Chapter 1: Units and Measurements
In December 1998, NASA launched the Mars Climate Orbiter, a spacecraft built to circle Mars and study its atmosphere. After a journey of more than nine months and hundreds of millions of kilometres, the orbiter reached Mars in September 1999. It fired its engines to slip into orbit around the planet — and was never heard from again.
The investigation that followed did not find a broken engine, a cracked circuit, or a failed computer. It found a mismatch of units. One team had supplied thruster data using the pound-second, a unit belonging to the British engineering system. The team writing the navigation software expected the same data in newton-second, the SI unit. Neither number was wrong by itself. Each was a perfectly good measurement — but they were measurements in different units, and nobody converted between them.
Because of that silent mismatch, the spacecraft’s path was slightly off. It approached Mars far lower than planned, entered the thin upper atmosphere, and was destroyed. A mission costing hundreds of crores was lost not to bad physics, but to careless handling of units.
Physics is a quantitative science. Every measurement we make is a comparison with a standard, and that standard must be one that everybody agrees upon. This chapter is about those standards — how we build them, how we use them to measure things as large as a galaxy and as small as an atom, and how we honestly state what we do not know.
Diagram
Fig. 1.1 – A spacecraft approaching Mars, with two data labels feeding into its trajectory: one tagged “pound-second (FPS)” and one tagged “newton-second (SI)”, the mismatch bending the path too close to the planet.
This chapter will answer each of these questions.
1.1 Introduction
Physics is a quantitative science. This single word carries a lot of meaning. It tells us that physics is not satisfied with saying that an object is “heavy” or that a journey is “long” — it insists on saying how heavy and how long, in numbers. Every experiment in physics therefore ends in a measurement of some physical quantity.
You already measure things constantly, even outside the laboratory. You judge the size of objects, the volume of a liquid, the amount of matter in a packet, the weight of vegetables or fruits at a shop, your body temperature when you are unwell, and the length of cloth needed for a shirt. Each of these is a measurement, whether or not you call it one.
What a Measurement Really Is
A measurement is always a comparison. When you measure something, you are silently comparing it with a standard that has been agreed upon and accepted internationally.
Consider buying fruit. To find the mass of the fruit, the shopkeeper places standard masses of 1 kg, 500 g and so on in the other pan of the balance. The fruit is not being measured against nothing — it is being measured against those standards. These accepted standards of comparison are what we call units.
Why a Number Alone Is Never Enough
Because measurement is a comparison, the result of a measurement always has two parts: a number, and the unit in which that number is counted.
For example, the length of a wire is written as 5 m. Here ‘m’ (metre) is the unit, and 5 is the value of the length in that unit. The same wire could equally well be written as 500 cm — the wire has not changed, only the standard we compared it with has changed, so the number changed too.
Different quantities are measured against different standards, because you cannot compare a length with a mass. Length is measured in metre (m), time in second (s), mass in kilogram (kg), and so on for every quantity.
Diagram
Fig. 1.2 – A beam balance with a fruit in one pan and standard 1 kg and 500 g masses in the other, labelled to show that measurement is a comparison with a standard.
With this idea in place — that every measurement is a comparison with a standard unit — the natural next question is: which standards should the whole world use? That is what the next section takes up.
1.2 System of Units
Once we accept that every measurement is a comparison with a standard, a practical problem appears immediately. Different regions of the world, at different times, chose different standards. A group of standard units chosen together for length, mass and time is called a system of units.
In your earlier standards you have already come across several such systems:
- CGS: Centimetre Gram Second system
- MKS: Metre Kilogram Second system
- FPS: Foot Pound Second system
- SI: System International
Notice how each name is simply a list of the standards it uses for length, mass and time, in that order. In the CGS system, length is measured in centimetre, mass in gram and time in second. In the MKS system the same three quantities are measured in metre, kilogram and second. The FPS system, used mainly in Britain and the United States, measures length in foot and mass in pound.
From Many Systems to One
The first three systems — CGS, MKS and FPS — were used extensively till recently. Having several systems in use at the same time is workable, but clumsy: every number crossing from one system to another has to be converted, and every conversion is an opportunity for error.
To settle the matter, in 1971 the 14th International General Conference on Weights and Measures recommended the use of the ‘International system’ of units. This international system of units is called the SI units, and it is the system used throughout this book.
The great practical advantage of SI is that it uses the decimal system. Larger and smaller units are related to the base unit by simple powers of ten — a kilometre is \(10^{3}\) m, a millimetre is \(10^{-3}\) m. Conversion within the system therefore needs no awkward factors and is very simple and convenient.
Having chosen SI, we must now decide which quantities deserve their own standards, and which can simply be built out of the others. That is the distinction taken up next.
1.2.1 Fundamental Quantities and Units
Look carefully at the quantities used in physics and you will find that they are not all independent of one another. Speed, for instance, cannot be defined without first knowing what length and time mean. But length itself does not lean on anything else — you can define it on its own, without borrowing from any other quantity.
Quantities of this second kind, which stand on their own feet, are called fundamental quantities.
There are seven fundamental quantities in all: length, mass, time, temperature, electric current, luminous intensity and amount of substance. Every other quantity in the whole of physics — force, energy, pressure, resistance, charge — is built out of these seven.
The Units That Go With Them
Since these seven quantities are independent, each one needs its own independently chosen standard. The units used to measure the fundamental quantities are called fundamental units.
The seven fundamental quantities, their SI units and the symbols for those units are listed in Table 1.1. You should learn this table thoroughly, because the symbols in the last column are the building blocks of every dimensional formula you will write later in this chapter.
Table 1.1: Fundamental Quantities with their SI Units and Symbols
| Fundamental quantity | SI units | Symbol |
|---|---|---|
| 1) Length | metre | m |
| 2) Mass | kilogram | kg |
| 3) Time | second | s |
| 4) Temperature | kelvin | K |
| 5) Electric current | ampere | A |
| 6) Luminous Intensity | candela | cd |
| 7) Amount of substance | mole | mol |
Diagram
Fig. 1.3 – Seven labelled blocks (length, mass, time, temperature, current, luminous intensity, amount of substance) forming the base of a pyramid, with derived quantities such as force, energy and pressure built on top of them.
1.2.2 Derived Quantities and Units
The seven fundamental quantities are only the beginning. In physics we constantly meet quantities such as speed, momentum, resistance and conductivity. None of these is on the list of seven — and none of them needs to be, because each can be built out of some or all of the seven.
Speed, for example, is length divided by time. Momentum is mass multiplied by velocity, which is itself length divided by time. These quantities depend on the fundamental ones and can be expressed entirely in terms of them.
Naturally, if the quantity is built out of fundamental quantities, its unit must be built out of fundamental units in exactly the same way.
Building a Derived Unit
The rule for constructing a derived unit is simple: write down the defining relation of the quantity, and replace each quantity in it by its unit.
Take velocity. Velocity is displacement divided by time, so
\[\text{SI unit of velocity} = \frac{\text{Unit of displacement}}{\text{Unit of time}} = \frac{\text{m}}{\text{s}} = \text{m s}^{-1}\]
Here m is the metre (unit of length) and s is the second (unit of time). Notice how the division of the two quantities has become a division of the two units.
Now take momentum, which is mass multiplied by velocity:
\[\text{Unit of momentum} = (\text{Unit of mass}) \times (\text{Unit of velocity})\] \[= \text{kg m/s} = \text{kg m s}^{-1}\]
Both of the above units — m s\(^{-1}\) and kg m s\(^{-1}\) — are derived units, because each is written purely in terms of the fundamental units metre, kilogram and second.
Supplementary Units
Besides the seven fundamental or basic units, there are two more units called supplementary units: (i) plane angle \(d\theta\) and (ii) solid angle \(d\Omega\).
These two are placed in a category of their own because they are angles — pure ratios of one length to another, or of an area to an area. Being ratios, they have no fundamental unit hiding inside them, and yet we still need standard names for measuring them.
(i) Plane angle (dθ): This is the ratio of the length of an arc of a circle to the radius of the circle, as shown in Fig. 1.4 (a). Thus
\[d\theta = \frac{ds}{r}\]
is the angle subtended by the arc at the centre of the circle. Here \(ds\) is the length of the small arc AB and \(r\) is the radius, both measured in metre. It is measured in radian (rad). An angle \(\theta\) in radian is denoted as \(\theta^{c}\).
Diagram
Fig. 1.4 (a) – A circular arc AB of length ds subtending a small plane angle dθ at the centre O, with radius r marked.
(ii) Solid angle (dΩ): This is the three-dimensional analogue of \(d\theta\). Where a plane angle opens out in a flat plane, a solid angle opens out as a cone in space — think of the cone of sky that a telescope collects light from. It is defined as the ratio of the area of a portion of the surface of a sphere to the square of the radius of the sphere. Thus
\[d\Omega = \frac{dA}{r^{2}}\]
is the solid angle subtended by area \(ds\) at O, as shown in Fig. 1.4 (b). Here \(dA\) is the patch of area on the sphere (in m\(^2\)) and \(r\) is the radius of the sphere (in m). It is measured in steradians (sr).
Diagram
Fig. 1.4 (b) – A patch of area dA on the surface of a sphere of radius r subtending a solid angle dΩ at the centre O.
A useful check follows at once. A sphere of radius \(r\) has surface area \(4\pi r^{2}\). Therefore the solid angle subtended by the entire sphere at its centre is
\[\Omega = \frac{4\pi r^{2}}{r^{2}} = 4\pi \ \text{sr}\]
This is the largest solid angle that can exist, just as \(2\pi\) radian is the largest plane angle around a point.
1.2.3 Conventions for the use of SI Units
A unit is a piece of information, and like any piece of information it can be corrupted by careless writing. Over the years the international bodies that maintain the SI have therefore fixed a set of writing conventions. These are not decorative rules of neatness — each one exists to remove a specific ambiguity that has confused people in the past.
The conventions are as follows.
(1) The unit of every physical quantity should be represented by its symbol. So we write 5 m, not 5 metres, in a calculation.
(2) The full name of a unit always starts with a small letter, even if the unit is named after a person — for example, 1 newton, 1 joule. But the symbol for a unit named after a person is written with a capital letter — for example, N after the scientist Newton, and J after the scientist Joule.
There is a quiet logic here. The lower-case full name reminds us that the unit is a public standard, not a personal possession; the capital symbol preserves the honour paid to the scientist.
(3) Symbols for units do not take the plural form. We write a force of 20 N, and not 20 newtons or 20 Ns. This one matters a great deal, because “Ns” already has a meaning of its own — newton second, the unit of impulse. Adding a plural ‘s’ would silently turn a force into an impulse.
(4) Symbols for units do not contain any full stop at the end of the recommended letter — for example, 25 kg and not 25 kg. (A full stop after a symbol can easily be mistaken for a decimal point in a printed table.)
(5) The units of physical quantities appearing in the numerator and the denominator should be written as one ratio. For example, the SI unit of acceleration is written m/s\(^{2}\) or m s\(^{-2}\), but not m/s/s. A double division sign is ambiguous — it is never clear which division is meant to be carried out first.
(6) Use of a combination of the full name and the symbol is avoided when a physical quantity is expressed as a combination of two units. For example, the unit J/kg K is correct, while joule/kg K is not correct. Either write the whole thing in symbols or write the whole thing in words; do not mix the two languages in one expression.
Prefixes and the Powers of Ten
(7) A prefix symbol is used before the symbol of the unit. The prefix symbol and the unit symbol together constitute a new symbol for the unit, which can be raised to a positive or negative power of 10.
For example, \[1\ \text{ms} = 1 \ \text{millisecond} = 10^{-3}\ \text{s}\] \[1\ \mu\text{s} = 1 \ \text{microsecond} = 10^{-6}\ \text{s}\] \[1\ \text{ns} = 1 \ \text{nanosecond} = 10^{-9}\ \text{s}\]
Use of double prefixes is avoided when a single prefix is available. Thus \[10^{-6}\ \text{s} = 1\ \mu\text{s} \quad \text{and not 1 mms}\] \[10^{-9}\ \text{s} = 1\ \text{ns} \quad \text{and not 1 m}\mu\text{s}\]
The reason is simply that a chain of prefixes forces the reader to do arithmetic before they can even understand the number. If a single prefix says the same thing, it should be used.
(8) A space or a hyphen must be introduced while indicating the multiplication of two units. For example, m/s should be written as m s\(^{-1}\) or m-s\(^{-1}\), and not as ms\(^{-1}\) — because “ms” will be read as millisecond.
This is the most dangerous of all the conventions, and it is worth pausing on. The symbol ms\(^{-1}\) could be read in two completely different ways: as metre per second (a speed), or as “per millisecond” (a frequency). One missing space changes the meaning of the quantity entirely.
Diagram
Fig. 1.5 – A two-column chart of “Correct” and “Incorrect” ways of writing units, illustrating conventions (2), (3), (5), (6) and (8) side by side.
1.3 Measurement of Length
One fundamental quantity which we have already discussed is length. To measure length or distance, the SI unit used is the metre (m).
But what exactly is one metre? A unit is only as good as the standard behind it, and the standard for the metre has been sharpened twice within living memory.
From a Coloured Light to a Beam of Light
In 1960, a standard for the metre based on the wavelength of orange-red light emitted by atoms of krypton was adopted. The advantage of such a standard is that it is reproducible anywhere: a krypton atom in Paris and a krypton atom in Pune emit exactly the same light, so any properly equipped laboratory in the world could recreate the metre for itself without borrowing anyone’s metal bar.
By 1983 a still more precise measurement was developed. It says that a metre is the length of the path travelled by light in vacuum during a time interval of 1/299792458 second.
Notice how the standard has become steadily less physical. A metal bar can be scratched, a wavelength must be produced by a real lamp, but a definition written in terms of light in vacuum and a fixed constant can never be lost, stolen or damaged.
The Range of Lengths in Nature
Lengths in physics do not come in a comfortable, human-sized range. They stretch from the almost unimaginably vast to the almost unimaginably tiny, and the same unit — the metre — must cover all of it. Some typical distances and lengths are given in Table 1.2.
Table 1.2: Some Useful Distances
| Measurement | Length in metre |
|---|---|
| Distance to Andromeda Galaxy (from Earth) | \(2 \times 10^{22}\) m |
| Distance to nearest star (after Sun) Proxima Centuari (from Earth) | \(4 \times 10^{16}\) m |
| Distance to Pluto (from Earth) | \(6 \times 10^{12}\) m |
| Average Radius of Earth | \(6 \times 10^{6}\) m |
| Height of Mount Everest | \(9 \times 10^{3}\) m |
| Thickness of this paper | \(1 \times 10^{-4}\) m |
| Length of a typical virus | \(1 \times 10^{-8}\) m |
| Radius of hydrogen atom | \(5 \times 10^{-11}\) m |
| Radius of proton | \(1 \times 10^{-15}\) m |
Read the table from top to bottom and notice the span: from \(10^{22}\) m down to \(10^{-15}\) m — a range of some thirty-seven powers of ten. This is why every entry is written in scientific notation, in powers of ten, rather than as a string of zeros.
Diagram
Fig. 1.6 – A single logarithmic scale line running from \(10^{-15}\) m to \(10^{22}\) m, with the entries of Table 1.2 marked as labelled points along it.
This enormous range also explains what follows. No single instrument can measure across all of it. A metre scale is useless for the distance to Pluto and equally useless for the radius of a proton. Different regions of the scale demand entirely different methods — and the next few subsections take them one at a time.
1.3.1 Measurements of Large Distance
Parallax method
Large distances, such as the distance of a planet or a star from the Earth, cannot be measured directly with a metre scale. Nobody can lay a scale from here to Jupiter. For such distances, an indirect method called the parallax method is used.
The remarkable thing about this method is that it converts a distance we cannot reach into an angle we can measure comfortably from where we stand.
An Experiment You Can Do Right Now
Let us do a simple experiment to understand what parallax is.
Hold your hand in front of you and look at it with your left eye closed. Now open the left eye and close the right eye instead. You will find that your hand appears to shift sideways against the background — a distant wall, a window, a tree.
Your hand has not moved. The background has not moved. Only the position of the observer — the eye doing the looking — has changed. This apparent shift is called parallax.
Now look at Fig. 1.7. The two eyes are at \(E_1\) and \(E_2\), and the object is at P. The angle \(\theta\) at P, between the two lines of sight, is called the parallax angle. By measuring this parallax angle \(\theta\) and knowing the distance \(E_1E_2\) between the two eyes, we can determine the distance of the object from us, that is, OP as \(E_1E_2/\theta\).
Diagram
Fig. 1.7 – Parallax method for determining distance: an object at P viewed from two eye positions E₁ and E₂, with the parallax angle θ marked at P and O the midpoint of E₁E₂.
Scaling the Method Up to the Sky
There is an obvious limitation. The distances of planets from the Earth are enormous, and the gap between your two eyes is a few centimetres. The parallax angle produced would be far too small to detect. So the two-eyes method, as it stands, cannot be used.
The remedy is to make the “eyes” much farther apart. In order to make simultaneous observations of an astronomical object, we select two distant points on the Earth and use two observers instead of two eyes.
Consider two positions A and B on the surface of the Earth, separated by a straight line of length \(b\), as shown in Fig. 1.8. Two observers at these two points observe a distant planet S at the same moment. We measure the angle \(\angle ASB\) between the two directions along which the planet is viewed at these two points. This angle, represented by the symbol \(\theta\), is the parallax angle.
Diagram
Fig. 1.8 – Measurement of distances of planets: points A and B on the Earth separated by basis b, both sighting a planet S at distance D, with parallax angle θ at S.
The observations must be simultaneous, because both the Earth and the planet are in motion. If one observer looks in the morning and the other in the evening, the geometry of the triangle has already changed and the angle measured is meaningless.
Turning the Triangle into an Arc
Now comes the step that makes the mathematics simple. The planet is far away — that is, the distance of the planet from the Earth is very large in comparison to \(b\). In symbols,
\[\frac{b}{D} \ll 1\]
and therefore \(\theta\) is very small.
When the angle at the apex of a triangle is very small, the base of that triangle is almost indistinguishable from a small arc of a circle drawn with the apex as centre. We can thus consider AB as the arc of length \(b\) of a circle, and \(D\) as its radius.
Once we do that, we are allowed to use the definition of plane angle from Section 1.2.2 — angle = arc/radius:
\[AB = b \quad \text{and} \quad AS = BS = D \quad \text{and} \quad \theta \cong \frac{AB}{D},\]
where \(\theta\) is in radian.
\[\therefore \ D = \frac{b}{\theta} \qquad \text{--- (1.1)}\]
Here \(b\) is the distance between the two observation points on Earth, called the basis (in metre), \(\theta\) is the measured parallax angle (in radian), and \(D\) is the distance of the planet from the Earth (in metre).
Equation (1.1) says something worth stating in words: for a fixed basis \(b\), the distance is inversely proportional to the parallax angle. A small angle means a large distance. This is the same fact you noticed with the distant tree, now written as an equation.
1.3.2 Measurement of Distance to Stars
The Sun is the star which is closest to the Earth. The next closest star is at a distance of 4.29 light years — a distance so large that the parallax method, as used for planets, runs into trouble.
Recall equation (1.1): \(D = b/\theta\), so \(\theta = b/D\). For a fixed basis \(b\), the parallax angle shrinks as the distance \(D\) grows. Stars are hundreds of thousands of times farther than the planets, so the parallax measured from even the two most distant points on the Earth will be too small to be measured.
Using the Earth’s Orbit as the Basis
If the angle is too small, there is only one thing left to do: make the basis \(b\) bigger. And the largest basis available to us is not on the Earth at all — it is the Earth’s own orbit.
For this purpose we measure the parallax between two farthest points along the orbit of the Earth around the Sun. These two points, observed six months apart, are 2 AU apart (see the box below for what an AU is). The basis has jumped from a few thousand kilometres to about \(3 \times 10^{11}\) m, and the parallax angle becomes large enough to detect.
Diagram
Fig. 1.9 – The Earth at two diametrically opposite points A and B of its orbit around the Sun S, each 1 AU from the Sun, both sighting a distant star; the parallax angle marked at the star.
Diagram
Fig. 1.10 – A right triangle with the Sun at one corner, a baseline of 1 AU, and the opposite vertex at a distance of 1 pc subtending an angle of 1″.
1.3.3 Measurement of the Size of a Planet or a Star
Knowing how far away a planet is only half the story. We would also like to know how big it is — and once again, no scale can be laid across it.
The way out is the same idea used for distance, but turned around. Instead of viewing one point of the object from two places, we now view two opposite points of the object from one place.
Angular Diameter
If \(d\) is the diameter of a planet, the angle subtended by it at any single point on the Earth is called the angular diameter of the planet.
Let \(\alpha\) be the angle between the two directions along which two diametrically opposite points of the planet are viewed through a telescope, as shown in Fig. 1.11. The telescope is turned from one edge of the planet’s disc to the other, and the angle through which it has been turned is \(\alpha\).
Diagram
Fig. 1.11 – Measurement of the size of a planet: a planet of diameter d at distance D from the Earth, subtending angular diameter α at the observer.
Since the planet is very far away, the same small-angle reasoning of Section 1.3.1 applies: the diameter \(d\) can be treated as a small arc, and the distance \(D\) as the radius of the circle. Using angle = arc/radius,
\[\alpha = \frac{d}{D}\]
The distance \(D\) of the planet is large, but we assume it has already been measured — by the parallax method of the previous subsections. With \(D\) known and \(\alpha\) measured, the diameter of the planet follows at once:
\[\therefore \ d = \alpha D \qquad \text{--- (1.2)}\]
Here \(d\) is the diameter of the planet (in metre), \(D\) is its distance from the Earth (in metre), and \(\alpha\) is the angular diameter in radian.
1.3.4 Measurement of Very Small Distances
At the other end of Table 1.2 lies the opposite problem. When we intend to measure the size of atoms and molecules, the conventional apparatus like a Vernier calliper or a screw gauge will not be useful.
The reason is a matter of scale. A Vernier calliper can resolve about \(10^{-4}\) m and a screw gauge about \(10^{-5}\) m, but an atom has a radius of the order of \(10^{-10}\) m. Asking such an instrument to measure an atom is like asking a metre scale to measure the thickness of a soap film — the object is smaller than the smallest division the instrument can register. Worse, these instruments work by touching the object, and there is no jaw or screw fine enough to grip a single atom.
Therefore, to measure the size of atoms we use an electron microscope or a tunnelling electron microscope.
Diagram
Fig. 1.12 – Comparison of measuring ranges: a metre scale, a Vernier calliper, a screw gauge, an optical microscope and an electron microscope placed along a logarithmic length scale from \(10^{0}\) m down to \(10^{-10}\) m.
For measuring sizes using a microscope, we need to select the wavelength of light to be used in the microscope such that it is smaller than the size of the object to be measured. Thus visible light, whose wavelength ranges from \(4000\ \overset{\circ}{\text{A}}\) to \(7000\ \overset{\circ}{\text{A}}\), can measure sizes down to about \(4000\ \overset{\circ}{\text{A}}\).
If we want to measure even smaller sizes, we need to use an even smaller wavelength, and so the use of an electron microscope becomes necessary. As you will study in the XII\(^{\text{th}}\) standard, each material particle corresponds to a wave. The approximate wavelength of the electrons in an electron microscope is about \(0.6\ \overset{\circ}{\text{A}}\), so that one can measure atomic sizes \(\approx 1\ \overset{\circ}{\text{A}}\) using this microscope.
1.4 Measurement of Mass
The SI unit of mass is the kilogram, and for well over a century its standard was an object you could actually hold.
Since 1889, a kilogram was the mass of a shiny piece of platinum–iridium alloy kept in a special glass case at the International Bureau of Weights and Measures. Every kilogram used anywhere on the Earth was, in the end, a copy of a copy of that one cylinder.
Why a Metal Cylinder Was Not Good Enough
This definition of mass has been modified on 20th May 2019. The reason is unsettling in its simplicity: the carefully kept platinum–iridium piece is seen to pick up micro particles of dirt, and is also affected by the atmosphere, causing its mass to change.
Think about what that means. The cylinder is the kilogram, by definition. So if the cylinder gains a few micrograms of dirt, the cylinder cannot be said to have become heavier — instead, the kilogram itself has quietly changed, and with it every mass measurement in the world. A standard that drifts is no standard at all.
The New Kilogram
The new measure of the kilogram is defined in terms of the magnitude of electric current. At first this sounds strange — how can a current define a mass? The link is the electromagnet.
We know that electric current can be used to make an electromagnet. An electromagnet attracts magnetic materials, and is thus used in research and in industrial applications such as cranes to lift heavy pieces of iron and steel. The larger the current, the stronger the magnetic pull.
Now imagine an extremely sensitive balance. One side holds one standard kilogram mass. On the other side, instead of placing metal weights, we let an electromagnet pull the pan down. We then ask: exactly how much current must be passed through the electromagnet so that its pull balances the standard kilogram?
That current is a precisely measurable electrical quantity. Thus the kilogram mass can be described in terms of the amount of current which has to be passed through an electromagnet so that it can pull down one side of an extremely sensitive balance to balance the other side which holds one standard kg mass.
Diagram
Fig. 1.13 – A sensitive balance with a standard 1 kg mass in one pan and a current-carrying electromagnet pulling down the other pan, with an ammeter in the circuit.
Mass on the Atomic Scale
While dealing with the mass of atoms and molecules, the kilogram is an inconvenient unit. An atom has a mass of the order of \(10^{-26}\) kg, and writing every atomic mass with such an exponent is clumsy and invites error.
It is much easier to compare the mass of any atom in terms of the mass of some standard atom. The standard atom has been decided internationally to be the C\(^{12}\) atom. The \((1/12)^{\text{th}}\) mass of an unexcited atom of C\(^{12}\) is called the atomic mass unit (amu).
\[1 \ \text{amu} = 1.6605402 \times 10^{-27}\ \text{kg}\]
with an uncertainty of 10 in the last two decimal places.
The one-twelfth is not arbitrary. A carbon-12 atom contains 6 protons and 6 neutrons, so taking a twelfth of it makes 1 amu come out very close to the mass of a single proton or neutron. On this scale, the mass number of an atom is very nearly its mass in amu — which is exactly what makes the unit convenient.
1.5 Measurement of Time
The SI unit of time is the second (s). As with length and mass, the interesting question is not what the unit is called, but what natural event we choose to count.
The Second from the Turning Earth
For many years, the duration of one mean solar day was considered as the reference. A mean solar day is the average time interval from one noon to the next noon.
The word “average” is doing quiet work here. The interval from noon to noon is not exactly the same on every day of the year, so the standard is built from the average taken over the year rather than from any single day.
The average duration of a day is taken as 24 hours. One hour is of 60 minutes and each minute is of 60 seconds. Thus a mean solar day
\[= 24 \ \text{hours} = 24 \times 60 \times 60 = 86400\ \text{s}\]
Accordingly, a second was defined as 1/86400 of a mean solar day.
This is a satisfying definition, because it ties our unit of time to the rotation of the Earth — the most familiar clock human beings have ever had.
Why the Earth Was Dismissed as a Clock
It was later observed that the length of a solar day varies gradually, due to the gradual slowing down of the Earth’s rotation. The Earth is very slightly losing spin, so the day is very slightly lengthening.
The consequence is the same as with the platinum–iridium cylinder: if the second is defined as a fraction of a day, and the day is stretching, then the second itself is stretching. The unit is drifting, and we cannot detect the drift from inside the definition.
Hence, to get a more standard and non-varying (constant) unit for the measurement of time, a cesium atomic clock is used. It is based on the periodic vibrations produced in the cesium atom. In a cesium atomic clock, a second is taken as the time needed for 9,192,631,770 vibrations of the radiation (wave) emitted during a transition between two hyperfine states of the Cs\(^{133}\) atom.
Diagram
Fig. 1.14 – Two clocks side by side: the Earth rotating from one noon to the next (mean solar day, 86400 s) and a caesium atom emitting radiation whose 9,192,631,770 vibrations make one second.
1.6 Dimensions and Dimensional Analysis
As mentioned earlier, a derived physical quantity can be expressed in terms of some combination of the seven basic or fundamental quantities. Velocity is length divided by time; force is mass times length divided by time squared. Strip away the numbers and the names, and what is left is this skeleton of fundamental quantities. That skeleton is what we call the dimensions of the quantity.
For convenience, the basic quantities are represented by symbols:
- ‘L’ for length
- ‘M’ for mass
- ‘T’ for time
- ‘K’ for temperature
- ‘I’ for current
- ‘C’ for luminous intensity
- ‘mol’ for amount of substance
Read that definition slowly, because every word in it is doing work. Dimensions are powers — not units, not numbers. To say that force has dimensions \([L^{1}M^{1}T^{-2}]\) is to say that its unit is obtained by raising metre to the power 1, kilogram to the power 1 and second to the power \(-2\), and multiplying them together.
Writing a Dimensional Formula
When we represent any derived quantity with appropriate powers of the symbols of the fundamental quantities, such an expression is called a dimensional formula.
Two points of notation must be respected. The dimensional formula is expressed inside a square bracket, and no comma is written in between any of the symbols. So we write \([L^{1}M^{0}T^{-1}]\) and never \([L^{1}, M^{0}, T^{-1}]\).
Diagram
Fig. 1.15 – The dimensional formula \([L^{1}M^{1}T^{-2}]\) of force, with callouts labelling the square bracket, the base symbols L, M, T and the powers 1, 1, −2.
Illustration
The method is always the same: write the defining relation of the quantity, replace each quantity by its dimensions, and simplify the powers.
(i) Dimensional formula of velocity
\[\text{Velocity} = \frac{\text{displacement}}{\text{time}}\]
\[\text{Dimensions of velocity} = \frac{[L]}{[T]} = [L^{1}M^{0}T^{-1}]\]
Notice that mass does not enter into velocity at all. We do not simply omit M — we write it with the power zero. Recording the missing quantities as zero powers keeps every dimensional formula in the same standard shape, so that two of them can be compared term by term.
(ii) Dimensional formula of velocity gradient
\[\text{velocity gradient} = \frac{\text{velocity}}{\text{distance}}\]
\[\text{Dimensions of velocity gradient} = \frac{[LT^{-1}]}{[L]} = [L^{0}M^{0}T^{-1}]\]
Here the two lengths have cancelled, leaving a quantity that depends only on time. Its SI unit is therefore s\(^{-1}\).
(iii) Dimensional formula for charge
\[\text{charge} = \text{current} \times \text{time}\]
\[\text{Dimensions of charge} = [I][T] = [L^{0}M^{0}T^{1}I^{1}]\]
This is the first quantity we have met whose dimensional formula needs a symbol beyond L, M and T. Electrical quantities bring in the current I, exactly as thermal quantities bring in the temperature K.
Table 1.3 gives the dimensions of various physical quantities commonly used in mechanics. Study it carefully and confirm each entry for yourself from its formula, rather than memorising the last column.
Table 1.3: Some Common Physical Quantities, their SI Units and Dimensions
| S. No | Physical quantity | Formula | SI unit | Dimensional formula |
|---|---|---|---|---|
| 1 | Density | \(\rho = M/V\) | kilogram per cubic metre (kg/m\(^{3}\)) | \([L^{-3}M^{1}T^{\circ}]\) |
| 2 | Acceleration | \(a = v/t\) | metre per second square (m/s\(^{2}\)) | \([L^{1}M^{\circ}T^{-2}]\) |
| 3 | Momentum | \(P = mv\) | kilogram metre per second (kg m/s) | \([L^{1}M^{1}T^{-1}]\) |
| 4 | Force | \(F = ma\) | kilogram metre per second square (kg m/s\(^{2}\)) or newton (N) | \([L^{1}M^{1}T^{-2}]\) |
| 5 | Impulse | \(J = F.t\) | newton second (Ns) | \([L^{1}M^{1}T^{-1}]\) |
| 6 | Work | \(W = F.s\) | joule (J) | \([L^{2}M^{1}T^{-2}]\) |
| 7 | Kinetic Energy | \(KE = 1/2\ mv^{2}\) | joule (J) | \([L^{2}M^{1}T^{-2}]\) |
| 8 | Pressure | \(P = F/A\) | kilogram per metre second square (kg/ms\(^{2}\)) | \([L^{-1}M^{1}T^{-2}]\) |
Two entries in this table deserve a second look. Momentum and impulse have the same dimensional formula, \([L^{1}M^{1}T^{-1}]\); so do work and kinetic energy, \([L^{2}M^{1}T^{-2}]\). This is not a coincidence — impulse is the change in momentum, and work is the change in kinetic energy. Quantities that can be equated to one another must share the same dimensions.
The converse, however, does not hold: two quantities having the same dimensions need not be the same physical quantity. Work and torque, for instance, share a dimensional formula but mean quite different things. Dimensions tell us about the structure of a quantity, not about its identity.
1.6.1 Uses of Dimensional Analysis
Dimensions are not merely a bookkeeping device. Once a quantity’s skeleton of L, M and T is known, that skeleton can be put to work in three distinct ways.
(i) To Check the Correctness of Physical Equations
In any equation relating different physical quantities, if the dimensions of all the terms on both the sides are the same, then that equation is said to be dimensionally correct. This is called the principle of homogeneity of dimensions.
The reasoning behind it is common sense dressed in symbols. You cannot add a length to a time, any more than you can add three metres to five seconds. Terms may only be added, subtracted or equated if they are the same kind of quantity — and “the same kind of quantity” is precisely what having the same dimensions means.
Consider the first equation of motion.
\[v = u + at\]
Dimension of L.H.S \(= [v] = [LT^{-1}]\)
Now examine the right-hand side, term by term:
\[[u] = [LT^{-1}]\] \[[at] = [LT^{-2}][T] = [LT^{-1}]\]
Dimension of R.H.S \(= [LT^{-1}] + [LT^{-1}]\)
\[[L.H.S] = [R.H.S]\]
As the dimensions of L.H.S and R.H.S are the same, the given equation is dimensionally correct.
Note carefully that we did not add the two right-hand terms to get \(2[LT^{-1}]\). Dimensions are not added like numbers; we merely observe that both terms carry the same dimensions, and that is enough.
(iii) To Find the Conversion Factor Between the Units of the Same Physical Quantity in Two Different Systems of Units
The third use is the most practical of all. A quantity keeps its dimensions whichever system you measure it in; only the size of the base units changes. That single fact lets us convert between systems without knowing anything about the quantity itself.
Let us use dimensional analysis to determine the conversion factor between the joule (SI unit of work) and the erg (CGS unit of work).
Let \(1\ \text{J} = x\) erg
The dimensional formula for work is \([L^{2}M^{1}T^{-2}]\). Substituting in the above equation, we can write
\[[L_{1}^{2}M_{1}^{1}T_{1}^{-2}] = x\ [L_{2}^{2}M_{2}^{1}T_{2}^{-2}]\]
\[x = \frac{[L_{1}^{2}M_{1}^{1}T_{1}^{-2}]}{[L_{2}^{2}M_{2}^{1}T_{2}^{-2}]}\]
\[\text{or,} \quad x = \left(\frac{L_{1}}{L_{2}}\right)^{2}\left(\frac{M_{1}}{M_{2}}\right)^{1}\left(\frac{T_{1}}{T_{2}}\right)^{-2}\]
where suffix 1 indicates SI units and 2 indicates CGS units.
In SI units, L, M, T are expressed in m, kg and s, and in the CGS system L, M, T are represented in cm, g and s respectively.
\[\therefore \ x = \left(\frac{\text{kg}}{\text{g}}\right)^{1}\left(\frac{\text{m}}{\text{cm}}\right)^{2}\left(\frac{\text{s}}{\text{s}}\right)^{-2}\]
\[\text{or}\quad x = \left(10^{3}\ \frac{\text{g}}{\text{g}}\right)^{1}\left((100)\ \frac{\text{cm}}{\text{cm}}\right)^{2}(1)^{-2}\]
\[\therefore \ x = (10^{3})(10^{4}) = 10^{7}\]
\[\therefore \ 1\ \text{joule} = 10^{7}\ \text{erg}\]
Observe the logic of the ratio. Since 1 kg contains 1000 g, the number measuring a given mass becomes 1000 times larger when expressed in grams. Each base unit is converted in this way, raised to the power that the dimensional formula demands, and the product of all three gives the conversion factor.
1.6.2 Limitations of Dimensional Analysis
Powerful as it is, dimensional analysis is not a substitute for physics. It knows only about the powers of L, M and T, and is entirely blind to everything else. That blindness produces four definite limitations, and an examiner is as likely to ask about these as about the uses.
1) The value of a dimensionless constant can be obtained with the help of experiments only.
We have already met this. The method gave us \(T = k\sqrt{l/g}\) for the simple pendulum, but the value \(k = 2\pi\) had to come from the laboratory. Pure numbers have no dimensions, so no amount of dimensional reasoning can pin them down.
2) Dimensional analysis cannot be used to derive relations involving trigonometric, exponential and logarithmic functions, as these quantities are dimensionless.
The argument of \(\sin\theta\), of \(e^{x}\) or of \(\log x\) must itself be a dimensionless number — you cannot take the sine of three metres. And the value these functions return is likewise a pure number. Since dimensional analysis works only by balancing powers of L, M and T, an equation such as \(y = a\sin(\omega t)\) lies completely outside its reach.
3) This method is not useful if the constant of proportionality is not a dimensionless quantity.
Illustration: The gravitational force between two point masses is directly proportional to the product of the two masses and inversely proportional to the square of the distance between the two.
\[\therefore F \propto \frac{m_{1}m_{2}}{r^{2}}\]
Let
\[F = G\ \frac{m_{1}m_{2}}{r^{2}}\]
The constant of proportionality ‘G’ is NOT dimensionless. Thus the earlier method will not work.
Look at what goes wrong. In the pendulum derivation we could equate dimensions on both sides only because \(k\) contributed nothing. Here, \(G\) carries dimensions of its own, and those unknown dimensions sit on the right-hand side of the equation exactly where we needed clarity. The balancing of powers can no longer be completed, and the relation cannot be derived by dimensions alone — it had to be discovered from observation.
4) If the correct equation contains some more terms of the same dimension, it is not possible to know about their presence using a dimensional equation.
For example, with standard symbols, the equation
\[S = \frac{1}{2}at^{2}\]
is dimensionally correct. However, the complete equation is
\[S = ut + \frac{1}{2}at^{2}\]
Both \(ut\) and \(\frac{1}{2}at^{2}\) have the dimension of length, so the homogeneity test is passed whether or not the first term is present. Dimensional analysis therefore cannot tell us that a term has gone missing, and an incomplete equation can sail through the check unnoticed.
Diagram
Fig. 1.16 – A two-column summary chart: “Dimensional analysis CAN” (check equations, find relations up to a constant, convert units) and “Dimensional analysis CANNOT” (find pure numbers, handle sin/exp/log, handle dimensional constants like G, detect missing terms).
1.7 Accuracy, Precision and Uncertainty in Measurement
Physics is a science based on observations and experiments. Observations of various physical quantities are made during an experiment. For example, during an atmospheric study we measure atmospheric pressure, wind velocity, humidity, and so on.
Every one of these observations is a number produced by a real instrument, handled by a real person, under real conditions. And no such number is ever perfect. Two separate questions must therefore be asked about any measurement: is it close to the truth? and is it repeatable? These are two different questions, and physics gives them two different names.
Accuracy
All the measurements may be accurate, meaning that the measured values are the same as the true values. Accuracy is how close a measurement is to the actual value of that quantity.
Precision
These measurements may be precise, meaning that multiple measurements give nearly identical values — that is, reproducible results.
Notice what precision does not mention: the true value. A precise instrument is one that agrees with itself, not necessarily one that agrees with nature.
In actual measurements, an observation may be both accurate and precise, or neither accurate nor precise. All four combinations are possible, as shown in Fig. 1.17.
Diagram
Fig. 1.17 – Four target boards showing shots that are (a) accurate and precise, (b) precise but not accurate, (c) accurate but not precise, (d) neither accurate nor precise.
The goal of the observer should be to get accurate as well as precise measurements. Precision alone gives a confident wrong answer; accuracy alone gives an answer that cannot be trusted to repeat.
Where the Uncertainty Comes From
Possible uncertainties in an observation may arise due to the following reasons:
Quality of the instrument used. A metre scale marked only in centimetres cannot deliver a reading to the nearest millimetre, however careful the observer.
Skill of the person doing the experiment. Two students using the same Vernier calliper may not tighten it identically or read the coincidence of the divisions with equal care.
The method used for measurement. Some methods are inherently blunter than others; measuring the volume of an irregular solid by displacement is more reliable than estimating it from its shape.
External or internal factors affecting the result of the experiment. Temperature, pressure, humidity, vibration or a fluctuating power supply can all shift a reading without the observer noticing.
Study this list carefully, because it is really a preview of the next section. Reasons 1, 2 and 3 tend to push the reading the same way every time and can, in principle, be identified and removed. Reason 4 pushes it now this way, now that. This distinction is the seed of the classification of errors that follows.
1.8 Errors in Measurements
Faulty measurements of a physical quantity can lead to errors. In physics, the word “error” does not mean a blunder or a mistake in arithmetic. It means the unavoidable difference between the value we obtain and the value that is actually there.
Errors are broadly divided into the following two categories.
Systematic Errors
a) Systematic errors: Systematic errors are errors that are not determined by chance but are introduced by an inaccuracy — involving either the observation or the measurement process — inherent to the system.
Sources of systematic error may be due to imperfect calibration of the instrument, and sometimes an imperfect method of observation.
The defining feature of these errors is stated in a single sentence, and it is worth underlining: each of these errors tends to be in one direction, either positive or negative. They do not scatter. They push every single reading the same way by the same amount, which is why repeating the experiment a hundred times will not reveal them.
The sources of systematic errors are as follows.
(i) Instrumental error: This type of error arises due to defective calibration of an instrument. For example, an incorrect zeroing of an instrument will lead to such a kind of error — the ‘zero’ of a thermometer not graduated at the proper place; the pointer of a weighing balance in the laboratory already indicating some value instead of showing zero when no load is kept on it; an ammeter showing a current of 0.5 amp even when it is not connected in a circuit.
In each of these examples, the instrument is lying by a fixed amount before the experiment has even begun, and it will go on telling the same lie in every reading.
(ii) Error due to imperfection in experimental technique: This is an error due to defective setting of an instrument. For example, the measured volume of a liquid in a graduated tube will be inaccurate if the tube is not held vertical.
Here the instrument itself is perfectly sound. It is the way it has been set up that introduces the fault — and since the tube stays tilted throughout the experiment, the error again repeats identically in every observation.
(iii) Personal error: Such errors are introduced due to the fault of the observer. Bias of the observer, carelessness in taking observations, and so on, could result in such errors.
For example, while measuring the length of an object with a ruler, it is necessary to look at the ruler directly from above. If the observer looks at it from an angle, the measured length will be wrong due to parallax — the same apparent shift of position with the observer’s position that was used to advantage in Section 1.3.1, but which here works against us. And because an observer usually sits in the same place throughout an experiment, the viewing angle stays the same, and so does the error.
Diagram
Fig. 1.18 – An observer’s eye viewing a scale from directly above (correct reading) and from an angle (parallax error), with the two apparent readings marked on the scale.
Systematic errors can be minimized by using the correct instrument, following the proper experimental procedure, and removing personal error. Notice the phrasing: they are removed by thinking about the setup, never by taking more readings.
Random Errors
b) Random errors: These are the errors which are introduced even after following all the procedures to minimize systematic errors.
These types of errors may be positive or negative. They cannot be eliminated completely, but we can minimize them by taking repeated observations and then taking their mean (average).
This is exactly the opposite of the situation with systematic errors, and the reason lies in the word “random”. Since a random error is as likely to push a reading up as down, the deviations tend to cancel one another when many readings are averaged. The more readings we take, the more thoroughly they cancel.
Random errors occur due to variation in the conditions in which the experiment is performed. For example, the temperature may change during the course of an experiment, the pressure of any gas used in the experiment may change, or the voltage of the power supply may change randomly.
The next subsection takes up the mathematical treatment of random errors: given a set of scattered readings, what single value do we report, and how far from the truth might it be?
1.8.1 Estimation of error
Suppose we have done the honest thing — checked the instrument, set it up properly, removed personal bias — and yet the readings still refuse to agree with one another. Random error remains. What number do we then report, and how much can we trust it?
Suppose the readings recorded repeatedly for a physical quantity during a measurement are
\[a_{1},\ a_{2},\ a_{3},\ \ldots\ldots\ldots\ a_{n}\]
The Most Probable Value
The arithmetic mean \(a_{mean}\) is given by
\[a_{mean} = \frac{a_{1} + a_{2} + a_{3} + \ldots\ldots\ldots + a_{n}}{n}\]
\[a_{mean} = \frac{1}{n}\sum_{i=1}^{n} a_{i} \qquad \text{--- (1.3)}\]
This is the most probable value of the quantity. The justification comes straight from the previous section: random errors are as likely to be positive as negative, so when the readings are added, the positive deviations tend to cancel the negative ones. What survives the averaging is our best estimate of the truth.
Absolute Error
We now ask how far each individual reading has strayed from this most probable value. The magnitude of the difference between the mean value and each individual value is called the absolute error in the observation.
Thus, for ‘\(a_{1}\)’, the absolute error \(\Delta a_{1}\) is given by
\[\Delta a_{1} = |\ a_{mean} - a_{1}\ |,\]
for \(a_{2}\),
\[\Delta a_{2} = |\ a_{mean} - a_{2}\ |\]
and so for \(a_{n}\) it will be
\[\Delta a_{n} = |\ a_{mean} - a_{n}\ |\]
The modulus signs are essential. We are interested in how far a reading has strayed, not in which side it strayed to — and if the signs were kept, the deviations would cancel and give zero, which would tell us nothing.
Mean Absolute Error
The arithmetic mean of all the absolute errors is called the mean absolute error in the measurement of the physical quantity.
\[\Delta a_{mean} = \frac{\Delta a_{1} + \Delta a_{2} + \ldots\ldots + \Delta a_{n}}{n}\]
\[= \frac{1}{n}\sum_{i=1}^{n}\Delta a_{i} \qquad \text{--- (1.4)}\]
The measured value of the physical quantity \(a\) can then be represented by
\[a = a_{mean} \pm \Delta a_{mean}\]
which tells us that the actual value of ‘\(a\)’ could be between \(a_{mean} - \Delta a_{mean}\) and \(a_{mean} + \Delta a_{mean}\).
This is the honest way to report a measurement. The single number \(a_{mean}\) is our best guess; the \(\pm\ \Delta a_{mean}\) is the width of the band inside which the truth is expected to lie.
Relative Error and Percentage Error
An error of 1 cm means very different things when you are measuring a pencil and when you are measuring a cricket pitch. To judge how serious an error is, it must be compared with the size of the quantity being measured.
The ratio of the mean absolute error to its arithmetic mean value is called the relative error.
\[\text{Relative error} = \frac{\Delta a_{mean}}{a_{mean}} \qquad \text{--- (1.5)}\]
Since it is a ratio of two quantities of the same kind, the relative error has no unit and no dimension — it is a pure number, and this is precisely what allows the quality of two completely different measurements to be compared.
When the relative error is represented as a percentage, it is called the percentage error.
\[\text{Percentage error} = \frac{\Delta a_{mean}}{a_{mean}} \times 100 \qquad \text{--- (1.6)}\]
1.8.2 Combination of errors
When we do an experiment and measure various physical quantities associated with the experiment, we must know how the errors from the individual measurements combine to give the error in the final result.
For example, in the measurement of the resistance of a conductor using Ohm’s law, there will be an error in the measurement of the potential difference and also in that of the current. Neither quantity is measured perfectly, and the resistance is calculated from both. It is important to study how these errors combine to give the error in the measurement of resistance.
The rules that follow are always derived on the same principle: we ask for the worst case. Errors could partially cancel if we are lucky, but a careful physicist never counts on luck. We therefore quote the largest error the result could possibly carry.
a) Errors in sum and in difference
Suppose two physical quantities A and B have measured values \(A \pm \Delta A\) and \(B \pm \Delta B\) respectively, where \(\Delta A\) and \(\Delta B\) are their mean absolute errors. We wish to find the absolute error \(\Delta Z\) in their sum.
\[Z = A + B\] \[Z \pm \Delta Z = (A \pm \Delta A) + (B \pm \Delta B)\] \[= (A + B) \pm \Delta A \pm \Delta B\] \[\pm \Delta Z = \pm \Delta A \pm \Delta B,\]
For the difference, i.e., if \(Z = A - B\),
\[Z \pm \Delta Z = (A \pm \Delta A) - (B \pm \Delta B)\] \[= (A - B) \pm \Delta A \mp \Delta B\] \[\pm \Delta Z = \pm \Delta A \pm \Delta B,\]
There are four possible values for \(\Delta Z\), namely \((+\Delta A - \Delta B)\), \((+\Delta A + \Delta B)\), \((-\Delta A - \Delta B)\) and \((-\Delta A + \Delta B)\). Hence the maximum value of the absolute error is \(\Delta Z = \Delta A + \Delta B\) in both the cases.
The signs must be allowed to take every combination, because we do not know in which direction each individual measurement has gone wrong. Among the four possibilities, the largest magnitude of \(\Delta Z\) is obtained when both errors happen to act the same way — and that worst case is the one we report.
Read the statement again and note the surprise buried in it: even for a difference, the errors are added. Subtracting two measured quantities does not subtract their uncertainties; it accumulates them.
b) Errors in product and in division
Suppose \(Z = AB\) and the measured values of A and B are \((A \pm \Delta A)\) and \((B \pm \Delta B)\). Then
\[Z \pm \Delta Z = (A \pm \Delta A)\ (B \pm \Delta B)\] \[= AB \pm A\Delta B \pm B\Delta A \pm \Delta A \Delta B\]
Dividing the L.H.S by \(Z\) and the R.H.S by \(AB\) — which is legitimate because \(Z = AB\) — we get
\[\left(1 \pm \frac{\Delta Z}{Z}\right) = \left(1 \pm \frac{\Delta B}{B} \pm \frac{\Delta A}{A} \pm \left(\frac{\Delta A}{A}\right)\left(\frac{\Delta B}{B}\right)\right)\]
Since \(\Delta A/A\) and \(\Delta B/B\) are very small, we shall neglect their product. If each relative error is of the order of one hundredth, their product is of the order of one ten-thousandth — utterly negligible beside the terms we are keeping. Hence the maximum relative error in Z is
\[\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \qquad \text{--- (1.7)}\]
This formula also applies to the division of two quantities.
c) Errors due to the power (index) of a measured quantity
Suppose
\[Z = A^{3} = A.A.A\]
This is just a product of three identical quantities, so the multiplication rule above applies three times over:
\[\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta A}{A} + \frac{\Delta A}{A}\]
Hence the maximum relative error in \(Z = A^{3}\) is three times the relative error in A. So if \(Z = A^{n}\),
\[\frac{\Delta Z}{Z} = n\ \frac{\Delta A}{A} \qquad \text{--- (1.8)}\]
In general, if \(Z = \dfrac{A^{p}B^{q}}{C^{r}}\),
\[\frac{\Delta Z}{Z} = p\ \frac{\Delta A}{A} + q\ \frac{\Delta B}{B} + r\ \frac{\Delta C}{C} \qquad \text{--- (1.9)}\]
Look carefully at equation (1.9). The power \(r\) of the quantity in the denominator enters with a plus sign, exactly like \(p\) and \(q\). Errors never subtract, whether the quantity divides or multiplies; every one of them makes the result worse.
The quantity in the formula which has a large power is responsible for the maximum error. This is a practical instruction for the laboratory: if a quantity appears cubed in your formula, its measurement deserves three times the care, because its error is tripled before it reaches the answer.
Diagram
Fig. 1.19 – A flow chart: “Are the quantities added/subtracted?” → add absolute errors; “Are they multiplied/divided/raised to powers?” → add relative errors weighted by their powers.
1.9 Significant Figures
In the previous sections, we have studied various types of errors, their origins and the ways to minimize them. One source of limitation, however, can never be argued away: our accuracy is limited to the least count of the instrument used during the measurement.
Least count is the smallest measurement that can be made using the given instrument. For example, with the usual metre scale, one can measure 0.1 cm as the least value. Hence its least count is 0.1 cm.
How Many Digits Are We Entitled To?
Suppose we measure the length of a metal rod using a metre scale of least count 0.1 cm. The measurement is done three times and the readings are 15.4, 15.4, and 15.5 cm. The most probable length, which is the arithmetic mean as per our earlier discussion, is 15.43.
Now look hard at that number. Out of it, we are certain about the digits 1 and 5, but we are not certain about the last 2 digits because of the least count limitation. The scale itself could not resolve beyond 0.1 cm, so the 4 was already an estimate, and the 3 that appeared from the averaging is beyond any claim we can honestly make.
This leads to the definition that governs how a measured value must be written.
Thus in the above example, we have 3 significant digits: 1, 5 and 4.
Note what is being permitted and what is being forbidden. We are allowed one doubtful digit — the first estimated one — because it still carries genuine information. We are not allowed a second, because it carries none. Writing 15.43 claims a precision the metre scale never possessed.
The larger the number of significant figures obtained in a measurement, the greater is the accuracy of the measurement. If one uses an instrument of smaller least count, the number of significant digits increases. Significant figures are therefore not a rule of arithmetic; they are an honest statement about the quality of the instrument that produced the number.
Rules for determining significant figures
1) All the nonzero digits are significant. For example, if the volume of an object is 178.43 cm\(^{3}\), there are five significant digits, which are 1, 7, 8, 4 and 3.
2) All the zeros between two nonzero digits are significant. For example, m = 165.02 g has 5 significant digits. Such a zero is a genuine measured value — it says the instrument read zero in that place, and that is real information.
3) If the number is less than 1, the zero or zeroes on the right of the decimal point and to the left of the first nonzero digit are not significant. For example, in 0.001405, the underlined zeros are not significant. Thus the above number has four significant digits.
These leading zeros are only placeholders that fix the position of the decimal point. Writing the same length as \(1.405 \times 10^{-3}\) makes this obvious: the zeros vanish, but nothing has been lost.
4) The zeros on the right hand side of the last nonzero number are significant — but for this, the number must be written with a decimal point. For example, 1.500 or 0.01500 have 4 significant figures each.
A trailing zero after the decimal point is a deliberate statement. By writing 1.500 rather than 1.5, the experimenter is declaring that the instrument was good enough to establish those last two zeros.
On the contrary, if a measurement yields a length L given as
\[L = 125\ \text{m} = 12500\ \text{cm} = 125000\ \text{mm},\]
it has only three significant digits.
This is the ambiguity that the whole system stumbles on. The zeros in 125000 mm arrived only because the unit was changed; they were never measured. Yet the way the number is written, no one can tell whether they were measured or not — and the quality of a measurement obviously cannot depend on whether we chose to report it in metre or in millimetre.
Scientific Notation and Order of Magnitude
To avoid the ambiguities in determining the number of significant figures, it is necessary to report every measurement in scientific notation (i.e., in powers of 10), that is, by using the concept of order of magnitude.
The magnitude of any physical quantity can be expressed as \(A \times 10^{n}\), where ‘A’ is a number such that \(0.5 \le A < 5\), and ‘n’ is an integer called the order of magnitude.
The advantage is immediate: the digits of \(A\) carry the significant figures, and the power of ten carries the size. The two jobs are separated, and no zero has to do both at once.
(i) radius of Earth \(= 6400\) km \[= 0.64 \times 10^{7}\ \text{m}\]
The order of magnitude is 7 and the number of significant figures is 2.
Note why the mantissa is written as 0.64 and not 6.4 — the definition here requires \(A\) to lie between 0.5 and 5, and 6.4 lies outside that band, so one more power of ten is moved into the exponent.
(ii) Magnitude of the charge on an electron \[e = 1.6 \times 10^{-19}\ \text{C}\]
Here the order of magnitude is \(-19\) and the number of significant digits is 2.
Diagram
Fig. 1.20 – New SI: a network of the seven base units (s, m, kg, A, K, mol, cd) with arrows from the defining constants Δν_Cs, c, h, e, k_B, N_A and K_cd.
Diagram
Fig. 1.21 – Old SI: the corresponding network of base units before 20 May 2019, with arrows from Δν_Cs, c, M_IPK, μ₀, T_TPW, m(¹²C) and K_cd.
1.10 Summary
Units and Systems of Units
- Physics is a quantitative science. Every measurement is a comparison of a quantity with an internationally accepted standard called a unit, and a measured value is always written as a number followed by its unit.
- The systems of units used earlier are CGS (centimetre, gram, second), MKS (metre, kilogram, second) and FPS (foot, pound, second). In 1971 the 14th International General Conference on Weights and Measures recommended the SI (System International), which uses the decimal system and so makes conversion simple.
Fundamental, Derived and Supplementary Units
- Fundamental quantities do not depend on any other physical quantity for their measurement. There are seven: length (metre, m), mass (kilogram, kg), time (second, s), temperature (kelvin, K), electric current (ampere, A), luminous intensity (candela, cd) and amount of substance (mole, mol).
- Derived quantities depend on some or all of the seven fundamental quantities, and their units are derived units, e.g. velocity in m s\(^{-1}\) and momentum in kg m s\(^{-1}\).
- Two supplementary units exist:
- Plane angle, \(d\theta = \dfrac{ds}{r}\), measured in radian (rad).
- Solid angle, \(d\Omega = \dfrac{dA}{r^{2}}\), measured in steradian (sr). The whole sphere subtends \(\Omega = \dfrac{4\pi r^{2}}{r^{2}} = 4\pi\) sr at its centre.
- Key angle conversions: \(\pi^{c} = 180^{\circ}\); \(1\) radian \(= 57.297^{\circ}\); \(1^{\circ} = 1.745 \times 10^{-2}\) rad; \(1' = 2.91 \times 10^{-4}\) rad; \(1'' = 4.847 \times 10^{-6}\) rad.
Conventions for SI Units
- Units are written by their symbols; full names begin with a small letter (newton), but symbols of units named after persons are capitalised (N, J).
- Symbols take no plural and no full stop: 20 N, 25 kg.
- Numerator and denominator are written as one ratio: m s\(^{-2}\), not m/s/s.
- Do not mix names and symbols: J/kg K is correct, joule/kg K is not.
- Prefixes stand for powers of ten: 1 ms \(= 10^{-3}\) s, 1 µs \(= 10^{-6}\) s, 1 ns \(= 10^{-9}\) s. Double prefixes are avoided.
- A space or hyphen shows multiplication of units: m s\(^{-1}\), not ms\(^{-1}\) (ms means millisecond).
Measurement of Length, Mass and Time
- Metre: the length of the path travelled by light in vacuum during a time interval of 1/299792458 second, the speed of light being fixed as \(c = 299792458\) m/s.
- Parallax is the apparent change in the position of an object due to a change in the position of the observer. For a basis \(b\) and parallax angle \(\theta\) (in radian), \[D = \frac{b}{\theta} \qquad \text{--- (1.1)}\]
- For stars, the parallax is measured between two farthest points on the Earth’s orbit, 2 AU apart.
- The angular diameter \(\alpha\) of a planet of diameter \(d\) at distance \(D\) gives \[d = \alpha D \qquad \text{--- (1.2)}\]
- Astronomical units of distance: \[1\ \text{AU} = 1.496 \times 10^{11}\ \text{m}, \quad 1\ \text{light year} = 9.467 \times 10^{15}\ \text{m},\] \[1\ \text{parsec (pc)} = 3.08 \times 10^{16}\ \text{m} \cong 3.26\ \text{light years}\]
- Very small distances use the fermi (\(1\ \text{F} = 10^{-15}\) m) and the Angstrom (\(1\ \overset{\circ}{\text{A}} = 10^{-10}\) m). Atomic sizes (\(\approx 1\ \overset{\circ}{\text{A}}\)) are measured with an electron microscope, whose electron wavelength is about \(0.6\ \overset{\circ}{\text{A}}\).
- Kilogram: from 1889 it was the mass of a platinum–iridium cylinder; since 20 May 2019 it is defined through the magnitude of electric current needed in an electromagnet to balance one standard kilogram.
- For atoms, \(1\) amu \(= (1/12)^{\text{th}}\) of the mass of an unexcited C\(^{12}\) atom \(= 1.6605402 \times 10^{-27}\) kg.
- Second: earlier 1/86400 of a mean solar day (24 × 60 × 60 = 86400 s); now the time needed for 9,192,631,770 vibrations of the radiation emitted in a transition between two hyperfine states of the Cs\(^{133}\) atom.
Dimensions and Dimensional Analysis
- The dimensions of a physical quantity are the powers to which the concerned fundamental units must be raised to obtain the unit of that quantity. The dimensional formula is written in square brackets with no commas, e.g. velocity \(= [L^{1}M^{0}T^{-1}]\), velocity gradient \(= [L^{0}M^{0}T^{-1}]\), charge \(= [L^{0}M^{0}T^{1}I^{1}]\).
- Common dimensional formulae: force \([L^{1}M^{1}T^{-2}]\), momentum and impulse \([L^{1}M^{1}T^{-1}]\), work and kinetic energy \([L^{2}M^{1}T^{-2}]\), pressure \([L^{-1}M^{1}T^{-2}]\), density \([L^{-3}M^{1}T^{\circ}]\), acceleration \([L^{1}M^{\circ}T^{-2}]\).
- Uses of dimensional analysis:
- To check the correctness of an equation, using the principle of homogeneity of dimensions — all terms on both sides must have the same dimensions (e.g. \(v = u + at\)).
- To establish a relation between related quantities, e.g. for a simple pendulum \(T = k\sqrt{l/g}\), with \(k = 2\pi\) found experimentally, so that \(T = 2\pi\sqrt{l/g}\).
- To find the conversion factor between two systems, e.g. \(1\ \text{joule} = 10^{7}\ \text{erg}\).
- Limitations: dimensionless constants can be found only by experiment; relations with trigonometric, exponential or logarithmic functions cannot be derived; the method fails when the constant of proportionality has dimensions (e.g. G in \(F = G\,m_{1}m_{2}/r^{2}\)); and missing terms of the same dimension go undetected (e.g. \(S = \frac{1}{2}at^{2}\) passes the test although the complete equation is \(S = ut + \frac{1}{2}at^{2}\)).
Accuracy, Precision and Errors
- Accuracy is closeness of a measurement to the true value; precision is closeness of repeated measurements to one another. A measurement should be both.
- Uncertainty arises from the quality of the instrument, the skill of the observer, the method used, and external or internal factors.
- Systematic errors act in one direction and arise from instrumental error (faulty zero/calibration), imperfection in experimental technique, and personal error. They are removed by better instruments and procedures, not by averaging.
- Random errors may be positive or negative and arise from varying conditions. They are minimized by taking repeated observations and their mean.
Estimation and Combination of Errors
- Most probable value: \[a_{mean} = \frac{1}{n}\sum_{i=1}^{n} a_{i} \qquad \text{--- (1.3)}\]
- Absolute error of a reading: \(\Delta a_{i} = |\ a_{mean} - a_{i}\ |\)
- Mean absolute error: \[\Delta a_{mean} = \frac{1}{n}\sum_{i=1}^{n}\Delta a_{i} \qquad \text{--- (1.4)}\]
- The result is reported as \(a = a_{mean} \pm \Delta a_{mean}\).
- Relative error and percentage error: \[\text{Relative error} = \frac{\Delta a_{mean}}{a_{mean}} \qquad \text{--- (1.5)}\] \[\text{Percentage error} = \frac{\Delta a_{mean}}{a_{mean}} \times 100 \qquad \text{--- (1.6)}\]
- Sum or difference: the maximum absolute error is the sum of the absolute errors, \(\Delta Z = \Delta A + \Delta B\) — errors add even for a difference.
- Product or division: the maximum relative error is the sum of the relative errors, \[\frac{\Delta Z}{Z} = \frac{\Delta A}{A} + \frac{\Delta B}{B} \qquad \text{--- (1.7)}\]
- Powers: if \(Z = A^{n}\), \[\frac{\Delta Z}{Z} = n\ \frac{\Delta A}{A} \qquad \text{--- (1.8)}\]
- In general, if \(Z = \dfrac{A^{p}B^{q}}{C^{r}}\), \[\frac{\Delta Z}{Z} = p\ \frac{\Delta A}{A} + q\ \frac{\Delta B}{B} + r\ \frac{\Delta C}{C} \qquad \text{--- (1.9)}\]
- The quantity carrying the largest power contributes the largest error.
Significant Figures and Order of Magnitude
- Least count is the smallest measurement possible with an instrument (0.1 cm for a metre scale).
- Significant figures are the digits about which we are certain, plus one additional doubtful digit.
- Rules: all nonzero digits are significant; zeros between nonzero digits are significant; for numbers less than 1, zeros to the right of the decimal point and to the left of the first nonzero digit are not significant (0.001405 has four); zeros to the right of the last nonzero digit are significant only if a decimal point is written (1.500 and 0.01500 have four each). \(L = 125\) m \(= 12500\) cm \(= 125000\) mm still has only three.
- Every measurement should be reported in scientific notation as \(A \times 10^{n}\), where \(0.5 \le A < 5\) and the integer \(n\) is the order of magnitude. For the radius of the Earth, \(6400\) km \(= 0.64 \times 10^{7}\) m — order of magnitude 7, two significant figures. For the electronic charge, \(e = 1.6 \times 10^{-19}\) C — order of magnitude \(-19\), two significant figures.
1.11 Check Your Concepts
Why is it said that a measurement is essentially a comparison? Explain with the example of finding the mass of a fruit on a beam balance.
A student reports the length of a rod simply as “12.5”. Explain why this statement carries no physical information at all.
Why did the international community finally adopt the SI in preference to the CGS, MKS and FPS systems? State the single feature of SI that makes conversion within it easy.
Temperature can be explained in terms of the motion of molecules. Why, then, is it still classified as a fundamental quantity and not as a derived one?
The newton is the SI unit of force, and force is not one of the seven fundamental quantities. Is the newton a fundamental unit or a derived unit? Justify your answer by writing the unit in terms of base units.
Why are the plane angle and the solid angle placed in a separate category called supplementary units rather than among the seven fundamental units?
A student writes the speed of sound as 340 ms\(^{-1}\). What has gone wrong with this notation, and how should it be corrected? Why does this convention matter?
Why can a Vernier calliper or a screw gauge not be used to measure the size of an atom? What property of the measuring “probe” decides the smallest size that can be resolved?
What happens to the parallax angle of a star as its distance from the Earth increases? Why is the Earth’s orbit, and not the Earth’s diameter, used as the basis while measuring stellar distances?
Both a coin held at arm’s length and the Moon subtend nearly the same angle at the eye. What does this tell you about the information contained in an angular diameter, and why must the distance be known independently before a size can be found?
The kilogram was defined for over a century by a platinum–iridium cylinder. Explain why the slow contamination of that cylinder was a far more serious problem than it might appear at first sight.
Why was the mean solar day abandoned as the standard for the second, and why is a caesium atom considered a better clock than the rotating Earth?
“A dimensionally correct equation need not be physically correct, but a dimensionally incorrect equation is certainly wrong.” Justify this statement with one example of each kind.
Dimensional analysis gives the period of a simple pendulum as \(T = k\sqrt{l/g}\), but cannot supply the value of \(k\). Why not? What does this failure tell you about the nature of dimensionless constants?
Explain why the relation \(F = G\,m_{1}m_{2}/r^{2}\) cannot be derived using dimensional analysis, even though it is a perfectly correct physical law.
State whether the following can be derived by dimensional analysis, giving a reason in each case: (a) the displacement of a particle in simple harmonic motion, \(y = a\sin(\omega t)\); (b) the relation between the velocity of a wave on a string, its tension and its mass per unit length.
An ammeter reads 0.5 A even when it is disconnected from the circuit. A student takes fifty readings with it and carefully averages them. Has the averaging improved the result? Explain, and classify the error involved.
Distinguish between accuracy and precision. Can a set of measurements be highly precise and yet highly inaccurate? Illustrate your answer with a laboratory example.
Why are random errors reduced by taking the mean of many readings, while systematic errors are not? What must be done instead to remove a systematic error?
Why are the absolute errors added — and never subtracted — when two measured quantities are subtracted? Explain why finding the difference of two nearly equal measured quantities is considered a poor experimental design.
In the determination of \(g\) using a simple pendulum, \(g = 4\pi^{2}l/T^{2}\). Which measurement, that of \(l\) or that of \(T\), must be made with greater care, and why?
Explain why the relative error, and not the absolute error, is used to compare the quality of two measurements of different quantities.
A calculator displays the mean of three readings taken with a metre scale as 15.43333 cm. How many of these digits may be reported, and why is writing all of them scientifically dishonest?
A length is recorded as \(L = 125\) m \(= 12500\) cm \(= 125000\) mm. Explain the ambiguity this creates in counting significant figures, and how scientific notation removes it.
Why is a trailing zero significant in 1.500 but not in 1500? What is the writer of “1.500” claiming about the instrument used?
What are the dimensions of the quantity \(l\sqrt{l/g}\), \(l\) being the length and \(g\) the acceleration due to gravity?
Define absolute error, mean absolute error, relative error and percentage error.
Describe what is meant by significant figures and order of magnitude.
If the measured values of two quantities are \(A \pm \Delta A\) and \(B \pm \Delta B\), \(\Delta A\) and \(\Delta B\) being the mean absolute errors, what is the maximum possible error in \(A \pm B\)? Show that if \(Z = \dfrac{A}{B}\), then \(\dfrac{\Delta Z}{Z} = \dfrac{\Delta A}{A} + \dfrac{\Delta B}{B}\).
Derive the formula for kinetic energy of a particle having mass \(m\) and velocity \(v\) using dimensional analysis.
1.12 Practice with Numericals
Express the following in SI base units: (a) 25 km h\(^{-1}\) (b) 5 g cm\(^{-3}\) (c) 72 µs.
A circular arc of length 0.44 m is drawn on a circle of radius 0.20 m. Calculate the plane angle subtended by the arc at the centre, in radian and in degrees.
A patch of area \(2.4 \times 10^{-2}\) m\(^{2}\) lies on the surface of a sphere of radius 0.40 m. Calculate the solid angle it subtends at the centre of the sphere, in steradian.
Two observers separated by a distance of \(8.0 \times 10^{6}\) m on the Earth’s surface observe a planet simultaneously and measure a parallax angle of 1 minute of arc. Calculate the distance of the planet from the Earth.
A star shows a parallax of 0.5 arcsec when observed from two opposite points of the Earth’s orbit, 2 AU apart. Calculate its distance from the Earth in metre and in parsec. (Take 1 AU = \(1.496 \times 10^{11}\) m.)
The angular diameter of the Moon as seen from the Earth is \(9.05 \times 10^{-3}\) rad. If the Moon is at a distance of \(3.84 \times 10^{8}\) m, calculate its diameter.
Convert 4.5 light years into (a) metre (b) astronomical units. (Take 1 light year = \(9.467 \times 10^{15}\) m and 1 AU = \(1.496 \times 10^{11}\) m.)
The mass of one molecule of a compound is 44 amu. Calculate its mass in kilogram, and the number of such molecules present in 1 kg of the compound. (Take 1 amu = \(1.6605402 \times 10^{-27}\) kg.)
Obtain the dimensional formulae of (a) frequency (b) angular velocity (c) power (d) surface tension (force per unit length) (e) coefficient of viscosity, given \(F = \eta A \dfrac{dv}{dx}\).
Check the dimensional correctness of the equation \(S = ut + \dfrac{1}{2}at^{2}\), where the symbols have their usual meanings.
The frequency \(n\) of a stretched wire is assumed to depend on its length \(l\), the tension \(F\) and the mass per unit length \(\mu\). Using dimensional analysis, obtain the relation between them up to a dimensionless constant.
Using dimensional analysis, find the conversion factor between the SI unit of force (newton) and the CGS unit of force (dyne), and hence show how many dyne make one newton.
A distant civilisation uses units in which the unit of mass is 5 kg, the unit of length is 2 m and the unit of time is 4 s. Express 1 joule of energy in their units.
The period of oscillation of a liquid drop is assumed to depend on its density \(\rho\), its radius \(r\) and the surface tension \(S\) (dimensions \([L^{0}M^{1}T^{-2}]\)). Obtain, by dimensional analysis, the form of the relation for the period, up to a dimensionless constant.
Five measurements of the diameter of a wire give 0.51 mm, 0.53 mm, 0.52 mm, 0.55 mm and 0.54 mm. Calculate the most probable value, the mean absolute error, the relative error and the percentage error.
Two lengths are measured as \((4.50 \pm 0.02)\) cm and \((3.20 \pm 0.03)\) cm. Calculate (a) their sum and (b) their difference, each with the maximum absolute error, and find the percentage error in the difference.
The current through a resistor is measured as \((2.5 \pm 0.1)\) A and the potential difference across it as \((10.0 \pm 0.2)\) V. Calculate the resistance and the maximum percentage error in it.
The side of a cube is measured as \((2.00 \pm 0.02)\) cm. Calculate the percentage error in its volume and in its total surface area.
In an experiment, a physical quantity is given by \(Y = \dfrac{a^{2}b^{3}}{c\sqrt{d}}\). If the percentage errors in \(a\), \(b\), \(c\) and \(d\) are 1%, 2%, 3% and 4% respectively, calculate the maximum percentage error in Y.
The mass of a body is \((20.0 \pm 0.2)\) g and its velocity is \((5.0 \pm 0.1)\) m s\(^{-1}\). Calculate the percentage error in the kinetic energy of the body.
State the number of significant figures in each of the following: 2.008 kg, 0.00450 m, 6.400 × 10\(^{3}\) s, 0.0250 A, 1200 m.
The length, breadth and height of a rectangular block are 5.24 cm, 3.1 cm and 2.06 cm. Calculate its volume to the correct number of significant figures.
Express the following in scientific notation and state the order of magnitude of each: (a) the mass of the Earth, \(5.97 \times 10^{24}\) kg (b) the radius of a hydrogen atom, \(5 \times 10^{-11}\) m (c) the speed of light, 299792458 m s\(^{-1}\).
A thin sheet of area 1.0 m\(^{2}\) is made by rolling out a wire of circular cross section of diameter 0.20 mm. Neglecting gaps, estimate the total length of wire used, to the nearest order of magnitude.
The acceleration due to gravity is determined with a simple pendulum of length \(l = (50.0 \pm 0.1)\) cm and time period \(T = (1.42 \pm 0.01)\) s. Calculate the maximum percentage error in the value of \(g\), and state which measurement limits the accuracy of the experiment.
The masses of two bodies are measured to be \(15.7 \pm 0.2\) kg and \(27.3 \pm 0.3\) kg. What is the total mass of the two and the error in it?
The distance travelled by an object in time \((100 \pm 1)\) s is \((5.2 \pm 0.1)\) m. What is the speed and its relative error?
An electron with charge \(e\) enters a uniform magnetic field \(\vec{B}\) with a velocity \(\vec{v}\). The velocity is perpendicular to the magnetic field. The force on the charge \(e\) is given by \(|\vec{F}| = Bev\). Obtain the dimensions of \(\vec{B}\).
A large ball 2 m in radius is made up of a rope of square cross section with edge length 4 mm. Neglecting the air gaps in the ball, what is the total length of the rope to the nearest order of magnitude?
Nuclear radius \(R\) has a dependence on the mass number \((A)\) as \(R = 1.3 \times 10^{-16} A^{1/3}\) m. For a nucleus of mass number \(A = 125\), obtain the order of magnitude of \(R\) expressed in metre.
In a workshop a worker measures the length of a steel plate with a Vernier callipers having a least count 0.01 cm. Four such measurements of the length yielded the following values: 3.11 cm, 3.13 cm, 3.14 cm, 3.14 cm. Find the mean length, the mean absolute error and the percentage error in the measured value of the length.
Find the percentage error in the kinetic energy of a body having mass \(60.0 \pm 0.3\) g moving with a velocity \(25.0 \pm 0.1\) cm/s.
In Ohm’s experiments, the values of the unknown resistances were found to be 6.12 Ω, 6.09 Ω, 6.22 Ω, 6.15 Ω. Calculate the mean absolute error, relative error and percentage error in these measurements.
An object is falling freely under the gravitational force. Its velocity after travelling a distance \(h\) is \(v\). If \(v\) depends upon gravitational acceleration \(g\) and distance, prove with dimensional analysis that \(v = k\sqrt{gh}\), where \(k\) is a constant.
\(v = at + \dfrac{b}{t + c} + v_{0}\) is a dimensionally valid equation. Obtain the dimensional formula for \(a\), \(b\) and \(c\), where \(v\) is velocity, \(t\) is time and \(v_{0}\) is initial velocity.
The length, breadth and thickness of a rectangular sheet of metal are 4.234 m, 1.005 m, and 2.01 cm respectively. Give the area and volume of the sheet to correct significant figures.
If the length of a cylinder is \(l = (4.00 \pm 0.001)\) cm, radius \(r = (0.0250 \pm 0.001)\) cm and mass \(m = (6.25 \pm 0.01)\) gm, calculate the percentage error in the determination of density.
When the planet Jupiter is at a distance of 824.7 million kilometer from the Earth, its angular diameter is measured to be 35.72″ of arc. Calculate the diameter of Jupiter.
If the formula for a physical quantity is \(X = \dfrac{a^{4}b^{3}}{c^{1/3}d^{1/2}}\), and if the percentage errors in the measurements of \(a\), \(b\), \(c\) and \(d\) are 2%, 3%, 3% and 4% respectively, calculate the percentage error in X.
Write down the number of significant figures in the following: 0.003 m\(^{2}\), 0.1250 gm cm\(^{-2}\), 6.4 × 10\(^{6}\) m, 1.6 × 10\(^{-19}\) C, 9.1 × 10\(^{-31}\) kg.
The diameter of a sphere is 2.14 cm. Calculate the volume of the sphere to the correct number of significant figures.
1.13 Objective Questions
- \([L^{1}M^{1}T^{-2}]\) is the dimensional formula for
- Velocity
- Acceleration
- Force
- Work
- The error in the measurement of the sides of a rectangle is 1%. The error in the measurement of its area is
- 1%
- (½)%
- 2%
- None of the above.
- Light year is a unit of
- Time
- Mass
- Distance
- Luminosity
- Dimensions of kinetic energy are the same as that of
- Force
- Acceleration
- Work
- Pressure
- Which of the following is not a fundamental unit?
- cm
- kg
- centigrade
- volt