Numerical Problems

CHAPTER 01

Question 1.1

Speed of light, v = 3.0 x 10⁸ ms^-1

Travel time, t = 1 year

                         = 365 x 24 x 60 x 60 sec

                         = 31536000 sec

Since

            S = v . t

Therefore

            S = 3.0 x 10⁸ x 31536000

S = 9.5 x 10¹⁵ m                     Answer

Question 1.2

Question 1.3

Length, L = 15.3 cm

Width, W = 12.80 cm

Volume, V = ?

Since

Volume = Length x Width

V = L x W

V = 15.3 x 12.80

V = 196 cm²               Answer

Question 1.4

Question 1.5

Here  

Length of simple pendulum, l = 100 cm

Time for 20 vibrations = 40.2 s

Meter scale accuracy = 1 mm

Stop watch accuracy = 0.1 s

Now

l = 100 cm

  = 100/100  m

  = 1 m

T = 40.2/20

   = 2.01 s

FOR LENGTH

            Absolute uncertainty = 1 mm = 0.1 cm

            %age uncertainty = (0.1/100)  x  (100/100) = 0.1 %

FOR TIME

            Absolute uncertainty = 0.1 sec

Average uncertainty = 0.1/20 = 0.005 sce

            %age uncertainty = (0.005/2.01)  x  (100/100) = 0.25 %

Total uncertainty = 2 x 0.025 + 0.1 = 0.6 %

Total uncertainty for g = 9.76 x 0.6 /100 = 0.06

Thus g = 9.76 ± 0.06 ms^-2

Question 1.6

Question 1.7

v_f = v_i + a t

[LT^-1] = [LT^-1] + [LT^-2 T]

[LT^-1] = [LT^-1] + [LT^-1]

[LT^-1] = [LT^-1]

LHS = RHS                           Hence Proved

Question 1.8

According to question

            v ; ρ^a E^b

            v = constant . ρ^a E^b              - Eq (1)

For ρ

            ρ = m/v = [ML^-3]                 - Eq (2)

For E

            E = stress/strain

            E = (F/A)/(∆l/l) = [MLT^-2/L²] / [L/L] = [ML^-1T^-2]                   - Eq (3)

For v

            v = [LT^-1]                               - Eq (4)

Putting the value of ρ, E and v from Eq (2), (3) and (4) in Eq (1), we get

            v = constant . ρ^a E^b 

            [LT^-1] = constant [ML^-3]^a [ML^-1T^-2]^b

            [LT^-1] = constant M^a+b L^-3a-b T^-2b

Equating powers of corresponding quantities on both sides

a + b =0; - 3a - b = 1; -2b = -1

By solving, we get,

            a = -½, b = ½

Therefore from Eq (1)

            v = constant . ρ^a E^b

            v = constant . ρ^-½ E^½

 Question 1.9

            E = m C²

            mgh = m v²

            [M LT^-2 L] = [M (L T^-1)²]

            [M L² T^-2] = [M L² T^-2]

LHS = RHS                           Hence Proved

Question 1.10

            a ; rⁿ v^m

            a = constt. rⁿ v^m

            [L T^-2] = constt. [L]ⁿ [L T^-1]^m

            L T^-2 = constt. L^n+m  T^-m

Equating powers of corresponding quantities on both sides

n + m = 1; -m = -2

Þ        m = 2; n = -1                         Answer

Chapter 1 - BASIC CONCEPTS - ISOTOPES

ISOTOPES AND THEIR RELATIVE ABUNDANCE

Atoms of same element having same atomic number but different atomic weights are called isotopes. This phenomenon is called isotropy. It was discovered by Soddy. Isotopes have same number of elections, protons and same electronic configuration. They differ in number of neutrons present in nucleus. The isotopes have same chemical properties. Consider example of hydrogen, it has three isotopes, i.e.

Thus we can define isotopes as atoms of same element which differ in number of neutrons in nucleus. Carbon has three isotopes, ₆C¹², ₆C¹³, ₆C¹⁴. Each one of them have 6 electrons and 6 protons but they have 6, 7, 8 neutrons respectively. The number of isotopes of some other elements are O = 3, Ni = 5, Ca = 6, Pd = 6, Cd = 9, Sn = 11.

RELATIVE ABUNDANCE OF ISOTOPES

The isotopes of all elements have their own natural abundance. This relative abundance is determined from mass spectrometry. The properties of element which are written in books are that of most abundant isotope of an element.

There are 280 stable naturally occurring isotopes in nature. There are 40 radioactive isotopes. Besides 300 unstable radioactive isotopes have been produced by artificial radioactivity.

The number of isotopes of an element is a complex property. Some general information is given over hare.

MONOISOTOPIC ELEMENTS: These elements only single isotope for example Gold, Iodine, Fluorine, Arsenic.

ODD ATOMIC NUMBER: Elements having odd atomic number never possess more than two stable isotopes.

EVEN ATOMIC NUMBER: Elements with even atomic no have large number of isotopes.

MASS NUMBER MULTIPLE OF FOUR: The isotopes with mass number which is multiple of four are quite abundant. For example ₈O¹⁶, Mg²⁴, Si³², Ca⁴⁰, Fe⁵⁶ are almost 50% of earth crust.

EVEN MASS NO AND ATOMIC NO: Out of 280 naturally occurring isotopes, 154 have even mass number and even atomic number.

MASS SPECTROMETRY: Determination of relative atomic masses of isotopes.

Mass spectrometer is an instrument which is used to measure the exact mass of different isotopes of an element.

BASIC PRINCIPLE: The substance whose mass is to be determined is first charged into vapours. These vapours are then ionized in an ionization chamber, with the beam of high energy electrons. The positive ions are separated on the basis of their m/e ratio in a magnetic analyzer. These ions are detected by ion collector. The results are represented in the form of a spectrum. m/e is plotted on x-axis and relative no. of ions along y-axis.

Chapter 1 - BASIC CONCEPTS (Contd.)

MOLECULE, ITS ATOMICITY AND CHARACTERISTICS

The smallest particle of a pure substance which can exist independently is called molecule.

ATOMICITY: The numbers of atoms present in a molecule is called its atomicity. Thus a molecule can be:

• MONO ATOMIC: It contains only one atom like He, Ne, Ar etc.
• DI ATOMIC: It contains two atoms like H₂, O₂, N_2, etc.
• TRI ATOMIC: It contains three atoms like H₂O, CO₂, etc.
• POLY ATOMIC: It contains many atoms. Such molecules are called Macromolecules.

MACROMOLECULES: These are very large molecules containing very large number of atoms. For example HAEMOGLOBIN contains 10,000 atoms in one molecule and it is 68,000 times heavier than H-atom.

A molecule may contain atoms of same element or atoms of different elements.

• HOMOATOMIC molecules: H₂, O₂, P₄, S₈
• HETEROATOMIC MOLECULES: HCL, NH₃, C₆H₁₂O₆, H₂SO₄

PROPERTIES OF MOLECULES

1. Molecules of same substance are similar in all respects.
2. Molecules have empty spaces between them. These are maximum in gases and minimum in solids.
3. The molecules are in the state of constant motion. These movements are maximum in gases, lesser in liquids while solids have no vibration motion.
4. Molecules have attractive forces for each other
5. The molecules have definite kinetic and potential energy.
6. For a chemical reaction, molecules must collide with each other. The molecules exchange atoms as a result of collision and new compounds are formed.

WHAT IS ION? ENERGY CHANGES DURING FORMATION OF A POSITIVE AND NEGATIVE ION

ION: the species which carry positive or negative charge are called ions. There are two types of ions:

1. Positive ion (cations)
2. Negative ion (anion)

POSITIVE ION: positive ion is formed when one or more electrons are removed from a neutral atom.

A – e^- → A¹⁺

If an atom loses one electron, unipositive ion is formed.

If an atom loses two electrons, dipositive ion is formed.

If an atom loses three electrons, tripositive ion is formed.

Energy is required to remove an electron from an atom to produce positive ion. It is called ionization energy. Thus formation of positive ion is an endothermic process. The most common positive ions are Na¹⁺, K¹⁺, Mg²⁺, Al³⁺, Fe³⁺, Sn⁴⁺, etc.

NEGATIVE ION (ANION): Negative ION is formed when an atom gains one, two or more electrons.

B + e^- → B^1-

The formation of negative ion is exothermic process. the most common negative ions are Cl^1-, Br^1-, F^1-, S^2-, etc. uninegative, dinegative and trinegative ions are formed due to addition of one, two or three electrons in an atom.

Some negative ion consists of groups of atoms. For example OH^1-, SO₄^2-, PO₄^3-, MnO₄^1-, CrO₄^2- etc. Positive ions having group of atoms are less common. For example NH₄¹⁺ and some carbonations (carbon having +ve charge). Properties of ions are different from corresponding atoms.

MOLECULAR ION AND ITS SIGNIFICANCE

When a molecule gains or losses electron, a molecular ion is formed. For example CH₄¹⁺, CO₃^2-, N₂¹⁺ etc. Cationic (+ve) molecular ions are more abundant than anionic (-ve) molecular ions.

These molecular ions are produced by passing high energy electron beam or alpha-particles through a gas.

SIGNIFICANCE: The molecular ions are quite unstable. The breaking of molecular ion gives useful information about structure of natural products.

RELATIVE ATOMIC MASS: The mass of an atom of an element as compared to mass of carbon atom taken as 12 is called relative atomic mass. For example on C-12 scale mass of H = 1/12*12=1.0078 amu and mass of carbon is 12.0000 amu. The masses of atoms are extremely small. We don't have a balance to weight atoms. Thus we use relative atomic mass unit scale. The elements have fractional relative atomic masses due to their different isotopic abundance.

Chapter 1 - BASIC CONCEPTS

what is an atom? brief history of atom.

Definition: Atom is defined as smallest particle of an element which may or may not exist independently. e.g He, Ne, Ar etc. but Hydrogen, Oxygen atoms cannot exist independently.

The modern researches have clearly shown that atom is further composed of subatomic particles like electrons, protons, neutrons, hyperons, neutrino, anti-neutrons etc. more than 100 particles are present in atom.

history of atom

Greek philosopher thought that when matter is broken into smaller and smaller particles, finally a smallest particle is obtained which cannot be further subdivided. Democritus called these particles exiled atoms. This term has been derived from word "atomis" which means indivisible.

In late 17^th century it was found that atoms of same element are present in different substances. It was also discovered how atoms of different elements combine to form compounds and how to break a given compound into its elements.

atomic theory of matter

Dalton showed that law of conservation of matter and law of constant proportions could be explained on the basis of atoms. He developed an atomic theory according to which all matter is composed of atoms of different elements differ in their properties.

J. Berzelius determined atomic masses of elements. Berzelius also developed a system of giving symbols to each element.

EVIDENCE OF ATOMS

It is not actually possible to see atoms. An optical microscope can measure size of an object of size 500nm or above. The size of an atom is almost 0.2nm. The objects of size of an atom could be observed in an electron microscope. It uses a beam of electrons instead of visible light. The wavelength of electrons is much shorter than that of visible light. An electron microscope photograph of a piece of graphite is shown over here. The bright bands in fig are layers of c-atoms.

SIZE AND MASS OF ATOM:

In twentieth century x-ray work has shown that the diameter of atoms are of the order of 2x10 ^-10 meter or 0.2nm.

Masses of atoms range from 10^-25 kg to 10^-27 kg. The masses of atom are generally expressed in terms of amu. 1 amu = 1.661x10^-27 kg. A full stop "." can bear two million atoms present in it. It gives an idea about very small size of an atom.

Chapter 1 Measurements - International system of units

International system of units
In 1960, an international committee agreed on a set of definitions and standards to describe the physical quantities. The system that was established is called the system international (SI).

Due to simplicity and convenience this system of units is being used by the world’s scientific community and by most nations.

The SI units are build-up from three kinds of units:

1. Base units
2. Supplementary units
3. Derived units

Base units
“The units defined arbitrarily for the measurement of seven base quantities in comparison with them are called base units.” The names of base units along with respective physical quantities and symbols are given below:

The standard definitions of base units are given below:

1.       Meter: The unit of length is meter. Before 1960, it was defined as:

a.      Definition 1: “The distance between two lines marked on the bar of an alloy of platinum (90%) and iridium (10%) kept under controlled conditions at the international bureau of weights and measures in France”.

The 11^th general conference on weights and measures (1960) redefined the standard as follows:

b.      Definition 2: “One meter is a length equal to 1,650,763.73 wavelength in vacuum of the orange red radiation emitted by the krypton-86 atom”.

In 1983, the meter was redefined as:

c.       Definition 3: “It is the distance travelled by light in vacuum during a time of 1/299,792,458 second”.

A few sub-multiple and multiples of meter are:

  

2.       Kilogram: The unit of mass is kilogram. It is defined as: “the mass of a platinum (90%) and iridium (10%) alloy cylinder, 3.9cm in diameter and 3.9cm in height, kept at the international bureau of weights and measures in France”.

This mass standard was established in 1901.

A few conversion relations of kg are:

3.       Second: The unit of time is termed as second.

a.      Definition 1: “One second is equal to 1/86400 part of an average day of the year 1900 A.D”.

In 1967, an international committee redefined second as

b.      Definition 2: “One second is equal to the duration in which the outer most electron of the cesium-133 atom makes 9,192,631,770 vibrations”.

A few sub-multiples and multiples of second are as given below:

4.       Kelvin: The unit of temperature is Kelvin.

Definition: “It is the fraction 1/273.16 of the thermodynamic temperature of the triple point of water”.

The triple point of a substance means the temperature at which solid, liquid and water vapour phases are in equilibrium. The triple point of water is taken as 273.16K. This standard was adopted in 1967.

5.      Ampere: The unit of electric current is ampere.

Definition: “One ampere is that constant current which if maintained in two straight parallel conductors of infinite length, of negligible circular cross-section and placed a meter apart in vacuum, would produce between those conductors a force equal to 2x10-7 Newton per meter of length”.

This unit was established in 1971.

6.      Candela: The unit of light intensity (or luminous intensity) is candela, which is defined as:

Definition: “One candela is the luminous intensity in the perpendicular direction of a surface of 1/600,000 square meter of a black body radiator at the solidification temperature of platinum under standard atmospheric pressure”.

This definition was adopted in 1967.

7.      Mole: The unit of amount of substance is mole.

Definition: “One mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kg of carbon-12”. This standard was adopted in 1971. When the unit mole is used, the element entities must be specified. There may be atoms, molecules, ions, electrons, other particles or specified groups of such particles. One mole of any substance contain 6.0225x10²³ entities.

Supplementary units

Those SI units which are not included either in base units or in derived units are called supplementary units. These are two in number as given below:

Radian: The radian is the plane angle between two radii of a circle which cut off on the circumference on arc, equal in length to the radius of the circle (see Figure). 

Total angle of the circumference = 2 π rad

Steradian: The steradian is the solid angle (three dimensional angle) subtended at the centre of a sphere by an area of its surface equal to the square of radius of the sphere (see Figure)

Derived units

SI-units for measuring all other physical quantities are derived from the base and supplementary units. These are called derived units. Some of the derived units are given as under:

Scientific notation

“When the numbers are expressed in standard form, it is called scientific notation, which employs power of ten”.

Rule: whiles writing the numbers in scientific notation, there should be only one non-zero digit to the left of decimal.

Examples:

Conventions for indicating units

Following points should be kept in mind while using units:

1. Full name of the units does not begin with a capital letter even if named after a scientist. e.g. newton.
2. The symbol of unit named after a scientist has initial capital letter, such as ‘N’ for newton.
3. The prefix should be written before the unit without any space, such as 1 x 10^-3 m = 1 mm
4. A combination of units is written each with one space apart. e.g. newton meter = N m
5. Compound prefixes are not allowed. e.g. 1µµF may be written as 1pF
6. A number such as 5.0 x 10⁴ cm may be expressed in scientific notation as 5.0 x 10² m
7. When a multiple of a base unit is raised to a power, the power applies to the whole multiple and not the base unit alone. e.g. 1 km² = 1 (km)² = 1 x 10⁶ m²
8. Measurement in practical work should be recorded immediately in the most convenient unit. e.g. reading of screw gauge in ‘mm’ mass in ‘grams’. But before calculation for the result, all measurements must be converted into SI-units.

Chapter 1 Measurements - Physical Quantities

Physical Quantities

Those quantities which can be measured accurately are called physical quantities. The foundation of physics rests upon physical quantities.

For Example: mass, length, time, velocity, force, density, temperature, electric current, etc.

Types: Physical quantities are of two types

Base Quantities: These are the minimum number of those physical quantities in terms of which other physical quantities can be defined.

For Example: length, mass, time, etc.

Derived Quantities: The quantities whose definitions are based on other physical quantities are called derived quantities.

For Example: velocity, acceleration, force, etc.

The measurement of a base quantity involves following two steps:

1. The choice of a standard
2. Estblishment of a procedure for comparing the quantity to be measured with two characteristics:
1. It is accessible
2. It is invariable

Chapter 1 Measurements - Introduction

Introduction

The earlier observations of man about the world around him and facts about the natural phenomena and material things resulted in the birth of single discipline of science called "Natural Philosophy".

It is further classified into two main branches:

(1) Biological Sciences

"The sciences which deals with the study of living things are called Biological sciences."
Examples are: Biology, Zoology, Botany, etc

(2) Physical Sciences

"the sciences which deals with non-living things are called physical sciences."

Examples are: Physics, Chemistry, Astronomy, Geology, etc.

Introduction to Physics

"physics is an important and basic part of physical sciences." It is an experimental science, required and applicable to nearly all disciplines of science.

At present there are following three main frontiers of fundamental science:

1. The world of extremely large body - the universe itself - its birth and expansion by "The Creator", from a very special "Cosmic Egg", with temperature of millions of degrees (Big-Bang Theory), which probably started 20 billion years ago.
2. The world of extremely small particles such as electron, protons, neutrons and other sub-atomic particles.
3. The world of complex matter. It is also the world of "middle sized" things, from molecule at one extreme, and bodies on the earth and the earth on other extreme. This is fundamental physics, which is the heart of science.

What is Physics?

It is the branch of science which deals with the study of matter, energy and the relationship between them.

The study of physics involves investigating such things as the laws of motion, structure of space and time, the nature and type of forces that hold different materials together, the interaction between different particles, the interaction of electromagnetic radiation with matter and so on.

Physics up-to 1890 AD is called classical physics and physics after 1890 AD up-till now and on-wards is called modern physics.

A few branches of classical physics are:

1. Mechanics: This branch of physics deals with the motion of particles under the action of different forces.
2. Heat & Thermodynamics: This branch of Physics deals with the nature of heat, its measurement, measurement of temperature and conversion of heat energy into mechanical energy.
3. Electromagnetism: This branch of physics deals with the interaction of electricity and magnetism, their development and use in practical life.
4. Optics: This branch deals with the study of nature of light, its properties and about optical instruments of different types.
5. Sound: this branch deals with production nature and different properties of sound.
6. Hydrodynamics: This branch deals with the study of the motion of fluids (liquids & gases).

A few branches of modern physics are:

1. Special relativity: In this branch of physics we study about non-accelerated frames of reference. This was stated by Einstein in 1905.
2. General Relativity: In this branch we study about the accelerated frames of reference. This theory came in 1915 (by Einstein).
3. Quantum Mechanics: It is the mathematical theory based on Plank's quantum hypothesis.
4. Atomic Physics: This branch deals with the behavior, structure and nature of atoms.
5. Molecular Physics: In this branch we study about the structure of matter on the basis of molecules.
6. Nuclear Physics: In this branch we study about the atomic nuclei, nuclear reactions and radioactivity.
7. Solid State Physics: This branch is concerned with the structure and properties of solids.
8. Particle Physics: This branch is concerned with "elementary particles" of matter.
9. Superconductivity: This branch deals with the study of superconductors and their development.
10. Superfluidity: This branch deals with the behavior and nature of super fluids.
11. Plasma Physics: This branch is concerned with highly ionized gases at very high temperatures. Plasma is called forth state of matter.
12. Magnetohydrodynamics: This branch deals with the behavior of a conducting fluid under the influence of a magnetic flux.
13. Space Physics: This branch deals with different aspects of space.

Physics is most fundamental of all sciences and provides other branches of science, basic principles and fundamental laws. This overlapping of physics gave birth to new branches such as physical chemistry, biophysics, astrophysics, health physics etc.