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This is an introductory chapter to physics which capsules the fundamental laws that we shall discuss in the rest of the chapters. We shall learn, in the form of basic laws, the factors that govern the calculations in physics. We shall begin by describing Reductionism as the process of deriving the properties of a bigger and a more complex system from the properties and interactions of its constituent simpler parts.

In the first section, we will glance through the various fields in physics, such as: The classical physics, Mechanics, Electrodynamics, Optics, Thermodynamics, Quantum mechanics and Applied physics.

Then, we shall learn about some of the fundamental forces in nature: The gravitational force (which is the force of mutual attraction between any two objects by virtue of their masses), the electromagnetic force (which is the force of attraction or repulsion between charges), the strong nuclear force (which was believed to bind protons and neutrons in a nucleus), the weak nuclear force (which appears only in certain nuclear processes such as the β-decay of a nucleus).

Under the head, “towards unification of forces”, we will learn how scientists unified their theories to explain the then new phenomena.

Some of the noted theories and discoveries shall then be studied in brief; such as, Einstein’s mass energy equation ($E = mc^2$) and Pauli’s prediction of new particles (now called, neutrino). In the subsequent section, we will define the basic terms used in measurements throughout the study of physics like, unit which is measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard; fundamental Units which are the units for the fundamental or base quantities; the units of all other physical quantities which can be expressed as combinations of the base units are called derived units and System of units which is a complete set of these units, both the base units and derived units.

By parallax method, we will learn to find large distances like the diameter of a planet. Successively, you will also see how short distances of the order $10^{-8}$ m to $10^{-10}$ m can be measured. We will then learn the standard units of measuring mass, which is Kilogram; length—meter; and the standard way of measuring time (then) which was using caesium atomic clocks.

Next, we shall look at some important terms associated with instruments— error, which is the result of every measurement by any measuring instrument, contains some uncertainty; accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity and Precision gives the value of the resolution or limit to which the quantity is measured.

Further, we will learn about the different kinds of errors like, Systematic errors, Instrumental errors, Personal errors, Random errors, Least- count error and we will learn to find Absolute error, Relative error and Percentage error. The concept of Significant figures shall then be studied, for, it is useful in reporting measurements with precisely following which, we will study some of the rules in writing significant figures.

Under the head, Dimensions of Physical Quantities, we will learn how any quantity can be expressed as a multiple of the basic seven dimensions, Length [L], Mass [M}]. Time [T], Electric current [A], Thermodynamic temperature [K], Lumnious intesnsity [cd] and amount of a substance [mol]. Finally, we will learn about homogeneity of quantities; that we can perform operations only on quantities with same dimensional formula.

1.1 Introduction:

What is Physics?

Physics, derived from the Greek word phusike means “knowledge of nature”. The Sanskrit word Vijnan and the Arabic word Ilm convey similar meaning, namely, knowledge.

Physics deals with the study of matter, motion of matter through space and time along with related concepts such as energy and force.

What is Science and what is the so-called Scientific Method?

Science is a systematic attempt to understand natural phenomena in as much detail and depth as possible, and use the knowledge so gained to predict, modify and control phenomena.

Science is exploring, experimenting and predicting from what we see around us. The curiosity to learn about the world, unravelling the secrets of nature is the first step towards the discovery of science.

Physics is a basic discipline in the category of Natural Sciences, which also includes other disciplines like Chemistry and Biology. We attempt to explain diverse physical phenomena in terms of a few concepts and laws. The effort is to see the physical world as manifestation of some universal laws in different domains and conditions. For example, the same law of gravitation (given by Newton) describes the fall of an apple to the ground, the motion of the moon around the earth and the motion of planets around the sun. Similarly, the basic laws of electromagnetism (Maxwell’s equations) govern all electric and magnetic phenomena. The attempts to unify fundamental forces of nature reflect this same quest for unification.

The scientific method involves several interconnected steps

• Systematic observations, controlled experiments, qualitative and experiment is basic to the progress of science. Science is ever dynamic. There is no final theory in science and no unquestioned authority among scientists.
• As observations improve in detail and precision or experiments yield new results, theories must account for them, if necessary, by introducing modifications. Sometimes the modifications may not be drastic and may lie within the framework of existing theory. For example, when Johannes Kepler (1571-1630) examined the extensive data on planetary motion collected by Tycho Brahe (1546-1601), the planetary circular orbits in heliocentric theory (sun at the centre of the solar system) imagined by Nicolas Copernicus (1473.1543) had to be replaced by elliptical orbits to fit the data better.
• Occasionally, however, the existing theory is simply unable to explain new observations. This causes a major upheaval in science. In the beginning of the twentieth century, it was realised that Newtonian mechanics, till then a very successful theory, could not explain some of the most basic features of atomic phenomena. Similarly, the then accepted wave picture of light failed to explain the photoelectric effect properly. This led to the development of a radically new theory (Quantum Mechanics) to deal with atomic and molecular phenomena.
• Just as a new experiment may suggest an alternative theoretical model, a theoretical advance may suggest what to look for in some experiments. The result of experiment of scattering of alpha particles by gold foil, in 1911 by Ernest Rutherford (1871.1937) established the nuclear model of the atom, which then became the basis of the quantum theory of hydrogen atom given in 1913 by Niels Bohr (1885.1962). On the other hand, the concept of antiparticle was first introduced theoretically by Paul Dirac (1902.1984) in 1930 and confirmed two years later by the experimental discovery of positron (antielectron) by Carl Anderson.
• The next most important insight was that the basic laws of physics are universal – the same laws apply in widely different contexts.
• Lastly, the strategy of approximation turned out to be very successful.
• Most observed phenomena in daily life are rather complicated manifestations of the basic laws. Scientists recognised the importance of extracting the essential features of a phenomenon from its less significant aspects. It is not practical to take into account all the complexities of a phenomenon in one go. A good strategy is to focus first on the essential features, discover the basic principles and then introduce corrections to build a more refined theory of the phenomenon.
• For example, a stone and a feather dropped from the same height do not reach the ground at the same time. The reason is that the essential aspect of the phenomenon, namely free fall under gravity, is complicated by the presence of air resistance. To get the law of free fall under gravity, it is better to create a situation wherein the air resistance is negligible. We can, for example, let the stone and the feather fall through a long evacuated tube. In that case, the two objects will fall almost at the same rate, giving the basic law that acceleration due to gravity is independent of the mass of the object. With the basic law thus found, we can go back to the feather, introduce corrections due to air resistance, modify the existing theory and try to build a more realistic theory of objects falling to the earth under gravity.

Reductionism:

The process of deriving the properties of a bigger and a more complex system from the properties and interactions of its constituent simpler parts is called reductionism. Reductionism is the heart of physics. For example, the subject of thermodynamics, developed in the nineteenth century, deals with bulk systems in terms of macroscopic quantities such as temperature, internal energy and entropy. Subsequently, the subjects of kinetic theory and statistical mechanics interpreted these quantities in terms of the properties of the molecular constituents of the bulk system. In particular, the temperature was seen to be related to the average kinetic energy of molecules of the system.

Questions for the section:

1.What is Reductionism?

1.2 Scope and Excitement of Physics

Range and Order of measurements:

Physics covers a tremendous range of magnitude of physical quantities like length, mass, time, energy, etc. At one end, it studies phenomena at the very small scale of length (10-14 m or even less) involving electrons, protons, etc.; at the other end, it deals with astronomical phenomena at the scale of galaxies or even the entire universe whose extent is of the order of 1026 m. The two length scales differ by a factor of 1040 or even more. The range of time scales can be obtained by dividing the length scales by the speed of light: 10-22 s to 1018 s. The range of masses goes from, say, 10-30 kg (mass of an electron) to 1055 kg (mass of known observable universe).

Classical Physics deals mainly with macroscopic phenomena and includes subjects like Mechanics, Electrodynamics, Optics and Thermodynamics.

Mechanics founded on Newton’s laws of motion and the law of gravitation is concerned with the motion (or equilibrium) of particles, rigid and deformable bodies, and general systems of particles. The propulsion of a rocket by a jet of ejecting gases, propagation of water waves or sound waves in air, the equilibrium of a bent rod under a load, etc., are problems of mechanics.

Electrodynamics deals with electric and magnetic phenomena associated with charged and magnetic bodies. Its basic laws were given by Coulomb, Oersted, Ampere and Faraday, and encapsulated by Maxwell in his famous set of equations. The motion of a current-carrying conductor in a magnetic field, the response of a circuit to an ac voltage (signal), the working of an antenna, the propagation of radio waves in the ionosphere, etc., are problems of electrodynamics.

Optics deals with the phenomena involving light. The working of telescopes and microscopes, colours exhibited by thin films, etc., are topics in optics.

Thermodynamics, in contrast to mechanics, does not deal with the motion of bodies as a whole. Rather, it deals with systems in macroscopic equilibrium and is concerned with changes in internal energy, temperature, entropy, etc., of the system through external work and transfer of heat. The efficiency of heat engines and refrigerators, the direction of a physical or chemical process, etc., are problems of interest in thermodynamics.

The microscopic domain of physics deals with the constitution and structure of matter at the minute scales of atoms and nuclei (and even lower scales of length) and their interaction with different probes such as electrons, photons and other elementary particles. This is explained under the quantum mechanics.

Applied physics deals with the application and exploitation of physical laws to make useful devices. It is considered to be one of the most interesting and exciting verticals of physics.

Note box 1: Quantitative measurement is central to the growth of science, especially physics, because the laws of nature happen to be expressible in precise mathematical equations.

Questions from section 1.2:

1. Describe the following:

a) Classical physics

b) Mechanics

c) Electrodynamics

d) Optics

e) Thermodynamics

f) Quantum mechanics

g) Applied physics

1.3 Fundamental forces in nature

1.3.1 Gravitational Force

Definition box 1: The gravitational force is the force of mutual attraction between any two objects by virtue of their masses.

The gravitational force is a universal force. Every object experiences this force due to every other object in the universe. In fact, even you exert a gravitational force upon all bodies around you. If the gravitational force you exerted would be great enough, bodies will zoom towards you instead of being stand-still on the ground. This is according to the Newton’s Universal law of gravitation.

It is gravity that governs the motion of the moon and artificial satellites around the earth, motion of the earth and planets around the sun, and, of course, the motion of bodies falling to the earth. It plays a key role in the large-scale phenomena of the universe, such as formation and evolution of stars, galaxies and galactic clusters.

Universal Gravitation Equation

$$ F= \frac {GMm}{r^2} $$
F is the force of gravity (measured in Newtons, N)
G is the gravitational constant of the universe and is always the same number
M is the mass of one object (measured in kilograms, kg)
m is the mass of the other object (measured in kilograms, kg)
r is the distance those objects are apart (measured in meters, m)

Note box 2: This is an inverse square law as halving the distance increases the factor by 4 and doubling the distance results in a slashing of the force to a quarter of the original force.

1.3.2 Electromagnetic Force

Electromagnetic force is the force between charged particles. In the simpler case when charges are at rest, the force is governed by Coulomb’s law: attractive for unlike charges and repulsive for like charges. Charges in motion produce magnetic effects and a magnetic field gives rise to a force on a moving charge.

The science of electromagnetic phenomena is defined in terms of the electromagnetic force, sometimes called the Lorentz force, which includes both electricity and magnetism as elements of one phenomenon.

Electric and magnetic effects are, in general, inseparable hence the name electromagnetic force. Like the gravitational force, electromagnetic force acts over large distances and does not need any intervening medium. It is enormously strong compared to gravity. For example, the electric force between two protons is $10^{36}$ times the gravitational force between them, for any fixed distance.

Matter, as we know, consists of elementary charged constituents like electrons and protons.

It is mainly the electromagnetic force that governs the structure of atoms and molecules, the dynamics of chemical reactions and the mechanical, thermal and other properties of materials. It underlies the macroscopic forces like tension, friction, normal force, spring force, etc.

Gravity is always attractive, while electromagnetic force can be attractive or repulsive. Another way of putting it is that mass comes only in one variety (there is no negative mass), but charge comes in two varieties: positive and negative charge. This is what makes all the difference. Matter is mostly electrically neutral (net charge is zero). Thus, electric force is largely zero and gravitational force dominates terrestrial phenomena.
Electric force manifests itself in atmosphere where the atoms are ionised and that leads to lightning.

The enormous strength of the electromagnetic force compared to gravity is evident in our daily life. When we hold a book in our hand, we are balancing the gravitational force on the book due to the huge mass of the earth by the normal (perpendicular) force provided by our hand. The latter is nothing but the net electromagnetic force between the charged constituents of our hand and the book, at the surface in contact. If electromagnetic force were not intrinsically so much stronger than gravity, the hand of the strongest man would crumble under the weight of a feather! Indeed, in that circumstance, we ourselves would crumble under our own weight!

1.3.3 Strong Nuclear Force

Characteristics:

• The strong nuclear force binds protons and neutrons in a nucleus. A nucleus will be unstable due to the electric repulsion between its protons is there are no other attractive forces present.
• The strong nuclear force is the strongest of all fundamental forces about 100 times the electromagnetic force in strength.
• It is charge-independent and acts equally between a proton and a proton, a neutron and a neutron, and a proton and a neutron.
• Its range is, however, extremely small, of about nuclear dimensions ($10^{-15}m). $
• The strong nuclear force is responsible for the stability of the nuclei.
• The electrons in an atom do not experience this force.

1.3.4 Weak Nuclear force

Characteristics:

• The weak nuclear force appears only in certain nuclear processes such as the β-decay of a nucleus. In β-decay, the nucleus emits an electron and an uncharged particle called neutrino.
• The weak nuclear force is not as weak as the gravitational force, but much weaker than the strong nuclear and electromagnetic forces.
• The range of weak nuclear force is exceedingly small, of the order of $10^{-16} m.$

1.3.5 Towards unification of forces

• Newton unified terrestrial and celestial domains under a common law of gravitation.
• The experimental discoveries of Oersted and Faraday showed that electric and magnetic phenomena are in general inseparable.
• Maxwell unified electromagnetism and optics with the discovery that light is an electromagnetic wave.
• Einstein attempted to unify gravity and electromagnetism but could not succeed in this venture.
• The electromagnetic and the weak nuclear force have now been unified and are seen as aspects of a single ‘electro-weak’ force.

Questions from section 1.3:

1. Define Gravitational force. Give the equation for universal law of gravitation.
2. Describe an electromagnetic force.
3. State the characteristics of:
   a) Strong Nuclear Force
   b) Weak nuclear force
4. Write a note on unification of forces.

1.4 Nature of Physical laws

In any physical phenomenon governed by different forces, several quantities may change with time. A remarkable fact is that some special physical quantities, however, remain constant in time. They are the conserved quantities of nature. Understanding these conservation principles is very important to describe the observed phenomena quantitatively.

For motion under an external conservative force, the total mechanical energy that is, the sum of kinetic and potential energy of a body is a constant. Consider the example of the free fall of an object under gravity. Both the kinetic energy of the object and its potential energy change continuously with time, but the sum remains fixed.

If the object is released from rest, the initial potential energy is completely converted into the kinetic energy of the object just before it hits the ground. This law restricted for a conservative force should not be confused with the general law of conservation of energy of an isolated system.

According to Einstein’s theory, mass m is equivalent to energy E given by the relation E = mc2 (popularly known as Einstein’s mass energy equation), where c is speed of light in vacuum. In a nuclear process mass gets converted to energy (or vice-versa). This is the energy which is released in a nuclear power generation and nuclear explosions.
Using the conservation laws of energy and momentum for β-decay, Wolfgang Pauli (1900 -1958) correctly predicted in 1931 the existence of a new particle (now called neutrino) emitted in β-decay along with the electron.

If you perform an experiment in your laboratory today and repeat the same experiment (on the same objects under identical conditions) after a year, the results are bound to be the same. It turns out that this symmetry of nature with respect to translation (i.e. displacement) in time is equivalent to the law of conservation of energy.

The phenomena of law of conservation of energy may differ from place to place because of differing conditions at different locations. For example, the acceleration due to gravity at the moon is one-sixth that at the earth, but the law of gravitation is the same both on the moon and the earth.

Questions from section 1.4:

1. Explain Einstein’s mass energy equation.

1.5 Units and Measurements

Unit: Measurement of any physical quantity involves comparison with a certain basic, arbitrarily chosen, internationally accepted reference standard called unit.

Fundamental Units: The units for the fundamental or base quantities are called fundamental or base units.

Derived units: The units of all other physical quantities which can be expressed as combinations of the base units. Such units obtained for the derived quantities are called derived units.

System of units: A complete set of these units, both the base units and derived units, is known as the system of units.
Three systems, the CGS, the FPS (or British) system and the MKS system were in use extensively. The base units for length, mass and time in these systems were as follows:

• In CGS system they were centimetre, gram and second respectively.
• In FPS system they were foot, pound and second respectively.
• In MKS system they were metre, kilogram and second respectively.

The system of units which is at present internationally accepted for measurement is the Système Internationale d. Unites (French for International System of Units), abbreviated as SI. The SI, with standard scheme of symbols, units and abbreviations, was developed and recommended by General Conference on Weights and Measures in 1971 for international usage in scientific, technical, industrial and commercial work. Because SI units used decimal system, conversions within the system are quite simple and convenient.

In SI, there are seven base units as given in the Table below:

Besides the seven base units, there are two more units that are defined for:

(a) plane angle dθ as the ratio of length of arc ds to the radius r

Description of (a) plane angle dθ and (b) solid angle dΩ

(b) solid angle dΩ as the ratio of the intercepted area dA of the spherical surface, described about the apex O as the centre, to the square of its radius r.

The unit for plane angle is radian with the symbol rad and the unit for the solid angle is steradian with the symbol sr. Both these are dimensionless quantities.

Some units retained for general use (Though outside SI):

When mole is used, the elementary entities must be specified. These entities may be atoms, molecules, ions, electrons, other particles or specified groups of such particles.

Questions from section 1.5:

1. Describe the terms:

a) Unit

b) Fundamental units

c) Derived units

d) System of units

e) Plane angle

f) Solid angle

1.6 Measurement and Length

1.6.1 Measurement of large distances

Hold a pencil in front of you against some specific point on the background (a wall) and look at the pencil first through your left eye ‘A’ (closing the right eye) and then look at the pencil through your right eye ‘B’ (closing the left eye), you would notice that the position of the pencil seems to change with respect to the point on the wall. This is called parallax.

The distance between the two points of observation is called the basis.

The Parallax Method:

To measure the distance D of a far away planet S by the parallax method, we observe it from two different positions (observatories) A and B on the Earth, separated by distance AB = b.

The ∠ASB (θ) is called the parallax angle or parallactic angle.

As the planet is far far away, b < < D;

$$ \frac {b}{D} < <1$$

This results in θ being rendered a very small value. Take AB as an arc of length b of a circle with centre at S and the distance D as the radius AS = BS so that AB = b = D θ where θ is in radians.
$$ D= \frac {b}{θ} $$

We can also determine the size or angular diameter of the planet with the data available and the results calculated. If d is the diameter of the planet and α the angular size of the planet (the angle subtended by d at the earth), we have

α = d/D
α is the angle between the two directions when two diametrically opposite points of the planet are viewed through the telescope. Since D is known, the diameter d of the planet can be determined using the above equation.

Example 1: Calculate the angle of(a) $1^0$ (degree) (b) 1′ (minute of arc or arcmin) and (c) 1″(second of arc or arc second) in radians. Use $360^0$=2π rad, $1^0$=60′ and 1′ = 60″.

Solution: (a) We have $360^0$ = 2π rad
$1^0 = (π /180) rad = 1.745×10^{–2} rad $
(b) $1^0 = 60′ = 1.745×10^{–2} rad $
$ 1′ = 2.908×10^{–4} rad ; 2.91×10^{–4} rad $
(c) $1′ = 60″ = 2.908×10^{–4} rad $
$ 1″ = 4.847×10^{–4} rad ; 4.85×10^{–6} rad $

Example 2: A man wishes to estimate the distance of a nearby tower from him. He stands at a point A in front of the tower C and spots a very distant object O in line with AC. He then walks perpendicular to AC up to B, a distance of 100 m, and looks at O and C again. Since O is very distant, the direction BO is practically the same as AO; but he finds the line of sight of C shifted from the original line of sight by an angle θ = $40^0$ (θ is known as ‘parallax’) estimate the distance of the tower C from his original position A.

Solution: We have, parallax angle θ = $40^0$
From Fig. 2.3, AB = AC tan θ
AC = AB/tanθ = 100 m/tan$ 40^0$
= 100 m/0.8391 = 119 m.

Example 3: The moon is observed from diametrically opposite points A and B on Earth. The angle θ subtended at the moon by the two directions of observation is $1^o$ 54′ . Given the diameter of the Earth to be about $1.276 × 10^7$ m, compute the distance of the moon from the Earth.

Solution: We have θ = 1° 54′ = 114′
$$= (114×60)′′×(4.85×10^{-6}) \space rad $$
$$= 3.32 × 10^{- 2} \space rad $$,
Since $ 1″=4.85×10^{−6} \space rad $.
Also b = AB =1.276 ×10$^7$m
Hence from Eq. (2.1), we have the earth-moon distance,
D=b/θ
= \frac {1.276× 10^7}{3.32 × 10^{-2}} $$
$$ = 3.84 ×10^8 \space m $$

Example 4: The Sun’s angular diameter is measured to be 1920′ ′. The distance D of the Sun from the Earth is 1.496 × 10$^{11}$ m. What is the diameter of the Sun?

Solution: Sun’s angular diameter α
= 1920″
$$= 1920 × 4.85 × 10^{-6} \space rad $$
$$=9.31×10^{−3} \space rad $$
Sun’s diameter
d =α D
$$ = \big ( 9.31×10^{-3} \big ) × \big (1.496×10^{11} \big ) m $$
$$=1.39 ×10^9 \space m $$

1.6.2 Estimation of Very Small Distances: Size of a Molecule

To measure a very small size like that of a molecule ($10^{-8}$ m to $10^{-10}$ m), we need specialized instruments like electron microscopes in which an electron beam is focussed on the object by carefully managing the electric and magnetic fields.
An optical microscope in which, visible radiation of wavelength 4 x $10^{-7}$m to 7 x $10^{-7}$m is used so that particles smaller than the size of the incident radiation cannot be measured.
However, the wavelength of an electron can be as small as a fraction of an angstrom. Such electron microscopes with a resolution of 0.6 Å (1 Å = 1 Angstorm = $10^{-10}$m) have been built. They can almost resolve atoms and molecules in a material.

Concept box 3: Tunnelling microscopy: A scanning tunneling microscope (STM) is an instrument for imaging surfaces at the atomic level.

Example 5: If the size of a nucleus (in the range of 10$^{–15}$ to 10$^{–14}$ m) is scaled up to the tip of a sharp pin, what roughly is the size of an atom ? Assume tip of the pin to be in the range 10$^{–5}$m to 10$^{–4}$m.

Solution: The size of a nucleus is in the range of 10$^{–15}$ m and 10$^{–14}$ m. The tip of a sharp pin is taken to be in the range of 10$^{–5}$ m and 10$^{–4}$ m.
Thus we are scaling up by a factor of 10$^{10}$. An atom roughly of size 10$^{–10}$ m will be scaled up to a size of 1 m. Thus, a nucleus in an atom is as small in size as the tip of a sharp pin placed at the centre of a sphere of radius about a metre long.

1.6.3 Range of lengths

The range of sizes that exist in the universe range from $10^{-14}$m (which is the size of the nucleus of an atom) to $10^{26}$ m (which is the extent of the observable universe).
We also use certain special length units for short and large lengths. These are: 1 fermi = 1 f = $10^{-15}$ m

1 angstrom = 1 Å = $10^{-10}$ m

1 astronomical unit = 1 AU (average distance of the Sun from the Earth) = 1.496 × $10^{11}$ m

1 light year = 1 ly = 9.46 × $10^{15 }$m (distance that light travels with velocity of 3 × $10^8$ m $s^{-1}$ in 1 year)

1 parsec = 3.08 × $10^{16}$ m (Parsec is the distance at which average radius of earth’s orbit subtends an angle of 1 arc second)

Questions from section 1.6:

1. What is parallax? Explain with an example. Also explain the parallax method to determine large distances.
2. Explain how small distances like the size of a molecule can be determined.

1.7 Measurement of Mass

Mass is a basic property of matter. It does not depend on the temperature, pressure or location of the object in space (Weight does). The SI unit of mass is kilogram (kg).

While dealing with atoms and molecules, the kilogram is an inconvenient unit. Unified atomic mass unit (u), is an important standard unit of mass established for expressing the mass of atoms as:

1 unified atomic mass unit = 1u

= (1/12) of the mass of an atom of carbon-12 isotope ($_{12}C^6$) including the mass of electrons

= 1.66 x $10^{-27}$ kg

Large masses in the universe like planets and stars, based on Newton’s law of gravitation can be measured by using gravitational method.

For measurement of small masses of atomic or subatomic particles, we make use of mass spectrograph in which radius of the trajectory is proportional to the mass of a charged particle moving in uniform electric and magnetic field.

Range of Masses

The masses vary from tiny mass of the order of $10^{-30}$ kg of an electron to the huge mass of about $10^{55}$ kg of the known universe.

Questions from section 1.7:

1. What is the SI unit of mass?

1.8 Measurement of time

Caesium atomic clock

• To measure any time interval we need a clock. We use an atomic standard of time, which is based on the periodic vibrations produced in a cesium atom. The resultant clock is the caesium clock, sometimes called the atomic clock, used in the national standards.
• In the cesium atomic clock, the second is taken as the time needed for 9,192,631,770 vibrations of the radiation corresponding to the transition between the two hyperfine levels of the ground state of cesium-133 atom.
• The vibrations of this cesium atom regulate the rate of this cesium atomic clock just as the vibrations of a small quartz crystal regulate a quartz wristwatch.
• The cesium atomic clocks are very accurate. In principle they provide portable standard.
• In our country, the National Physical Laboratory (NPL), New Delhi, has the responsibility of maintenance and improvement of physical standards, including that of time, frequency, etc. the Indian Standard Time (IST) is linked to a set of cesium atomic clocks in NPL.
• The cesium atomic clocks are so accurate that they impart the uncertainty in time realisation as ± 1 × $10^{-13}$ i.e. 1 part in $10^{13}$. This implies that the uncertainty gained over time by such a device is less than 1 part in $10^{13}$; they lose or gain no more than 3 μs in one year.

Note box 3: The SI unit of length (i.e meter) has been expressed in terms the path length light travels in certain interval of time (1/299, 792, 458 of a second).

Questions from section 1.8:

1. Write a note on caesium atomic clock.

1.9 Accuracy, Precision of instruments and errors in measurements

Error: The result of every measurement by any measuring instrument contains some uncertainty. This uncertainty is called error.

Accuracy: The accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity.

Precision: Precision tells us to what resolution or limit the quantity is measured.

The accuracy in measurement may depend on several factors, including the limit or the resolution of the measuring instrument.

Ex: Suppose the true value of a certain length is near 3.678 cm. In one experiment, using a measuring instrument of resolution 0.1 cm, the measured value is found to be 3.5 cm, while in another experiment using a measuring device of greater resolution, say 0.01 cm, and the length is determined to be 3.38 cm. The first measurement has more accuracy (because it is closer to the true value) but less precision (its resolution is only 0.1 cm), while the second measurement is less accurate but more precise.

There are two kinds of errors (speaking broadly):

1. Systematic errors – Systematic errors are those errors that tend to be in one direction, either positive or negative.

2. Instrumental errors – Instrumental errors arise from the errors due to imperfect design or calibration of the measuring instrument, zero error in the instrument, Imperfection in experimental technique or procedure.

Ex: The temperature graduations of a thermometer may be inadequately calibrated (it may read 104 °C at the boiling point of water at STP whereas it should read 100 °C)

Other types of errors:

1. Personal errors

Errors that arise due to an individual’s bias, lack of proper setting of the apparatus or individual’s carelessness in taking observations without observing proper precautions.

Ex: If someone who is measuring a value from a scale tilts his head towards his right, then he is introducing an error due to parallax.

2. Random Errors

Errors that occur irregularly and are absolutely random with respect to occurrence and size are called Radom errors.

Ex: Unpredictable fluctuations in temperature, voltage supply, mechanical vibrations of experimental set-ups, etc), personal (unbiased) errors by the observer taking readings, etc. When the same person repeats the same observation, it is very likely that he may get different readings every time.

3. Least count error

The smallest value that can be measured by the measuring instrument is called its least count.

The least count error is the error associated with the resolution of the instrument.

Ex: A vernier callipers has the least count as 0.01 cm. This means that for all values measured upto a hundredth of a cm, the vernier callipers is accurate. Any measurement that requires a precision greater than a hundredth of centimetre might not be accurate.

A spherometer may have a least count of 0.001 cm.

To reduce the least count error:

1) We can use instruments having higher precision.

2) We can repeat the observations several times and take the mean of all observtions (While this does not necessarily guarantee a greater accuracy, it still continues to be a standard practice).

4. Absolute error, Relative error and Percentage error

The standard practice while conducting most experiments is to repeat the observations multiple times so that a mean of the observed readings can be taken.
Suppose the values obtained in several measurements are $a_1, a_2, a_3…., a_n$. The arithmetic mean of these values is taken as:
$$a_{mean}=a_1+a_2+⋯+a_n$$

Or

$$a_{mean}= ∑{i=1}^n \frac {a_i}{n}$$ The magnitude of the difference between the individual measurement and the true value of the quantity is called the absolute error (Δa) of the measurement. $$∆a_1=a_1-a{mean}$$
$$∆a_n=a_n-a_{mean}$$

The absolute error (Δa) calculated may either be positive or negative. |Δa| is hence used which only denotes the magnitude of the error.

The arithmetic mean of all the absolute errors is taken as the final or mean absolute error $(Δa_{mean}$).

Δamean = (|Δa1|+|Δa2 |+|Δa3|+…+ |Δan|) / n

$$Δa_{mean} = (|Δa_1|+|Δa_2 |+|Δa_3|+…+ |Δa_n|) / n$$
$$Δa_{mean} = ∑{i=1}^n \frac {|Δa_i |}{n}$$

Any measurement of the physical quantity a is likely to lie between: $(a{mean}+ Δa_{mean}$ ) and ($a_{mean}− Δa_{mean})$

As, $a=a_{mean} ± Δa_{mean}$

$$ a_{mean} – ∆a_{mean} = ≤ a ≤ a_{mean} + ∆a_{mean}$$

The relative error is the ratio of the mean absolute error $Δa_{mean}$ to the mean value $a_{mean}$ of the quantity measured.
$$ Relative \space error= \frac {∆a_{mean}}{a_{mean}}$$

When the relative error is expressed in per cent, it is called the percentage error (δa).
$$Percentage \space error (δa)= \bigg (\frac {∆a_{mean}}{a_{mean}} \times \bigg) 100 \% $$

Example 6: Two clocks are being tested against a standard clock located in a national laboratory. At 12:00:00 noon by the standard clock, the readings of the two clocks are:

(If you are doing an experiment that requires precision time interval measurements, which of the two clocks will you prefer?

Solution: The range of variation over the seven days of observations is 162 s for clock 1, and 31 s for clock 2. The average reading of clock 1 is much closer to the standard time than the average reading of clock 2. The important point is that a clock’s zero error is not as significant for precision work as its variation, because a ‘zero-error’ can always be easily corrected. Hence, clock 2 is to be preferred to clock 1.

Example 7: We measure the period of oscillation of a simple pendulum. In successive measurements, the readings turn out to be 2.63 s, 2.56 s, 2.42 s, 2.71s and 2.80 s. Calculate the absolute errors, relative error or percentage error.

Solution: The mean period of oscillation of the pendulum
$$ T = \frac {(2.63+2.56+2.42+2.71+2.80 )s}{5} $$
$$ = \frac {13.12}{5}s $$
= 2.624 s
= 2.62 s
As the periods are measured to a resolution of 0.01 s, all times are to the second decimal; it is proper to put this mean period also to the second decimal.
The errors in the measurements are
$$ 2.63 \space s – 2.62 \space s = 0.01 \space s $$
$$ 2.56 \space s – 2.62 \space s = – 0.06 \space s$$
$$2.42 \space s – 2.62 \space s = – 0.20 \space s$$
$$2.71 \space s – 2.62 \space s = 0.09 \space s$$
$$2.80 \space s – 2.62 \space s = 0.18 \space s$$

Note that the errors have the same units as the quantity to be measured. The arithmetic mean of all the absolute errors (for arithmetic mean, we take only the magnitudes) is
$$ ΔΤ_{mean} = [(0.01+ 0.06+0.20+0.09+0.18)s]/5 $$
$$= 0.54 \space s/5 $$
$$ = 0.11 \space s $$

That means, the period of oscillation of the simple pendulum is (2.62 ± 0.11) s i.e. it lies between (2.62 + 0.11) s and (2.62 – 0.11) s or between 2.73 s and 2.51 s. As the arithmetic mean of all the absolute errors is 0.11 s, there is already an error in the tenth of a second. Hence there is no point in giving the period to a hundredth. A more correct way will be to write

T = 2.6 ± 0.1 s

Note that the last numeral 6 is unreliable, since it may be anything between 5 and 7. We indicate this by saying that the measurement has two significant figures. In this case, the two significant figures are 2, which is reliable and 6, which has an error associated with it. You will learn more about the significant figures in section 2.7. For this example, the relative error or the percentage error is
$$ δa= \frac {0.1}{2.6}×100 = 4 \% $$

Questions from section 1.9:

1. Describe the terms:

a) Error

b) Accuracy

c) Precision

2. Explain in detail all the types of errors.

1.10 Combination of errors

(a) Error of a sum or difference

If two physical quantities A and B have measured values A ± ΔA, B ± ΔB respectively where ΔA and ΔB are their absolute errors. We wish to find the error ΔZ in the sum: Z = A + B

We have by addition, Z ± ΔZ = (A ± ΔA) + (B ± ΔB)

The maximum possible error in Z: ΔZ = ΔA + ΔB

For the difference Z = A . B, we have

Z ± Δ Z = (A ± ΔA) . (B ± ΔB)

= (A . B) ± ΔA ± ΔB

or,

± ΔZ = ± ΔA ± ΔB

The maximum value of the error ΔZ is again: ΔA + ΔB

Hence the rule: When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities.

(b) Error of a product or a quotient

Suppose Z = AB and the measured values of A and B are A ± ΔA and B ± ΔB.

Then: Z ± ΔZ = (A ± ΔA) (B ± ΔB)

= AB ± B ΔA ± A ΔB ± ΔA ΔB

Dividing LHS by Z and RHS by AB we have,

1±(ΔZ/Z) = 1 ± (ΔA/A) ± (ΔB/B) ± (ΔA/A)(ΔB/B)

Since ΔA and ΔB are small, we shall ignore their product. Hence the maximum relative error:

ΔZ/ Z = (ΔA/A) + (ΔB/B)

You can easily verify that this is true for division also.

Hence the rule: When two quantities are multiplied or divided, the relative error in the result is the sum of the relative errors in the multipliers.

(c) Error in case of a measured quantity raised to a power Suppose $Z = A^2,$
Then, ΔZ/Z = (ΔA/A) + (ΔA/A) = 2 (ΔA/A)

Hence, the relative error in $A^2$ is two times the error in A.

In general, if $Z = A^p B^q/C^r$

Then, ΔZ/Z = p (ΔA/A) + q (ΔB/B) + r (ΔC/C).

Hence the rule: The relative error in a physical quantity raised to the power k is the k times the relative error in the individual quantity.

Questions from section 1.10:

1. Determine the equation for the following combination of errors:

(a) Error of a sum or difference

(b) Error of a product or a quotient

(c) Error in case of a measured quantity raised to a power

Example 8: The temperatures of two bodies measured by a thermometer are $t_1 = 20 ^0C ± 0.5 ^0C$ and $t_2 = 50 ^0C ± 0.5 ^0C$. Calculate the temperature difference and the error theirin.

Solution:

$$ t′ = t_2–t_1 = (50 ^0C±0.5 ^0C)– (20^0C±0.5 ^0C)$$
$$ t′ = 30 ^0C ± 1 ^0C $$

Example 9: The resistance R = V/I where V = (100 ± 5)V and I = (10 ± 0.2)A. Find the percentage error in R.

Solution: The percentage error in V is 5% and in I it is 2%. The total error in R would therefore be 5% + 2% = 7%.

Example 10: Two resistors of resistances $R_1$ = 100 ±3 ohm and $R_2$ = 200 ± 4 ohm are connected (a) in series, (b) in parallel. Find the equivalent resistance of the (a) series combination, (b) parallel combination. Use for (a) the relation $R =R_1 + R_2$, and for (b)
$$ \frac {1}{R’} = \frac {1}{R_1} + \frac {1}R_2} and $$

Solution: (a) The equivalent resistance of series combination
$$ R =R_1 + R_2 = (100 ± 3) \space ohm + (200 ± 4) \space ohm $$
$$ = 300 ± 7 \space ohm $$.
(b) The equivalent resistance of parallel combination
$$ R’ = \frac { R_1R_2}{R_1+R_2} = \frac {200}{3} = 66.7 \space ohm $$
Then, from $$ \frac {1}{R’} = \frac {1}{R_1} + \frac {1}R_2} and $$
we get,
$$ \frac {ΔR’}{R’^2} = \frac { ΔR_1}{R^2_1}+ \frac { ΔR_2}{R^2_2} $$
$$ ΔR’ = (R’^2) \frac { ΔR_1}{R^2_1}+(R’^2) \frac { ΔR_2}{R^2_2} $$
$$ = \bigg ( \frac {66.7}{100} \bigg)^2 3 +\bigg ( \frac {66.7}{200} \bigg)^2 4 $$
$$ = 1.8 $$
Then, R′=66.7±1.8 ohm
(Here, ΔR is expresed as 1.8 instead of 2 to keep in confirmity with the rules of significant figures.)

Example 11: Find the relative error in Z, if $Z = A^4B^{1/3}/CD^{3/2}$.

Solution: The relative error in Z is ΔZ/Z =

$$ 4(ΔA/A) +(1/3) (ΔB/B) + (ΔC/C) + (3/2) (ΔD/D) $$

Example 12: The period of oscillation of a simple pendulum is T= 2πL/g. Measured value of L is 20.0 cm known to 1 mm accuracy and time for 100 oscillations of the pendulum is found to be 90 s using a wrist watch of 1 s resolution. What is the accuracy in the determination of g?

Solution:

$$ g = 4π^2L/T^2 $$4.

Here, $ T = \frac {t}{n} $ and $Δ T = \frac {Δt}{n} . Therefore, \frac { Δ T}{T} = \frac { Δt}{t}$.
The errors in both L and t are the least count errors. Therefore,

$$ (Δg/g) = (ΔL/L) + 2(ΔT/T ) $$

$$ = \frac {0.1}{20.0} + 2 \bigg( \frac {1}{90} \bigg) = 0.032 $$

Thus, the percentage error in g is

$$ 100 (Δg/g) = 100(ΔL/L) + 2 × 100 (ΔT/T ) $$

$$ = 3 \% $$

1.11 Significant figures

Significant digits: Every measurement involves errors. The result of measurement should be reported in a way that indicates the precision of measurement. Normally, the reported result of measurement is a number that includes all digits in the number that are known reliably plus the first digit that is uncertain. The reliable digits plus the first uncertain digit are known as significant digits or significant figures.

Significant figures = Reliable digits + First uncertain digit

The location of decimal point is of no consequence in determining the number of significant figures.

“A choice of change of different units does not change the number of significant digits or figures in a measurement”. This important remark makes most of the following observations clear:

1) The length 2.308 cm has four significant figures. But in different units, the same value can be written as 0.02308 m or 23.08 mm or 23080 μm.

The example gives the following rules:

• All the non-zero digits are significant.
• All the zeros between two non-zero digits are significant, no matter where the decimal point is, if at all.
• If the number is less than 1, the zero(s) on the right of decimal point but to the left of the first non-zero digit are not significant. [In 0.00 2308, the underlined zeroes are not significant].
• The terminal or trailing zero(s) in a number without a decimal point are not significant.
• [Thus 123 m = 12300 cm = 123000 mm has three significant figures, the trailing zero(s) being not significant.]
• The trailing zero(s) in a number with a decimal point are significant. [The numbers 3.500 or 0.06900 have four significant figures each.]

2) Do not be confused regarding the trailing zero(s). Suppose a length is reported to be 4.700 m. It is evident that the zeroes here are meant to convey the precision of measurement and are, therefore, significant. [If these were not, it would be superfluous to write them explicitly, the reported measurement would have been simply 4.7 m]. Now suppose we change units, then 4.700 m = 470.0 cm = 4700 mm = 0.004700 km Since the last number has trailing zero(s) in a number with no decimal, we would conclude erroneously from observation (1) above that the number has two significant figures, while in fact, it has four significant figures and a mere change of units cannot change the number of significant figures.

3) To remove such ambiguities in determining the number of significant figures, the best way is to report every measurement in scientific notation (in the power of 10).

In this notation, every number is expressed as a $10^b$, where a is a number between 1 and 10, and b is any positive or negative exponent (or power) of 10. In order to get an approximate idea of the number, we may round off the number a to 1 (for a ≤ 5) and to 10 (for 5<a ≤ 10). Then the number can be expressed approximately as $10^b$ in which the exponent (or power) b of 10 is called order of magnitude of the physical quantity. When only an estimate is required, the quantity is of the order of $10^b$.

For example, the diameter of the earth (1.28107m) is of the order of $10^7$ m with the order of magnitude 7. The diameter of hydrogen atom (1.06 10.10m) is of the order of $10^{-10}$m, with the order of magnitude -10. Thus, the diameter of the earth is 17 orders of magnitude larger than the hydrogen atom.

It is often customary to write the decimal after the first digit. Now the confusion mentioned in (a) above disappears: 4.700 m = 4.700 x $10^2$ cm

= 4.700 $ 10^3$ mm

= 4.700 x $10^{-3}$ km

The power of 10 is irrelevant to the determination of significant figures. However, all zeroes appearing in the base number in the scientific notation are significant. Each number in this case has four significant figures. Thus, in the scientific notation, no confusion arises about the trailing zero(s) in the base number a. They are always significant.

(4) The scientific notation is ideal for reporting measurement. But if this is not adopted, we use the rules adopted in the preceding example:

• For a number greater than 1, without any decimal, the trailing zero(s) are not significant.
• For a number with a decimal, the trailing zero(s) are significant.

(5) The digit 0 conventionally put on the left of a decimal for a number less than 1 (like 0.1250) is never significant. However, the zeroes at the end of such number are significant in a measurement.

(6) The multiplying or dividing factors which are neither rounded numbers nor numbers representing measured values are exact and have infinite number of significant digits. For example in r=d/2 or s = 2πr, the factor 2 is an exact number and it can be written as 2.0, 2.00 or 2.0000 as required. Similarly, in T = t/n, n is an exact number.

Example 13: Each side of a cube is measured to be 7.203 m. What are the total surface area and the volume of the cube to appropriate significant figures?

Solution: The number of significant figures in the measured length is 4. The calculated area and the volume should therefore be rounded off to 4 significant figures.
$$ Surface \space area \space of \space the \space cube = 6(7.203)^2 \space m^2 $$
$$ = 311.299254 \space m^2 $$
$$= 311.3 \space m^2 $$
$$ Volume \space of \space the \space cube = (7.203)^3 \space m^3 $$
$$= 373.714754 \space m^3 $$
$$= 373.7 \space m^3 $$

Example 14: 5.74 g of a substance occupies 1.2 $cm^3$. Express its density by keeping the significant figures in view

Solution: There are 3 significant figures in the measured mass whereas there are only 2 significant figures in the measured volume. Hence the density should be expressed to only 2 significant figures.
$$ Density = \frac {5.74}{1.2} \space g \space cm^{-3} $$
$$= 4.8 \space g \space cm^{-3}$$.

1.11.1 Rules for Arithmetic Operations with Significant Figures

The result of a calculation involving approximate measured values of quantities (i.e. values with limited number of significant figures) must reflect the uncertainties in the original measured values. It cannot be more accurate than the original measured values themselves on which the result is based. In general, the final result should not have more significant figures than the original data from which it was obtained. Thus, if mass of an object is measured to be, say, 4.237 g (four significant figures) and its volume is measured to be 2.51 $cm^3$, then its density, by mere arithmetic division, is 1.68804780876 g/ cm^3$, upto 11 decimal places. It would be clearly absurd and irrelevant to record the calculated value of density to such a precision when the measurements on which the value is based, have much less precision. The following rules for arithmetic operations with significant figures ensure that the final result of a calculation is shown with the precision that is consistent with the precision of the input measured values:

(1) In multiplication or division, the final result should retain as many significant figures as are there in the original number with the least significant figures.

Thus, in the example above, density should be reported to three significant figures.
$$Density= \frac {4.237 \space g}{2.51 \space cm^{-3} }=1.69g.cm^{-3}$$
Similarly, if the speed of light is given as 3.00 $10^8$ m $s^{-1}$ (three significant figures) and one year (1y = 365.25 d) has 3.1557 $10^7$ s (five significant figures), the light year is 9.47 $10^{15}$ m (three significant figures).

(2) In addition or subtraction, the final result should retain as many decimal places as are there in the number with the least decimal places.

For example, the sum of the numbers 436.32 g, 227.2 g and 0.301 g by mere arithmetic addition, is 663.821 g. But the least precise measurement (227.2 g) is correct to only one decimal place. The final result should, therefore, be rounded off to 663.8 g.

Similarly, the difference in length can be expressed as:

0.307 m – 0.304 m = 0.003 m = 3 x $10^{-3 }$m.

Note that we should not use the rule (1) applicable for multiplication and division and write 664 g as the result in the example of addition and 3.00 x $10^{-3 }$ m in the example of subtraction. They do not convey the precision of measurement properly. For addition and subtraction, the rule is in terms of decimal places.

1.11.2 Rounding off the Uncertain Digits

The result of computation with approximate numbers, which contain more than one uncertain digit, should be rounded off. The rules for rounding off numbers to the appropriate significant figures are obvious in most cases. A number 2.746 rounded off to three significant figures is 2.75, while the number 2.743 would be 2.74.

The rule by convention is that the preceding digit is raised by 1 if the insignificant digit to be dropped (the underlined digit in this case) is more than 5, and is left unchanged if the latter is less than 5. But what if the number is 2.745 in which the insignificant digit is 5. Here, the convention is that if the preceding digit is even, the insignificant digit is simply dropped and, if it is odd, the preceding digit is raised by 1. Then, the number 2.745 rounded off to three significant figures becomes 2.74. On the other hand, the number 2.735 rounded off to three significant figures becomes 2.74 since the preceding digit is odd.

In any involved or complex multi-step calculation, you should retain, in intermediate steps, one digit more than the significant digits and round off to proper significant figures at the end of the calculation. Similarly, a number known to be within many significant figures, such as in 2.99792458 × 108 m/s for the speed of light in vacuum, is rounded off to an approximate value 3 × 108 ms—1, which is often employed in computations. Finally, remember that exact numbers that appear in formulae like 2 π in:

$$T=2π \sqrt { \frac {L}{g}}$$

have a large (infinite) number of significant figures. The value of π =3.1415926…. is known to a large number of significant figures. You may take the value as 3.142 or 3.14 for π, with limited number of significant figures as required in specific cases.of significant figures. The value of π = 3.1415926…. is known to a large number of significant figures. You may take the value as 3.142 or 3.14 for π, with limited number of significant figures as required in specific cases.

1.11.3 Rules for Determining the Uncertainty in the Results of Arithmetic Calculations

The rules for determining the uncertainty or error in the number/measured quantity in arithmetic operations can be understood from the following examples.

(1) If the length and breadth of a thin rectangular sheet are measured, using a metre scale as 16.2 cm and, 10.1 cm respectively, there are three significant figures in each measurement. It means that the length l may be written as

l = 16.2 ± 0.1 cm

= 16.2 cm ± 0.6 %.

Similarly, the breadth b may be written as:
b = 10.1 ± 0.1 cm
= 10.1 cm ± 1 %

Then, the error of the product of two (or more) experimental values, using the combination of errors rule, will be:
l b = 163.62 cm2 + 1.6%
= 163.62 + 2.6 $cm^2$

This leads us to quote the final result as:
l b = 164 + 3 $cm^2$
Here 3 $cm^2$ is the uncertainty or error in the estimation of area of rectangular sheet.

(2) If a set of experimental data is specified to n significant figures, a result obtained by combining the data will also be valid to n significant figures.
However, if data are subtracted, the number of significant figures can be reduced.
For example, 12.9 g – 7.06 g, both specified to three significant figures, cannot properly be evaluated as 5.84 g but only as 5.8 g, as uncertainties in subtraction or addition combine in a different fashion (smallest number of decimal places rather than the number of significant figures in any of the number added or subtracted).

(3) The relative error of a value of number specified to significant figures depends not only on n but also on the number itself.

For example, the accuracy in measurement of mass 1.02 g is ± 0.01 g whereas another measurement 9.89 g is also accurate to ± 0.01 g. The relative error in 1.02 g is:

= (± 0.01/1.02) × 100 %

= ± 1%

Similarly, the relative error in 9.89 g is

= (± 0.01/9.89) × 100 %

= ± 0.1 %

Finally, remember that intermediate results in a multi-step computation should be calculated to one more significant figure in every measurement than the number of digits in the least precise measurement.

These should be justified by the data and then the arithmetic operations may be carried out; otherwise rounding errors can build up. For example, the reciprocal of 9.58, calculated (after rounding off) to the same number of significant figures (three) is 0.104, but the reciprocal of 0.104 calculated to three significant figures is 9.62. However, if we had written 1/9.58 = 0.1044 and then taken the reciprocal to three significant figures, we would have retrieved the original value of 9.58. This example justifies the idea to retain one more extra digit (than the number of digits in the least precise measurement) in intermediate steps of the complex multi-step calculations in order to avoid additional errors in the process of rounding off the numbers.

Questions from section 1.11:

1. What are significant figures?
2. With an example, explain the rules in writing significant figures.
3. Explain the rules followed for obtaining significant figures which have arithmetic operations.
4. Explain the rules for rounding off uncertain digits with examples.
5. State the rules for determining the uncertainty in the results of arithmetic calculations.

1.12 Dimensions of Physical Quantities

All physical quantities can be expressed in terms of some combination of the seven fundamental or base quantities.

The seven dimensions (represented within []) of the physical World:

Length [L], Mass [M}]. Time [T], Electric current [A], Thermodynamic temperature [K], Lumnious intesnsity[cd], amount of a substance [mol].

The dimensions of a physical quantity are the powers (or exponents) to which the base quantities are raised to represent that quantity.

In mechanics, all the physical quantities can be written in terms of the dimensions [L], [M] and [T].

Ex: The volume occupied by an object is expressed as the product of length, breadth and height, or three lengths. Hence the dimensions of volume are [L] [L] [L] = [$L]^3 = [L^3]$. As the volume is independent of mass and time, it is said to possess zero dimension in mass [M°], zero dimension in time [T°].

Force is the product of mass and acceleration and can be expressed as: $ Force = mass \times acceleration = \frac {mass \times length}{time^2 }$

The dimensions of force are $[M] [L]/[T]^2 = [M L T^{—2}]$. This is called as the dimensional formula of force. The dimensional formula tells us which of the base quantities represent the dimensions of a physical quantity.

Similarly, this is a dimensional equation for volume:

[V] = [M0 L3 T0]
$[V] = [M^0 L^3 T^0]$

[Symbol of the Physical Quantity being represented] = [Dimensional formula]

The whole equation above is known as a dimensional equation.

1.12.1-Dimensional Analysis and its applications

Checking the Dimensional Consistency of Equations

We can only add or subtract similar physical quantities. Thus, velocity cannot be added to force, or an electric current cannot be subtracted from the thermodynamic temperature. This simple principle called the principle of homogeneity of dimensions in an equation is extremely useful in checking the correctness of an equation. If the dimensions of all the terms are not same, the equation is wrong.

Dimensional consistency does not guarantee correct equations. It is uncertain to the extent of dimensionless quantities or functions. A pure number, ratio of similar physical quantities, such as angle as the ratio (length/length), refractive index as the ratio (speed of light in vacuum/speed of light in medium) etc., has no dimensions.

The dimensions of each term may be written as:

[x] = [L]

[$x_0$ ] = [L]

[$v_0$ t] = [L $T^{-1}$] [T] = [L]

[(1/2) a $t^ 2$] = [L $T^{-2}$] [$T ^2$] = [L]

If an equation fails this consistency test, it is proved wrong, but if it passes, it is not proved right. Thus, a dimensionally correct equation need not be actually an exact (correct) equation, but a dimensionally wrong (incorrect) or inconsistent equation must be wrong.

1.12.2 Deducing Relation among the Physical Quantities

The method of dimensions can be used to deduce relation among the physical quantities. For this, we should know the dependence of the physical quantity on other quantities (upto three physical quantities or linearly independent variables) and consider it a product type of dependence. Let us take an example.

For example,

Let us consider a simple pendulum, having a bob attached to a string that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on its length (l), mass of the bob (m) and acceleration due to gravity (g). Let us derive the expression for its time period using method of dimensions.

The dependence of time period T on the quantities l, g and m as a product may be written as:

$$T = k \space l^x g^y m^z$$

where k is dimensionless constant and x, y and z are the exponents.

By considering dimensions on both sides, we have

$$[L^oM^oT^1]=[L^1][L^1T^{–2}]^y[M^1]^z$$
$$= L^{x+y} T^{–2y} M^z$$

On equating the dimensions on both sides, we have x + y = 0; –2y = 1; and z = 0

So that $ x = \frac {1}{2},y = -\frac {1}{2},z=0$

Then, T =$ k \space l^{½ }g^{- \frac {1}{2}}$

$$ T = k \sqrt { \frac {l}{g}}$$

Note box 4: Note that value of constant, k cannot be obtained by the method of dimensions. Here it does not matter if some number multiplies the right side of this formula, because that does not affect its dimensions. Actually, $k=2 \pi$ ,so that $ T = 2 \pi \sqrt { \frac {l}{g}}$

Example 15: Let us consider an equation $ \frac {1}{2} \space m \space v^2 = m g h $ where m is the mass of the body, v its velocity, g is the acceleration due to gravity and h is the height. Check whether this equation is dimensionally correct

Solution: The dimensions of LHS are
$$ [M] [L T^{–1} ]^2 = [M] [ L^2 T^{–2}] $$
$$ = [M L^2 T^{–2}] $$
The dimensions of RHS are
$$ [M][L T^{–2}] [L] = [M][L^2 T^{–2}]$$
$$ = [M L^2 T^{–2}]$$
The dimensions of LHS and RHS are the same and hence the equation is dimensionally correct.

Example 16: The SI unit of energy is $ J = kg m^2 s^{–2}$; that of speed v is m s$^{–1}$ and of acceleration a is m s$^{–2}$. Which of the formulae for kinetic energy (K) given below can you rule out on the basis of dimensional arguments (m stands for the mass of the body):
$$(a) K = m^2 v^3 $$
$$(b) K = (1/2)mv^2 $$
$$(c) K = ma $$
$$(d) K = (3/16)mv^2 $$
$$(e) K = (1/2)mv^2 + ma $$

Solution: Every correct formula or equation must have the same dimensions on both sides of the equation. Also, only quantities with the same physical dimensions can be added or subtracted. The dimensions of the quantity on the right side are $[M^2 L^3 T^{–3}]$ for (a); $[M L^2 T^{–2}]$ for (b) and (d); $[M L T^{–2}]$ for (c). The quantity on the right side of (e) has no proper dimensions since two quantities of different dimensions have been added. Since the kinetic energy K has the dimensions of $[M L^2 T^{–2}]$, formulas (a), (c) and (e) are ruled out. Note that dimensional arguments cannot tell which of the two, (b) or (d), is the correct formula. For this, one must turn to the actual definition of kinetic energy (see Chapter 6). The correct formula for kinetic energy is given by (b).

Example 17: Consider a simple pendulum, having a bob attached to a string, that oscillates under the action of the force of gravity. Suppose that the period of oscillation of the simple pendulum depends on its length (l), mass of the bob (m) and acceleration due to gravity (g). Derive the expression for its time period using method of dimensions.

Solution: The dependence of time period T on the quantities l, g and m as a product may be written as :
$$ T = k l^x g^y m^z $$
where k is dimensionless constant and x, y and z are the exponents.
By considering dimensions on both sides, we have
$$ [L^oM^oT^1]=[L^1][L^1T^{–2}]^y[M^1]^z $$
$$ = L^{x+y} T^{–2y} M^ z $$
On equating the dimensions on both sides, we have
x + y = 0; –2y = 1; and z = 0
So that,, $ x = \frac {1}{2} $ ,$ y = – \frac {1}{2}$,z=0.
Then, $ T = k l^½ g^{–½} $
Or, $ T = k \sqrt { \frac {l}{g} } $

Note that value of constant k cannot be obtained by the method of dimensions. Here it does not matter if some number multiplies the right side of this formula, because that does not affect its dimensions.

Actually, k = 2π so that $T =2π \sqrt { \frac {l}{g} } $

Advantage and Limitation of using Dimensional analysis:

Dimensional analysis is very useful in deducing relations among the interdependent physical quantities.

However, dimensionless constants cannot be obtained by this method. The method of dimensions can only test the dimensional validity, but not the exact relationship between physical quantities in any equation. It does not distinguish between the physical quantities having same dimensions.

Questions from section 1.12:

1. Give two examples for dimensional formula.
2. Explain the principle of homogeneity of dimensions.
3. State the advantage and disadvantage of using dimensional analysis.