Units and Measurements

Table of contents

• Introduction
• International System of units
• Dimensions of physical quantities
• Dimensional Formulae and Dimensional Equations
• Dimensional Analysis and Its Applications
• Accuracy and precision of instruments
• Errors in Measurement
• Significant Figures

Introduction

Science is all about curious observations and the related careful experiments. But we can appreciate a scientific observation more if it can provide us precise and quantitative details.

For example, Newton observed an apple falling from the tree and this led him to the discovery of gravitational force. But besides that he also summarised his discovery in the form of a mathematical formula which is known as Newton's law of gravitation. It made his observation more relevant because it not only explained why an apple falls to the ground but also tells about the force and the acceleration with which it falls. Not only this, we can also find the force with which moon is attracted towards earth, or earth is attracted towards sun using this formula.

This illustration tells us the importance of quantitative description and hence the need for measurement. This chapter deals with the measurements of various physical quantities, their units and errors in their measurements. We will also discuss significant figures in measured values, and also how dimensional analysis help us to understand the physical behaviour of a quantity.

Measurement is a process of determining how large or small a physical quantity is as compared to a basic reference standard. This reference standard is called the unit of the particular physical quantity. A unit can be chosen arbitrarily but then it should be accepted internationally and should not vary from place to place.

To express the measurement of a physical quantity, we need to know two things:

(i) The unit in which the quantity is measured.

(ii) The magnitude of the quantity i.e. the number of times that unit is contained in the given physical quantity.

We deal with a number of physical quantities in physics, but the units of all these quantities can be expressed in the units of few basic quantities. Thus physical quantities are of two types:

Fundamental or Base Quantities

These are treated as independent of other physical quantities and are not usually defined in terms of other physical quantities. The units for these quantities are called fundamental or base units. There are seven base quantities. These are length, mass, time, electric current, temperature, luminous intensity and amount of substance.

Derived Quantities

All physical quantities whose units can be expressed as combination of base units are called derived physical quantities. Thus all quantities other than seven base quantities are derived quantities e.g., velocity, acceleration, momentum, etc. Units of these quantities are called derived units.

For example, speed of an object is given by the relation

Speed = Distance / Time

∴ Unit of speed = Unit of distance / Unit of time = m/s or m s^–1

Other units can also be derived in a similar manner.

International System of Units

A complete set of both the base units and derived units, is called the system of units. Historically many systems of units were in use in different parts of the world. A few of them which were quite popular till recently are given below.

1. CGS System: In this system centimetre, gram and second are used as the base units for length, mass and time respectively.

2. FPS System: It uses foot, pound and second as base units for length, mass and time respectively.

3. MKS System: It uses metre, kilogram and second as the respective units for length, mass and time.

General conference on Weights and Measures developed and put forward a new system of units in the year 1971. This system, with units, their proper symbols and abbreviations was proposed to be used globally in the field of science, trade and commerce. This system of units is now accepted internationally and is called SI system. SI is an abbreviation for Systeme Internationale d′ unites (French name for International System of Units).

The SI system is a decimal system, also known as metric system, a modernised and extended form of metric systems like CGS and MKS. There are seven base units and two supplementary units in SI. These units with their names and symbols are given below.

Base Units

Supplementary Units

Definitions of Base Units

The base units used in SI system are defined as follows:

Metre (m)

It is defined by taking the fixed numerical value of the speed of light in vacuum c to be 299792458 when expressed in the unit m s^–1, where the second is defined in terms of the caesium frequency ∆ν_Cs.

Kilogram (kg)

It is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015 × 10^–34 when expressed in the unit J s, which is equal to kg m² s^–1, where the metre and the second are defined in terms of c and ∆ν_Cs.

Second (s)

It is defined by taking the fixed numerical value of the caesium frequency ∆ν_Cs, the unperturbed ground-state hyperfine transition frequency of the caesium-133 atom, to be 9192631770 when expressed in the unit Hz, which is equal to s^–1.

Ampere (A)

It is defined by taking the fixed numerical value of the elementary charge e to be 1.602176634 × 10^–19 when expressed in the unit C, which is equal to A s, where the second is defined in terms of ∆ν_Cs.

Kelvin (K)

It is defined by taking the fixed numerical value of the Boltzmann constant k to be 1.380649 × 10^–23 when expressed in the unit J K^–1, which is equal to kg m² s^–2 K^–1, where the kilogram, metre and second are defined in terms of h, c and ∆ν_Cs.

Mole (mol)

One mole contains exactly 6.02214076 × 10²³ elementary entities. This number is the fixed numerical value of the Avogadro constant, N_A, when expressed in the unit mol^–1 and is called the Avogadro number. The amount of substance, symbol n, of a system is a measure of the number of specified elementary entities. An elementary entity may be an atom, a molecule, an ion, an electron, any other particle or specified group of particles.

Candela (cd)

It is defined by taking the fixed numerical value of the luminous efficacy of monochromatic radiation of frequency 540 × 10¹² Hz, K_cd, to be 683 when expressed in the unit lm W^–1, which is equal to cd sr W^–1, or cd sr kg^–1 m^–2 s³, where the kilogram, metre and second are defined in terms of h, c and ∆ν_Cs.

Supplementary Unit Definitions

Plane Angle

The angle subtended by an arc of a circle at its centre is called a plane angle. Mathematically it is the ratio of the arc length ds of the circle to its radius r.

Thus plane angle dθ = ds / r

SI unit of plane angle is radian which is represented as rad.

One radian is defined as the plane angle subtended at the centre of a circle by an arc equal in length to the radius of the circle.

The total plane angle subtended by a circle at its centre is 2π radian.

Another unit in which angle is measured, is degree.

1° = π/180 rad

One degree is also equal to 60 minutes and 1 minute is equal to 60 seconds.

1° = 60′ and 1′ = 60″

Solid Angle

The angle subtended by a given surface area of a spherical surface at its centre is called a solid angle. Mathematically it is the ratio of the intercepted area dA of the spherical surface to the square of its radius r.

Thus solid angle dΩ = dA / r²

SI unit of a solid angle is steradian and is represented as 'sr.'

One steradian is defined as the solid angle subtended at the centre of a sphere by a surface of the sphere equal in area to that of a square, having each side equal to the radius of the sphere.

The SI units for all other physical quantities can be derived from the SI base units mentioned above.

Some derived SI units are given special names. For example SI unit of force is kg m s^–2. It is also called "newton" in honour of the scientist Isaac Newton. "newton" is denoted by the symbol 'N'. Some other such units having special names are joule (J) for energy, watt (W) for power, coulomb (C) for charge, volt (V) for potential difference, tesla (T) for the magnetic field strength etc. Some derived SI units use these units with special names and the seven base units. For example SI unit of work can also be written as "newton metre" or N m. Similarly SI unit of power can be written as "joule per second" or J s^–1. A few more such units are: V m^–1 for electric field strength, C m for dipole moment, N m^–1 for surface tension etc.

Guidelines for Writing SI Units in Symbols

One should keep following points in mind while using symbols for SI units.

1. Standard Unit Symbols are written in lower case roman type. e.g., 'metre' has the symbol 'm'; 'kilogram' has the symbol 'kg'.

2. Unit names are never capitalised. However a unit symbol is capitalised only if the unit is named after a scientist. e.g., units like 'newton', 'joule', 'volt', 'ampere' respectively have symbols as N, J, V, A.

3. If a unit, named after a scientist, contains two letters in its symbol, then the initial letter of the symbol is capital. For example, Hz for the unit 'hertz', Wb for the unit 'weber' and Pa for the unit 'pascal'.

4. Symbols are not followed by a fullstop.

5. Unit symbols are never used in plural form. Thus a length of 100 metres is expressed as 100 m and not as 100 ms.

6. Not more than one solidus (/) should be used in a unit symbol. For example, the SI unit of acceleration is written as m/s² and not as m/s/s. Pressure is expressed in the units of N/m² or N m^–2 but not as N/m/m. Thus writing 10 N/m² is correct, but 10 N/m/m is wrong.

7. For inconveniently small or large values of quantities prefixes are used with their units to indicate suitable multiples or submultiples, in powers of 10. For example, 10³ m can be written as kilometre or km, 10^–9 m can be written as nanometre or nm. Here 'kilo' and 'nano' are prefixes to respectively indicate the amounts 1000 and 10^–9.

8. Prefix symbol is written very close to the unit symbol without spacing between them. But if the unit of a physical quantity is obtained by multiplying the units of two or more quantities, then these unit symbols are written with spacing between them. Thus the symbol ms^–1 means per millisecond. But the symbol m s^–1 means metre per second. Thus ms^–1 and m s^–1 are two different physical quantities.

9. When a prefix is placed before a unit symbol, the combined prefix and symbol should be considered as one new symbol which can be raised to a positive or negative power without any bracket. For example, km³ means (10³ m)³ and not 10³ m³. Similarly, µs^–1 means (10^–6 s)^–1 but not 10^–6 s^–1.

10. The use of double prefixes should be avoided as far as practicable. Thus 10^–12 m should be written as picometre or pm instead of writing as µµm.

11. Unit names and unit symbols should not be used together while expressing a physical quantity. For example, unit of linear momentum should either be written as 'kg m s^–1' or 'kilogram metre per second' but not as 'kg metre s^–1'.

Common SI Prefixes and Symbols for Multiples and Sub-Multiples

Order of Magnitude

If the magnitude of a physical quantity is expressed as a × 10^b, where (a ≤ 5), then the exponent b is called the order of magnitude of the physical quantity. If 5 < a ≤ 10, then the order of magnitude of the physical quantity becomes b + 1, where b is any positive or negative exponent (or power) of 10.

For example, the speed of light is given as 3.00 × 10⁸ m s^–1. So the order of magnitude of the speed of light is 8. The order of magnitude gives an estimate of the magnitude of the quantity. The charge on an electron is 1.6 × 10^–19 C. Therefore, we can say that the charge possessed by an electron is of the order 10^–19 or its order of magnitude is –19. The expression of a quantity as a × 10^b is called scientific notation.