the radian (rad). One radian is the angle whose arc equals the radius. A full turn has an arc of $2\pi r$, so a full turn is $2\pi \mathrm{r}\mathrm{a}\mathrm{d}$, and $1\mathrm{r}\mathrm{a}\mathrm{d}={180}^{\circ }\mathrm{/}\pi \approx {57.3}^{\circ }$. Because $\theta$ is a ratio, it does not matter how big a circle you draw (Fig. 1.3).

A solid angle is the three-dimensional version: the opening of a cone, such as the beam of a torch. Put the tip of the cone at the centre of a sphere of radius $r$; the cone cuts a small patch of area $\mathrm{d}A$ out of the sphere's surface. The solid angle is

Its unit is the steradian (sr). A sphere twice as large has a patch four times larger in the same cone, so the ratio stays fixed (Fig. 1.5). The whole sphere has area $4\pi r^2$, so all directions around a point make up $4\pi \mathrm{s}\mathrm{r}\approx 12.6\mathrm{s}\mathrm{r}$. The radian and the steradian are ratios of two lengths and of two areas, so both are dimensionless: they carry a name but no dimension.

1.3 Significant figures

Measure the same pencil against two rulers and you will write down different numbers of digits (Fig. 1.6). On a scale marked only in whole centimetres, the end of the pencil lies between 7 and 8, a little before halfway, so you record $7.4\mathrm{c}\mathrm{m}$: the 7 is read off a mark and the 4 is judged by eye. On a scale with millimetre marks, the end lies between 7.4 and 7.5 cm, and you can judge one more digit: $7.46\mathrm{c}\mathrm{m}$. In each reading every digit except the last comes straight from the scale, and the last is a careful guess.

That is the rule for writing any measured value: all the digits you are sure of, then one estimated digit, and nothing after it. Together these are the significant figures (सार्थक अंक) of the measurement. How many you can honestly claim depends on the smallest division of the instrument, its least count (अल्पतमांक). Writing $7.4613\mathrm{c}\mathrm{m}$ for the pencil would claim a precision that neither ruler can give.

Because the count is fixed by the measurement, it cannot depend on the unit you happen to write. A coin's mass of $3.79\mathrm{g}$ is also $0.00379\mathrm{k}\mathrm{g}$: the extra zeros only shift the decimal point, and both forms have three significant figures. With that in mind, here are the rules for counting significant figures in a written number:

1. Every non-zero digit is significant. $3.47$ has three significant figures.
2. Zeros between non-zero digits are significant, wherever the decimal point is. $40.08$ and $4008$ both have four.
3. In a number less than 1, the zeros ahead of the first non-zero digit do not count, and neither does the zero usually written before the decimal point. They only locate the decimal point: $0.0036$ has two significant figures.
4. Trailing zeros in a whole number with no decimal point are not significant. By this convention $5600\mathrm{m}$ has two significant figures.
5. Trailing zeros after a decimal point are significant. $2.50$ has three significant figures and $0.07100$ has four.

Rule 4 can hide real information. A groundsman measures a boundary rope with a tape as 6500 cm, correct to the nearest centimetre, so all four digits were measured. Yet written as $6500\mathrm{c}\mathrm{m}$ the value seems to have only two significant figures, because its zeros sit at the end of a whole number. Change the unit to metres and the same length reads $65.00\mathrm{m}$, where rule 5 makes all four digits count. A whole number ending in zeros simply cannot tell you whether those zeros were measured.

Scientific notation removes the doubt. Write the value as $a\times {10}^b$, with $1\le a<10$ and $b$ a positive or negative whole number. Every digit of $a$ is significant, and the power of ten only sets the scale. The rope is $6.500\times {10}^3\mathrm{c}\mathrm{m}$, or $6.500\times {10}^{-2}\mathrm{k}\mathrm{m}$: four significant figures in any unit. Had the tape been read only to the nearest metre, you would write $6.5\times {10}^3\mathrm{c}\mathrm{m}$. Without scientific notation, fall back on rules 4 and 5.

Some numbers in formulas are not measurements at all. The 4 in the surface area of a sphere, $4\pi r^2$, the $\frac{1}{2}$ in $\frac{1}{2}m{v}^2$, and the 60 in $1\mathrm{m}\mathrm{i}\mathrm{n}=60\mathrm{s}$ are exact. Treat such numbers as having unlimited significant figures (4 = 4.000…), so they never limit the precision of a result. Constants such as $\pi =3.14159\dots$ can be taken to as many figures as the data need; 3.14 or 3.142 is usually enough.

Scientific notation also gives a quick sense of size. Round $a$ to 1 if $a\le 5$, or to 10 if $a>5$. The value is then roughly a single power of ten, ${10}^n$, and we say it is of the order of ${10}^n$; the exponent $n$ is its order of magnitude (परिमाण की कोटि).11Some books round at $\sqrt{10}\approx 3.16$ instead of 5. We follow the NCERT rule; the two agree except when $a$ lies between 3.16 and 5. Mount Everest rises $8.849\times {10}^3\mathrm{m}$; since $8.849>5$, it is of the order of ${10}^4\mathrm{m}$. The Moon is $3.84\times {10}^8\mathrm{m}$ away, of the order of ${10}^8\mathrm{m}$. So roughly ${10}^8\mathrm{/}{10}^4={10}^4$ Everests, stacked end to end, would reach the Moon. Fig. 1.8 places some other lengths on the same ladder of powers of ten.

1.3.1 Rules for arithmetic operations with significant figures

You time a toy car over $1.25\mathrm{m}$ and get $3.8\mathrm{s}$. A calculator reports the speed as $0.328947368\mathrm{m}\mathrm{/}\mathrm{s}$. Only the first two digits mean anything: the time was known to two figures, so the speed is $0.33\mathrm{m}\mathrm{/}\mathrm{s}$. A calculated result can never be more precise than the measurements that went into it. Two rules make sure it is reported that way.

A shopkeeper puts a $12.6\mathrm{k}\mathrm{g}$ sack of rice, weighed on a platform scale that reads to $0.1\mathrm{k}\mathrm{g}$, into a carton with a $0.845\mathrm{k}\mathrm{g}$ packet of dal, weighed on a kitchen scale that reads to $1\mathrm{g}$. The calculator gives $13.445\mathrm{k}\mathrm{g}$. But the sack's mass says nothing about hundredths or thousandths of a kilogram, so those columns of the sum are unknown too (Fig. 1.9). The total is $13.4\mathrm{k}\mathrm{g}$.

The rule really is about decimal places, not significant figures. On one balance a bottle weighs $152.37\mathrm{g}$ and a tablet $0.42\mathrm{g}$. Both are known to a hundredth of a gram, so the total $152.79\mathrm{g}$ is reliable to its last digit, even though $0.42\mathrm{g}$ has only two significant figures; rounding the sum to two figures would throw away three good digits. Subtraction can go the other way. Two clinical thermometers read ${36.8}^{\circ }\mathrm{C}$ and ${36.6}^{\circ }\mathrm{C}$, three significant figures each, yet their difference, ${0.2}^{\circ }\mathrm{C}$, has only one. The small difference of two nearly equal measurements is always far less certain, relatively, than either measurement.

1.3.2 Rounding off the uncertain digits

A calculation usually produces more digits than you can keep, so the result must be rounded. When the part you drop is clearly more or less than half a unit in the last kept place, the rule is the familiar one: to three significant figures, $4.837$ becomes $4.84$ and $4.832$ becomes $4.83$.

The interesting case is a dropped digit of exactly 5, which lies precisely halfway. Always rounding such values up would push results upwards on average. The convention is: if the digit before the 5 is even, leave it; if it is odd, raise it by one. So $6.125$ rounds to $6.12$ (2 is even) and $6.135$ rounds to $6.14$ (3 is odd). Over many roundings, half the ties go down and half go up. The tie rule applies only when the dropped part is exactly 5; $6.1251$ is more than halfway and rounds up to $6.13$ (Fig. 1.10).

Two habits complete the picture. In a calculation with several steps, carry one extra digit in the intermediate results and round only the final answer; Section 1.3.3 shows what goes wrong otherwise. And use constants to a sensible number of figures: the Avogadro constant $6.02214076\times {10}^{23}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$ is usually taken as $6.02\times {10}^{23}\mathrm{m}\mathrm{o}{\mathrm{l}}^{-1}$, and $\pi$ as 3.14 or 3.142, to match the precision of the data. The inspector below counts significant figures and rounds any value you type, using exactly these rules.

1.3.3 Rules for determining the uncertainty in the results of arithmetic calculations

Significant figures are a rough guide to precision. When we know the uncertainty of each measurement, we can do better and estimate the uncertainty of a calculated result. Three observations cover what you need.

(1) In a product, relative errors add. A trolley of mass $m=2.50\pm 0.01\mathrm{k}\mathrm{g}$ rolls at $v=1.20\pm 0.02\mathrm{m}\mathrm{/}\mathrm{s}$. The relative error (आपेक्षिक त्रुटि) of each value is its uncertainty divided by the value: $0.01\mathrm{/}2.50=0.4\mathrm{\%}$ for the mass and $0.02\mathrm{/}1.20=1.7\mathrm{\%}$ for the speed. The momentum $p=mv=3.00\mathrm{k}\mathrm{g}\mathrm{m}\mathrm{/}\mathrm{s}$ then carries a relative error of about $0.4\mathrm{\%}+1.7\mathrm{\%}=2.1\mathrm{\%}$, which is $0.06\mathrm{k}\mathrm{g}\mathrm{m}\mathrm{/}\mathrm{s}$. So we report $p=3.00\pm 0.06\mathrm{k}\mathrm{g}\mathrm{m}\mathrm{/}\mathrm{s}$. Fig. 1.12 shows where the rule comes from, using the simplest product to picture, the area of a rectangle.

(2) In sums and differences, absolute errors add. A rod of length $25.4\pm 0.1\mathrm{c}\mathrm{m}$ has $3.86\pm 0.01\mathrm{c}\mathrm{m}$ cut off. The remaining piece could be as long as $25.5-3.85=21.65\mathrm{c}\mathrm{m}$ or as short as $25.3-3.87=21.43\mathrm{c}\mathrm{m}$, so it is $21.54\pm 0.11\mathrm{c}\mathrm{m}$: only $21.5\mathrm{c}\mathrm{m}$ is worth writing. The larger uncertainty dominates, which is why sums and differences are rounded to the fewest decimal places, and why a difference can have fewer significant figures than either value.

(3) Equal significant figures do not mean equal precision. A stopwatch reads $2.05\mathrm{s}$ for one sprint start and $8.75\mathrm{s}$ for a short race, both to $\pm 0.01\mathrm{s}$ and both with three significant figures. The first has a relative error of $0.01\mathrm{/}2.05\approx 0.5\mathrm{\%}$, the second only $0.01\mathrm{/}8.75\approx 0.1\mathrm{\%}$. Counting significant figures is a rough guide; the relative error is the exact measure of precision.

Finally, here is why you carry an extra digit through intermediate steps. Find the volume of a cube of edge $1.27\mathrm{m}$ in two steps. The area of a face is $1.6129{\mathrm{m}}^2$; round it to $1.61{\mathrm{m}}^2$, multiply by the edge, and you get $2.0447\approx 2.04{\mathrm{m}}^3$. But the true volume, ${1.27}^3=2.048{\mathrm{m}}^3$, rounds to $2.05{\mathrm{m}}^3$. Keep four figures in the middle step, $1.613{\mathrm{m}}^2$, and the product $2.0485$ rounds correctly to $2.05{\mathrm{m}}^3$.

1.4 Dimensions of physical quantities

Two quantities can differ enormously in size and units and still be the same kind of thing. The wavelength of green light and the height of Mount Everest are both lengths. The speed of a snail and the speed of light are both a length divided by a time. The idea of dimensions captures this kind, and ignores the size.

Every physical quantity can be built from the seven SI base quantities, so we treat these seven as the basic dimensions and write each in square brackets: mass $[\mathrm{M}]$, length $[\mathrm{L}]$, time $[\mathrm{T}]$, amount of substance $[\mathrm{m}\mathrm{o}\mathrm{l}]$, electric current $[\mathrm{A}]$, temperature $[\mathrm{K}]$ and luminous intensity $[\mathrm{c}\mathrm{d}]$. Write a quantity as a product of base quantities; the exponent each one carries is its dimension (विमा) in that base quantity, and together these exponents are the dimensions of the quantity. Square brackets around a symbol, as in $[v]$, are read as “the dimensions of $v$”.

In mechanics three of them are enough: $[\mathrm{M}]$, $[\mathrm{L}]$ and $[\mathrm{T}]$. To find the dimensions of a mechanical quantity, trace its definition back to mass, length and time. Momentum is mass times velocity, and velocity is a length per unit time, so

Read the exponents straight off the brackets: momentum is of the first power in mass and in length, and of power $-1$ in time. Force, mass times acceleration, works out to $[\mathrm{M}\mathrm{L}{\mathrm{T}}^{-2}]$. Area, a length times a length, is $[{\mathrm{L}}^2]$; to stress that mass and time play no part, we may write it as $[{\mathrm{M}}^0{\mathrm{L}}^2{\mathrm{T}}^0]$. Two things are deliberately left out. Size does not appear: a glacier creeping a metre a day and a rifle bullet both move with dimensions $[\mathrm{L}{\mathrm{T}}^{-1}]$. Nor does the particular meaning of the quantity, as you will see in Section 1.6.2. Fig. 1.13 pictures the exponents for four common quantities.

1.5 Dimensional formulae and dimensional equations

Every derived quantity is defined by a relation, and the relation fixes its dimensions. Take power, the rate of doing work: $P=W\mathrm{/}t$, where the work $W$ is force times distance. Replace each symbol by its dimensions and add up the exponents of each base quantity (Fig. 1.14). The result, $[\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-3}]$, is the dimensional formula (विमीय सूत्र) of power: a compact list of the base quantities involved, each with its exponent. Written as an equation, $[P]=[\mathrm{M}{\mathrm{L}}^2{\mathrm{T}}^{-3}]$, it becomes a dimensional equation (विमीय समीकरण). Here are three more:

Table 1.5 collects the dimensional formulae of the quantities you will use most often in this book. You need not memorise it: each row follows from its defining relation in a line or two.

1.6 Dimensional analysis and its applications

Dimensions are more than labels. Only quantities of the same kind can be added, subtracted or compared, so dimensions put strict limits on what a correct equation can look like. Using these limits is called dimensional analysis (विमीय विश्लेषण). It lets us check equations, convert units, and sometimes predict the form of a relation before we have a full theory.

In any calculation, carry the units along with the numbers and let them cancel as symbols would: hours in a numerator cancel hours in a denominator, so $(\mathrm{k}\mathrm{m}\mathrm{/}\mathrm{h})\times \mathrm{h}=\mathrm{k}\mathrm{m}$. Dimensions behave in the same way, $[\mathrm{M}\mathrm{L}{\mathrm{T}}^{-2}]\mathrm{/}[\mathrm{M}]=[\mathrm{L}{\mathrm{T}}^{-2}]$. Handling units like this is also the safest way to convert them: replace each unit by its value in the new units, then simplify.

The same idea gives a general recipe. Suppose a quantity has dimensions $[{\mathrm{M}}^a{\mathrm{L}}^b{\mathrm{T}}^c]$ and numerical measure $n_1$ in a system whose units of mass, length and time are $M_1$, $L_1$ and $T_1$. Applying $n_1{u}_1=n_2{u}_2$ ((1.1)) to each base unit, its measure in a second system is

1.6.1 Checking the dimensional consistency of equations

Five kilograms plus three seconds is not eight of anything. A sum, a difference or an equality makes sense only between quantities of the same kind, so every term in a correct physical equation must have the same dimensions.

Applying the principle of homogeneity is quick. For a body moving with constant acceleration $a$, a student remembers $v^2=u^2+2as$. Every term should be a squared speed, $[{\mathrm{L}}^2{\mathrm{T}}^{-2}]$. The terms $v^2$ and $u^2$ clearly are, and $[as]=[\mathrm{L}{\mathrm{T}}^{-2}][\mathrm{L}]=[{\mathrm{L}}^2{\mathrm{T}}^{-2}]$, so the equation passes. Had she written $v=u+a{t}^2$, the last term would have dimensions $[\mathrm{L}]$ while the others have $[\mathrm{L}{\mathrm{T}}^{-1}]$, and the equation fails (Fig. 1.15).