Physics Ebook - Class 11 - Part 1 PDF

PHYSICS PART - 1 CLASS XI  SYLLABUS (180 periods) UNIT – 1 Nature of the Physical World and Measurement (7 periods) Physics – scope and excitement – physics in relation to technology and society. Forces in nature – gravitational, electromagnetic and nuclear...

PHYSICS PART - 1 CLASS XI  SYLLABUS (180 periods) UNIT – 1 Nature of the Physical World and Measurement (7 periods) Physics – scope and excitement – physics in relation to technology and society. Forces in nature – gravitational, electromagnetic and nuclear forces (qualitative ideas) Measurement – fundamental and derived units – length, mass and time measurements. Accuracy and precision of measuring instruments, errors in measurement – significant figures. Dimensions - dimensions of physical quantities - dimensional analysis – applications. UNIT – 2 Kinematics (29 periods) Motion in a straight line – position time graph – speed and velocity – uniform and non-uniform motion – uniformly accelerated motion – relations for uniformly accelerated motions. Scalar and vector quantities – addition and subtraction of vectors, unit vector, resolution of vectors - rectangular components, multiplication of vectors – scalar, vector products. Motion in two dimensions – projectile motion – types of projectile – horizontal and oblique projectile. Force and inertia, Newton’s first law of motion. Momentum – Newton’s second law of motion – unit of force – impulse. Newton’s third law of motion – law of conservation of linear momentum and its applications. Equilibrium of concurrent forces – triangle law, parallelogram law and Lami’s theorem – experimental proof. Uniform circular motion – angular velocity – angular acceleration – relation between linear and angular velocities. Centripetal force – motion in a vertical circle – bending of cyclist – vehicle on level circular road – vehicle on banked road. Work done by a constant force and a variable force – unit of work.  Energy – Kinetic energy, work – energy theorem – potential energy – power. Collisions – Elastic and in-elastic collisions in one dimension. UNIT – 3 Dynamics of Rotational Motion (14 periods) Centre of a two particle system – generalization – applications – equilibrium of bodies, rigid body rotation and equations of rotational motion. Comparison of linear and rotational motions. Moment of inertia and its physical significance – radius of gyration – Theorems with proof, Moment of inertia of a thin straight rod, circular ring, disc cylinder and sphere. Moment of force, angular momentum. Torque – conservation of angular momentum. UNIT – 4 Gravitation and Space Science (16 periods) The universal law of gravitation; acceleration due to gravity and its variation with the altitude, latitude, depth and rotation of the Earth. – mass of the Earth. Inertial and gravitational mass. Gravitational field strength – gravitational potential – gravitational potential energy near the surface of the Earth – escape velocity – orbital velocity – weightlessness – motion of satellite – rocket propulsion – launching a satellite – orbits and energy. Geo stationary and polar satellites – applications – fuels used in rockets – Indian satellite programme. Solar system – Helio, Geo centric theory – Kepler’s laws of planetary motion. Sun – nine planets – asteroids – comets – meteors – meteroites – size of the planets – mass of the planet – temperature and atmosphere. Universe – stars – constellations – galaxies – Milky Way galaxy - origin of universe. UNIT – 5 Mechanics of Solids and Fluids (18 periods) States of matter- inter-atomic and inter-molecular forces. Solids – elastic behaviour, stress – strain relationship, Hooke’s law – experimental verification of Hooke’s law – three types of moduli of elasticity – applications (crane, bridge). Pressure due to a fluid column – Pascal’s law and its applications (hydraulic lift and hydraulic brakes) – effect of gravity on fluid pressure.  Surface energy and surface tension, angle of contact – application of surface tension in (i) formation of drops and bubbles (ii) capillary rise (iii) action of detergents. Viscosity – Stoke’s law – terminal velocity, streamline flow – turbulant flow – Reynold’s number – Bernoulli’s theorem – applications – lift on an aeroplane wing. UNIT – 6 Oscillations (12 periods) Periodic motion – period, frequency, displacement as a function of time. Simple harmonic motion – amplitude, frequency, phase – uniform circular motion as SHM. Oscillations of a spring, liquid column and simple pendulum – derivation of expression for time period – restoring force – force constant. Energy in SHM. kinetic and potential energies – law of conservation of energy. Free, forced and damped oscillations. Resonance. UNIT – 7 Wave Motion (17 periods) Wave motion- longitudinal and transverse waves – relation between v, n, λ. Speed of wave motion in different media – Newton’s formula – Laplace’s correction. Progressive wave – displacement equation –characteristics. Superposition principle, Interference – intensity and sound level – beats, standing waves (mathematical treatment) – standing waves in strings and pipes – sonometer – resonance air column – fundamental mode and harmonics. Doppler effect (quantitative idea) – applications. UNIT – 8 Heat and Thermodynamics (17 periods) Kinetic theory of gases – postulates – pressure of a gas – kinetic energy and temperature – degrees of freedom (mono atomic, diatomic and triatomic) – law of equipartition of energy – Avogadro’s number. Thermal equilibrium and temperature (zeroth law of thermodynamics) Heat, work and internal energy. Specific heat – specific heat capacity of gases at constant volume and pressure. Relation between Cp and Cv. First law of thermodynamics – work done by thermodynamical system – Reversible and irreversible processes – isothermal and adiabatic processes – Carnot engine – refrigerator - efficiency – second law of thermodynamics. Transfer of heat – conduction, convection and radiation – Thermal conductivity of solids – black body radiation – Prevost’s theory – Kirchoff’s law – Wien’s displacement law, Stefan’s law (statements only). Newton’s law of cooling – solar constant and surface temperature of the Sun- pyrheliometer. UNIT – 9 Ray Optics (16 periods) Reflection of light – reflection at plane and curved surfaces. Total internal refelction and its applications – determination of velocity of light – Michelson’s method. Refraction – spherical lenses – thin lens formula, lens makers formula – magnification – power of a lens – combination of thin lenses in contact. Refraction of light through a prism – dispersion – spectrometer – determination of µ – rainbow. UNIT – 10 Magnetism (10 periods) Earth’s magnetic field and magnetic elements. Bar magnet - magnetic field lines Magnetic field due to magnetic dipole (bar magnet) along the axis and perpendicular to the axis. Torque on a magnetic dipole (bar magnet) in a uniform magnetic field. Tangent law – Deflection magnetometer - Tan A and Tan B positions. Magnetic properties of materials – Intensity of magnetisation, magnetic susceptibility, magnetic induction and permeability Dia, Para and Ferromagnetic substances with examples. Hysteresis.  EXPERIMENTS (12 × 2 = 24 periods) 1. To find the density of the material of a given wire with the help of a screw gauge and a physical balance. 2. Simple pendulum - To draw graphs between (i) L and T and (ii) L and T2 and to decide which is better. Hence to determine the acceleration due to gravity. 3. Measure the mass and dimensions of (i) cylinder and (ii) solid sphere using the vernier calipers and physical balance. Calculate the moment of inertia. 4. To determine Young’s modulus of the material of a given wire by using Searles’ apparatus. 5. To find the spring constant of a spring by method of oscillations. 6. To determine the coefficient of viscosity by Poiseuille’s flow method. 7. To determine the coefficient of viscosity of a given viscous liquid by measuring the terminal velocity of a given spherical body. 8. To determine the surface tension of water by capillary rise method. 9. To verify the laws of a stretched string using a sonometer. 10. To find the velocity of sound in air at room temperature using the resonance column apparatus. 11. To determine the focal length of a concave mirror 12. To map the magnetic field due to a bar magnet placed in the magnetic meridian with its (i) north pole pointing South and (ii) north pole pointing North and locate the null points.  CONTENTS Page No. Mathematical Notes................................ 1 1. Nature of the Physical World and Measurement................................... 13 2. Kinematics.............................................. 37 3. Dynamics of Rotational Motion.............. 120 4. Gravitation and Space Science............. 149 5. Mechanics of Solids and Fluids............ 194 Annexure................................................. 237 Logarithmic and other tables................ 252 (Unit 6 to 10 continues in Volume II)  1. Nature of the Physical World and Measurement The history of humans reveals that they have been making continuous and serious attempts to understand the world around them. The repetition of day and night, cycle of seasons, volcanoes, rainbows, eclipses and the starry night sky have always been a source of wonder and subject of thought. The inquiring mind of humans always tried to understand the natural phenomena by observing the environment carefully. This pursuit of understanding nature led us to today’s modern science and technology. 1.1 Physics The word science comes from a Latin word “scientia” which means ‘to know’. Science is nothing but the knowledge gained through the systematic observations and experiments. Scientific methods include the systematic observations, reasoning, modelling and theoretical prediction. Science has many disciplines, physics being one of them. The word physics has its origin in a Greek word meaning ‘nature’. Physics is the most basic science, which deals with the study of nature and natural phenomena. Understanding science begins with understanding physics. With every passing day, physics has brought to us deeper levels of understanding of nature. Physics is an empirical study. Everything we know about physical world and about the principles that govern its behaviour has been learned through observations of the phenomena of nature. The ultimate test of any physical theory is its agreement with observations and measurements of physical phenomena. Thus physics is inherently a science of measurement. 1.1.1 Scope of Physics The scope of physics can be understood if one looks at its various sub-disciplines such as mechanics, optics, heat and thermodynamics, electrodynamics, atomic physics, nuclear physics, etc. 13 Mechanics deals with motion of particles and general systems of particles. The working of telescopes, colours of thin films are the topics dealt in optics. Heat and thermodynamics deals with the pressure - volume changes that take place in a gas when its temperature changes, working of refrigerator, etc. The phenomena of charged particles and magnetic bodies are dealt in electrodynamics. The magnetic field around a current carrying conductor, propagation of radio waves etc. are the areas where electrodynamics provide an answer. Atomic and nuclear physics deals with the constitution and structure of matter, interaction of atoms and nuclei with electrons, photons and other elementary particles. Foundation of physics enables us to appreciate and enjoy things and happenings around us. The laws of physics help us to understand and comprehend the cause-effect relationships in what we observe. This makes a complex problem to appear pretty simple. Physics is exciting in many ways. To some, the excitement comes from the fact that certain basic concepts and laws can explain a range of phenomena. For some others, the thrill lies in carrying out new experiments to unravel the secrets of nature. Applied physics is even more interesting. Transforming laws and theories into useful applications require great ingenuity and persistent effort. 1.1.2 Physics, Technology and Society Technology is the application of the doctrines in physics for practical purposes. The invention of steam engine had a great impact on human civilization. Till 1933, Rutherford did not believe that energy could be tapped from atoms. But in 1938, Hann and Meitner discovered neutron-induced fission reaction of uranium. This is the basis of nuclear weapons and nuclear reactors. The contribution of physics in the development of alternative resources of energy is significant. We are consuming the fossil fuels at such a very fast rate that there is an urgent need to discover new sources of energy which are cheap. Production of electricity from solar energy and geothermal energy is a reality now, but we have a long way to go. Another example of physics giving rise to technology is the integrated chip, popularly called as IC. The development of newer ICs and faster processors made the computer industry to grow leaps and bounds in the last two decades. Computers have become affordable now due to improved production techniques 14 and low production costs. The legitimate purpose of technology is to serve poeple. Our society is becoming more and more science-oriented. We can become better members of society if we develop an understanding of the basic laws of physics. 1.2 Forces of nature Sir Issac Newton was the first one to give an exact definition for force. “Force is the external agency applied on a body to change its state of rest and motion”. There are four basic forces in nature. They are gravitational force, electromagnetic force, strong nuclear force and weak nuclear force. Gravitational force It is the force between any two objects in the universe. It is an attractive force by virtue of their masses. By Newton’s law of gravitation, the gravitational force is directly proportional to the product of the masses and inversely proportional to the square of the distance between them. Gravitational force is the weakest force among the fundamental forces of nature but has the greatest large−scale impact on the universe. Unlike the other forces, gravity works universally on all matter and energy, and is universally attractive. Electromagnetic force It is the force between charged particles such as the force between two electrons, or the force between two current carrying wires. It is attractive for unlike charges and repulsive for like charges. The electromagnetic force obeys inverse square law. It is very strong compared to the gravitational force. It is the combination of electrostatic and magnetic forces. Strong nuclear force It is the strongest of all the basic forces of nature. It, however, has the shortest range, of the order of 10−15 m. This force holds the protons and neutrons together in the nucleus of an atom. 15 Weak nuclear force Weak nuclear force is important in certain types of nuclear process such as β-decay. This force is not as weak as the gravitational force. 1.3 Measurement Physics can also be defined as the branch of science dealing with the study of properties of materials. To understand the properties of materials, measurement of physical quantities such as length, mass, time etc., are involved. The uniqueness of physics lies in the measurement of these physical quantities. 1.3.1 Fundamental quantities and derived quantities Physical quantities can be classified into two namely, fundamental quantities and derived quantities. Fundamental quantities are quantities which cannot be expressed in terms of any other physical quantity. For example, quantities like length, mass, time, temperature are fundamental quantities. Quantities that can be expressed in terms of fundamental quantities are called derived quantities. Area, volume, density etc. are examples for derived quantities. 1.3.2 Unit To measure a quantity, we always compare it with some reference standard. To say that a rope is 10 metres long is to say that it is 10 times as long as an object whose length is defined as 1 metre. Such a standard is called a unit of the quantity. Therefore, unit of a physical quantity is defined as the established standard used for comparison of the given physical quantity. The units in which the fundamental quantities are measured are called fundamental units and the units used to measure derived quantities are called derived units. 1.3.3 System International de Units (SI system of units) In earlier days, many system of units were followed to measure physical quantities. The British system of foot−pound−second or fps system, the Gaussian system of centimetre − gram − second or cgs system, the metre−kilogram − second or the mks system were the three 16 systems commonly followed. To bring uniformity, the General Conference on Weights and Measures in the year 1960, accepted the SI system of units. This system is essentially a modification over mks system and is, therefore rationalised mksA (metre kilogram second ampere) system. This rationalisation was essential to obtain the units of all the physical quantities in physics. In the SI system of units there are seven fundamental quantities and two supplementary quantities. They are presented in Table 1.1. Table 1.1 SI system of units Physical quantity Unit Symbol Fundamental quantities Length metre m Mass kilogram kg Time second s Electric current ampere A Temperature kelvin K Luminous intensity candela cd Amount of substance mole mol Supplementary quantities Plane angle radian rad Solid angle steradian sr 1.3.4 Uniqueness of SI system The SI system is logically far superior to all other systems. The SI units have certain special features which make them more convenient in practice. Permanence and reproduceability are the two important characteristics of any unit standard. The SI standards do not vary with time as they are based on the properties of atoms. Further SI system of units are coherent system of units, in which the units of derived quantities are obtained as multiples or submultiples of certain basic units. Table 1.2 lists some of the derived quantities and their units. 17  Table 1.2 Derived quantities and their units Physical Quantity Expression Unit Area length × breadth m2 Volume area × height m3 Velocity displacement/ time m s–1 Acceleration velocity / time m s–2 Angular velocity angular displacement / time rad s–1 Angular acceleration angular velocity / time rad s-2 Density mass / volume kg m−3 Momentum mass × velocity kg m s−1 Moment of intertia mass × (distance)2 kg m2 Force mass × acceleration kg m s–2 or N Pressure force / area N m-2 or Pa Energy (work) force × distance N m or J Impulse force × time N s Surface tension force / length N m-1 Moment of force (torque) force × distance N m Electric charge current × time A s Current density current / area A m–2 Magnetic induction force / (current × length) N A–1 m–1 1.3.5 SI standards Length Length is defined as the distance between two points. The SI unit of length is metre. One standard metre is equal to 1 650 763.73 wavelengths of the orange − red light emitted by the individual atoms of krypton − 86 in a krypton discharge lamp. Mass Mass is the quantity of matter contained in a body. It is independent of temperature and pressure. It does not vary from place 18 to place. The SI unit of mass is kilogram. The kilogram is equal to the mass of the international prototype of the kilogram (a plantinum − iridium alloy cylinder) kept at the International Bureau of Weights and Measures at Sevres, near Paris, France. An atomic standard of mass has not yet been adopted because it is not yet possible to measure masses on an atomic scale with as much precision as on a macroscopic scale. Time Until 1960 the standard of time was based on the mean solar day, the time interval between successive passages of the sun at its highest point across the meridian. It is averaged over an year. In 1967, an atomic standard was adopted for second, the SI unit of time. One standard second is defined as the time taken for 9 192 631 770 periods of the radiation corresponding to unperturbed transition between hyperfine levels of the ground state of cesium − 133 atom. Atomic clocks are based on this. In atomic clocks, an error of one second occurs only in 5000 years. Ampere The ampere is the constant current which, flowing through two straight parallel infinitely long conductors of negligible cross-section, and placed in vacuum 1 m apart, would produce between the conductors a force of 2 × 10 -7 newton per unit length of the conductors. Kelvin 1 The Kelvin is the fraction of of the thermodynamic 273.16 temperature of the triple point of water*. Candela The candela is the luminous intensity in a given direction due to a * Triple point of water is the temperature at which saturated water vapour, pure water and melting ice are all in equilibrium. The triple point temperature of water is 273.16 K. 19 source, which emits monochromatic radiation of frequency 540 × 1012 Hz 1 and of which the radiant intensity in that direction is watt per steradian. 683 Mole The mole is the amount of substance which contains as many elementary entities as there are atoms in 0.012 kg of carbon-12. 1.3.6 Rules and conventions for writing SI units and their symbols 1. The units named after scientists are not written with a capital initial letter. For example : newton, henry, watt 2. The symbols of the units named after scientist should be written by a capital letter. For example : N for newton, H for henry, W for watt 3. Small letters are used as symbols for units not derived from a proper name. For example : m for metre, kg for kilogram 4. No full stop or other punctuation marks should be used within or at the end of symbols. For example : 50 m and not as 50 m. 5. The symbols of the units do not take plural form. For example : 10 kg not as 10 kgs 6. When temperature is expressed in kelvin, the degree sign is omitted. For example : 273 K not as 273o K (If expressed in Celsius scale, degree sign is to be included. For example 100o C and not 100 C) 7. Use of solidus is recommended only for indicating a division of one letter unit symbol by another unit symbol. Not more than one solidus is used. For example : m s−1 or m / s, J / K mol or J K–1 mol–1 but not J / K / mol. 20  8. Some space is always to be left between the number and the symbol of the unit and also between the symbols for compound units such as force, momentum, etc. For example, it is not correct to write 2.3m. The correct representation is 2.3 m; kg m s–2 and not as kgms-2. 9. Only accepted symbols should be used. For example : ampere is represented as A and not as amp. or am ; second is represented as s and not as sec. 10. Numerical value of any physical quantity should be expressed in scientific notation. For an example, density of mercury is 1.36 × 104 kg m−3 and not as 13600 kg m−3. 1.4 Expressing larger and smaller physical quantities Once the fundamental Table 1.3 Prefixes for power of ten units are defined, it is easier Power of ten Prefix Abbreviation to express larger and smaller 10−15 femto f units of the same physical quantity. In the metric (SI) 10−12 pico p system these are related to the 10−9 nano n fundamental unit in multiples 10−6 micro µ of 10 or 1/10. Thus 1 km is 10−3 milli m 1000 m and 1 mm is 1/1000 metre. Table 1.3 lists the 10−2 centi c standard SI prefixes, their 10−1 deci d meanings and abbreviations. 101 deca da In order to measure very 102 hecto h large distances, the following 103 kilo k units are used. 106 mega M (i) Light year 109 giga G Light year is the distance 1012 tera T travelled by light in one year 1015 peta P in vacuum. 21  Distance travelled = velocity of light × 1 year ∴ 1 light year = 3 × 108 m s−1 × 1 year (in seconds) = 3 × 108 × 365.25 × 24 × 60 × 60 = 9.467 × 1015 m 1 light year = 9.467 × 1015 m (ii) Astronomical unit Astronomical unit is the mean distance of the centre of the Sun from the centre of the Earth. 1 Astronomical unit (AU) = 1.496 × 1011 m 1.5 Determination of distance For measuring large distances such as the distance of moon or a planet from the Earth, special methods are adopted. Radio-echo method, laser pulse method and parallax method are used to determine very large distances. Laser pulse method The distance of moon from the Earth can be determined using laser pulses. The laser pulses are beamed towards the moon from a powerful transmitter. These pulses are reflected back from the surface of the moon. The time interval between sending and receiving of the signal is determined very accurately. If t is the time interval and c the velocity of the laser pulses, then ct the distance of the moon from the Earth is d =. 2 1.6 Determination of mass The conventional method of finding the mass of a body in the laboratory is by physical balance. The mass can be determined to an accuracy of 1 mg. Now−a−days, digital balances are used to find the mass very accurately. The advantage of digital balance is that the mass of the object is determined at once. 1.7 Measurement of time We need a clock to measure any time interval. Atomic clocks provide better standard for time. Some techniques to measure time interval are given below. 22 Quartz clocks The piezo−electric property* of a crystal is the principle of quartz clock. These clocks have an accuracy of one second in every 109 seconds. Atomic clocks These clocks make use of periodic vibration taking place within the atom. Atomic clocks have an accuracy of 1 part in 1013 seconds. 1.8 Accuracy and precision of measuring instruments All measurements are made with the help of instruments. The accuracy to which a measurement is made depends on several factors. For example, if length is measured using a metre scale which has graduations at 1 mm interval then all readings are good only upto this value. The error in the use of any instrument is normally taken to be half of the smallest division on the scale of the instrument. Such an error is called instrumental error. In the case of a metre scale, this error is about 0.5 mm. Physical quantities obtained from experimental observation always have some uncertainity. Measurements can never be made with absolute precision. Precision of a number is often indicated by following it with ± symbol and a second number indicating the maximum error likely. For example, if the length of a steel rod = 56.47 ± 3 mm then the true length is unlikely to be less than 56.44 mm or greater than 56.50 mm. If the error in the measured value is expressed in fraction, it is called fractional error and if expressed in percentage it is called percentage error. For example, a resistor labelled “470 Ω, 10%” probably has a true resistance differing not more than 10% from 470 Ω. So the true value lies between 423 Ω and 517 Ω. 1.8.1 Significant figures The digits which tell us the number of units we are reasonably sure of having counted in making a measurement are called significant figures. Or in other words, the number of meaningful digits in a number is called 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. * When pressure is applied along a particular axis of a crystal, an electric potential difference is developed in a perpendicular axis. 23  For example, 2.868 cm has four significant figures. But in different units, the same can be written as 0.02868 m or 28.68 mm or 28680 µm. All these numbers have the same four significant figures. From the above example, we have the following rules. i) All the non−zero digits in a number are significant. ii) All the zeroes between two non−zeroes digits are significant, irrespective of the decimal point. iii) If the number is less than 1, the zeroes on the right of decimal point but to the left of the first non−zero digit are not significant. (In 0.02868 the underlined zeroes are not significant). iv) The zeroes at the end without a decimal point are not significant. (In 23080 µm, the trailing zero is not significant). v) The trailing zeroes in a number with a decimal point are significant. (The number 0.07100 has four significant digits). Examples i) 30700 has three significant figures. ii) 132.73 has five significant figures. iii) 0.00345 has three and iv) 40.00 has four significant figures. 1.8.2 Rounding off Calculators are widely used now−a−days to do the calculations. The result given by a calculator has too many figures. In no case the result should have more significant figures than the figures involved in the data used for calculation. The result of calculation with number containing more than one uncertain digit, should be rounded off. The technique of rounding off is followed in applied areas of science. A number 1.876 rounded off to three significant digits is 1.88 while the number 1.872 would be 1.87. The rule is that if the insignificant digit (underlined) is more than 5, the preceeding digit is raised by 1, and is left unchanged if the former is less than 5. If the number is 2.845, the insignificant digit is 5. In this case, the convention is that if the preceeding digit is even, the insignificant digit is simply dropped and, if it is odd, the preceeding digit is raised by 1. Following this, 2.845 is rounded off to 2.84 where as 2.815 is rounded off to 2.82. 24 Examples 1. Add 17.35 kg, 25.8 kg and 9.423 kg. Of the three measurements given, 25.8 kg is the least accurately known. ∴ 17.35 + 25.8 + 9.423 = 52.573 kg Correct to three significant figures, 52.573 kg is written as 52.6 kg 2. Multiply 3.8 and 0.125 with due regard to significant figures. 3.8 × 0.125 = 0.475 The least number of significant figure in the given quantities is 2. Therefore the result should have only two significant figures. ∴ 3.8 × 0.125 = 0.475 = 0.48 1.8.3 Errors in Measurement The uncertainity in the measurement of a physical quantity is called error. It is the difference between the true value and the measured value of the physical quantity. Errors may be classified into many categories. (i) Constant errors It is the same error repeated every time in a series of observations. Constant error is due to faulty calibration of the scale in the measuring instrument. In order to minimise constant error, measurements are made by different possible methods and the mean value so obtained is regarded as the true value. (ii) Systematic errors These are errors which occur due to a certain pattern or system. These errors can be minimised by identifying the source of error. Instrumental errors, personal errors due to individual traits and errors due to external sources are some of the systematic errors. (iii) Gross errors Gross errors arise due to one or more than one of the following reasons. (1) Improper setting of the instrument. 25  (2) Wrong recordings of the observation. (3) Not taking into account sources of error and precautions. (4) Usage of wrong values in the calculation. Gross errros can be minimised only if the observer is very careful in his observations and sincere in his approach. (iv) Random errors It is very common that repeated measurements of a quantity give values which are slightly different from each other. These errors have no set pattern and occur in a random manner. Hence they are called random errors. They can be minimised by repeating the measurements many times and taking the arithmetic mean of all the values as the correct reading. The most common way of expressing an error is percentage error. If the accuracy in measuring a quantity x is ∆x, then the percentage ∆x error in x is given by × 100 %. x 1.9 Dimensional Analysis Dimensions of a physical quantity are the powers to which the fundamental quantities must be raised. displacement We know that velocity = time [L] = [T ] = [MoL1T−1] where [M], [L] and [T] are the dimensions of the fundamental quantities mass, length and time respectively. Therefore velocity has zero dimension in mass, one dimension in length and −1 dimension in time. Thus the dimensional formula for velocity is [MoL1T−1] or simply [LT−1].The dimensions of fundamental quantities are given in Table 1.4 and the dimensions of some derived quantities are given in Table 1.5 26  Table 1.4 Dimensions of fundamental quantities Fundamental quantity Dimension Length L Mass M Time T Temperature K Electric current A Luminous intensity cd Amount of subtance mol Table 1.5 Dimensional formulae of some derived quantities Physical quantity Expression Dimensional formula Area length × breadth [L2] Density mass / volume [ML−3] Acceleration velocity / time [LT−2 ] Momentum mass × velocity [MLT−1] Force mass × acceleration [MLT−2 ] Work force × distance [ML2T−2 ] Power work / time [ML2T−3 ] Energy work [ML2T−2 ] Impulse force × time [MLT−1 ] Radius of gyration distance [L] Pressure force / area [ML−1T−2 ] Surface tension force / length [MT−2 ] Frequency 1 / time period [T−1] Tension force [MLT−2 ] Moment of force (or torque) force × distance [ML2T−2 ] Angular velocity angular displacement / time [T−1] Stress force / area [ML−1T−2] Heat energy [ML2T−2 ] Heat capacity heat energy/ temperature [ML2T-2K-1] Charge current × time [AT] Faraday constant Avogadro constant × elementary charge [AT mol-1] Magnetic induction force / (current × length) [MT-2 A-1] 27 Dimensional quantities Constants which possess dimensions are called dimensional constants. Planck’s constant, universal gravitational constant are dimensional constants. Dimensional variables are those physical quantities which possess dimensions but do not have a fixed value. Example − velocity, force, etc. Dimensionless quantities There are certain quantities which do not possess dimensions. They are called dimensionless quantities. Examples are strain, angle, specific gravity, etc. They are dimensionless as they are the ratio of two quantities having the same dimensional formula. Principle of homogeneity of dimensions An equation is dimensionally correct if the dimensions of the various terms on either side of the equation are the same. This is called the principle of homogeneity of dimensions. This principle is based on the fact that two quantities of the same dimension only can be added up, the resulting quantity also possessing the same dimension. The equation A + B = C is valid only if the dimensions of A, B and C are the same. 1.9.1 Uses of dimensional analysis The method of dimensional analysis is used to (i) convert a physical quantity from one system of units to another. (ii) check the dimensional correctness of a given equation. (iii) establish a relationship between different physical quantities in an equation. (i) To convert a physical quantity from one system of units to another Given the value of G in cgs system is 6.67 × 10−8dyne cm2 g−2. Calculate its value in SI units. In cgs system In SI system Gcgs = 6.67 × 10−8 G = ? M1 = 1g M2 = 1 kg L1 = 1 cm L2 = 1m T1 = 1s T2 = 1s 28  The dimensional formula for gravitational constant is ⎡⎣M −1L3T −2 ⎤⎦. In cgs system, dimensional formula for G is ⎡⎣M1x L1 T1z ⎤⎦ y In SI system, dimensional formula for G is ⎡M 2x Ly2 T2z ⎤ ⎣ ⎦ Here x = −1, y = 3, z = −2 ∴ G ⎡⎣M 2x L2yT2z ⎤⎦ = Gcgs ⎡M1x L1yT1z ⎤ ⎣ ⎦ x y z ⎡ M1 ⎤ ⎡ L1 ⎤ ⎡T1 ⎤ or G = Gcgs ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣M2 ⎦ ⎣ L2 ⎦ ⎣T2 ⎦ −1 3 −2 ⎡1g ⎤ ⎡1 cm ⎤ ⎡1 s ⎤ = 6.67 × 10−8 ⎢ ⎥ ⎢⎣ 1 m ⎥⎦ ⎢⎣1 s ⎥⎦ ⎣1 kg ⎦ −1 3 ⎡ 1g ⎤ ⎡ 1 cm ⎤ −2 = 6.67 × 10−8 ⎢ ⎥ ⎢100 cm ⎥ ⎣1000 g ⎦ ⎣ ⎦ = 6.67 × 10−11 ∴ In SI units, G = 6.67 × 10−11 N m2 kg−2 (ii) To check the dimensional correctness of a given equation Let us take the equation of motion s = ut + (½)at2 Applying dimensions on both sides, [L] = [LT−1] [T] + [LT−2] [T2] (½ is a constant having no dimension) [L] = [L] + [L] As the dimensions on both sides are the same, the equation is dimensionally correct. (iii) To establish a relationship between the physical quantities in an equation Let us find an expression for the time period T of a simple pendulum. The time period T may depend upon (i) mass m of the bob (ii) length l of the pendulum and (iii) acceleration due to gravity g at the place where the pendulum is suspended. 29  (i.e) T α mx l y gz or T = k mx l y gz...(1) where k is a dimensionless constant of propotionality. Rewriting equation (1) with dimensions, [T1] = [Mx] [L y] [LT−2]z [T1] = [Mx L y + z T−2z] Comparing the powers of M, L and T on both sides x = 0, y + z = 0 and −2z = 1 Solving for x, y and z, x = 0, y = ½ and z = –½ From equation (1), T = k mo l½ g−½ 1/2 ⎡l ⎤ l T = k ⎢ ⎥ = k g ⎣g ⎦ Experimentally the value of k is determined to be 2π. l ∴ T = 2π g 1.9.2 Limitations of Dimensional Analysis (i) The value of dimensionless constants cannot be determined by this method. (ii) This method cannot be applied to equations involving exponential and trigonometric functions. (iii) It cannot be applied to an equation involving more than three physical quantities. (iv) It can check only whether a physical relation is dimensionally correct or not. It cannot tell whether the relation is absolutely correct 1 2 or not. For example applying this technique s = ut + at is dimensionally 4 1 2 correct whereas the correct relation is s = ut + at. 2 30  Solved Problems 1.1 A laser signal is beamed towards a distant planet from the Earth and its reflection is received after seven minutes. If the distance between the planet and the Earth is 6.3 × 1010 m, calculate the velocity of the signal. Data : d = 6.3 × 1010 m, t = 7 minutes = 7 × 60 = 420 s Solution : If d is the distance of the planet, then total distance travelled by the signal is 2d. 2d 2 × 6.3 × 1010 ∴ velocity = = = 3 × 108 m s −1 t 420 1.2 A goldsmith put a ruby in a box weighing 1.2 kg. Find the total mass of the box and ruby applying principle of significant figures. The mass of the ruby is 5.42 g. Data : Mass of box = 1.2 kg Mass of ruby = 5.42 g = 5.42 × 10–3 kg = 0.00542 kg Solution: Total mass = mass of box + mass of ruby = 1.2 + 0.00542 = 1.20542 kg After rounding off, total mass = 1.2 kg h 1.3 Check whether the equation λ = is dimensionally correct mv (λ - wavelength, h - Planck’s constant, m - mass, v - velocity). Solution: Dimension of Planck’s constant h is [ML2 T–1] Dimension of λ is [L] Dimension of m is [M] Dimension of v is [LT–1] h Rewriting λ= using dimension mv ⎡ML2T −1 ⎤⎦ [L ] = ⎣ [M ] ⎡⎣LT −1 ⎤⎦ [L ] = [L ] As the dimensions on both sides of the equation are same, the given equation is dimensionally correct. 31 1.4 Multiply 2.2 and 0.225. Give the answer correct to significant figures. Solution : 2.2 × 0.225 = 0.495 Since the least number of significant figure in the given data is 2, the result should also have only two significant figures. ∴ 2.2 × 0.225 = 0.50 1.5 Convert 76 cm of mercury pressure into N m-2 using the method of dimensions. Solution : In cgs system, 76 cm of mercury pressure = 76 × 13.6 × 980 dyne cm–2 Let this be P1. Therefore P1 = 76 × 13.6 × 980 dyne cm–2 In cgs system, the dimension of pressure is [M1aL1bT1c] Dimension of pressure is [ML–1 T–2]. Comparing this with [M2aL2bT2c] we have a = 1, b = –1 and c = -2. a b c ∴ Pressure in SI system P2 = P1 ⎡⎢ M1 ⎤⎥ ⎡⎢ L1 ⎤⎥ ⎡⎢ T1 ⎤⎥ ⎣ M 2 ⎦ ⎣ L2 ⎦ ⎣T2 ⎦ 1 −1 −2 ⎡10-3 kg ⎤ ⎡10-2 m ⎤ ⎡1s ⎤ ie P2 = 76×13.6×980 ⎢ 1 kg ⎥ ⎢ ⎥ ⎢⎣1s ⎥⎦ ⎣ ⎦ ⎣ 1m ⎦ = 76 × 13.6 × 980 ×10–3 ×102 = 101292.8 N m-2 P2 = 1.01 × 105 N m-2 32  Self evaluation (The questions and problems given in this self evaluation are only samples. In the same way any question and problem could be framed from the text matter. Students must be prepared to answer any question and problem from the text matter, not only from the self evaluation.) 1.1 Which of the following are equivalent? (a) 6400 km and 6.4 × 108 cm (b) 2 × 104 cm and 2 × 106 mm (c) 800 m and 80 × 102 m (d) 100 µm and 1 mm 1.2 Red light has a wavelength of 7000 Å. In µm it is (a) 0.7 µm (b) 7 µm (c) 70 µm (d) 0.07 µm 1.3 A speck of dust weighs 1.6 × 10–10 kg. How many such particles would weigh 1.6 kg? (a) 10–10 (b) 1010 (c) 10 (d) 10–1 1.4 The force acting on a particle is found to be proportional to velocity. The constant of proportionality is measured in terms of (a) kg s-1 (b) kg s (c) kg m s-1 (d) kg m s-2 1.5 The number of significant digits in 0.0006032 is (a) 8 (b) 7 (c) 4 (d) 2 1.6 The length of a body is measured as 3.51 m. If the accuracy is 0.01 m, then the percentage error in the measurement is (a) 351 % (b) 1 % (c) 0.28 % (d) 0.035 % 1.7 The dimensional formula for gravitational constant is 1 3 –2 –1 3 –2 (a) M L T (b) M L T –1 –3 –2 1 –3 2 (c) M L T (d) M L T 33 1.8 The velocity of a body is expressed as v = (x/t) + yt. The dimensional formula for x is (a) MLoTo (b) MoLTo (c) MoLoT (d) MLTo 1.9 The dimensional formula for Planck’s constant is 3 2 (a) MLT (b) ML T o 4 2 –1 (c) ML T (d) ML T 1.10 _____________have the same dimensional formula (a) Force and momentum (b) Stress and strain (c) Density and linear density (d) Work and potential energy 1.11 What is the role of Physics in technology? 1.12 Write a note on the basic forces in nature. 1.13 Distinguish between fundamental units and derived units. 1.14 Give the SI standard for (i) length (ii) mass and (iii) time. 1.15 Why SI system is considered superior to other systems? 1.16 Give the rules and conventions followed while writing SI units. 1.17 What is the need for measurement of physical quantities? 1.18 You are given a wire and a metre scale. How will you estimate the diameter of the wire? 1.19 Name four units to measure extremely small distances. 1.20 What are random errors? How can we minimise these errors? 1 1.21 Show that gt2 has the same dimensions of distance. 2 1.22 What are the limitations of dimensional analysis? 1.23 What are the uses of dimensional analysis? Explain with one example. Problems 1.24 How many astronomical units are there in 1 metre? 34 1.25 If mass of an electron is 9.11 × 10–31 kg how many electrons would weigh 1 kg? 1.26 In a submarine fitted with a SONAR, the time delay between generation of a signal and reception of its echo after reflection from an enemy ship is observed to be 73.0 seconds. If the speed of sound in water is 1450 m s–1, then calculate the distance of the enemy ship. 1.27 State the number of significant figures in the following: (i) 600900 (ii) 5212.0 (iii) 6.320 (iv) 0.0631 (v) 2.64 × 1024 1.28 Find the value of π2 correct to significant figures, if π = 3.14. 1.29 5.74 g of a substance occupies a volume of 1.2 cm3. Calculate its density applying the principle of significant figures. 1.30 The length, breadth and thickness of a rectanglar plate are 4.234 m, 1.005 m and 2.01 cm respectively. Find the total area and volume of the plate to correct significant figures. 1.31 The length of a rod is measured as 25.0 cm using a scale having an accuracy of 0.1 cm. Determine the percentage error in length. 1.32 Obtain by dimensional analysis an expression for the surface tension of a liquid rising in a capillary tube. Assume that the surface tension T depends on mass m of the liquid, pressure P of the liquid and 1 radius r of the capillary tube (Take the constant k = 2 ). 1.33 The force F acting on a body moving in a circular path depends on mass m of the body, velocity v and radius r of the circular path. Obtain an expression for the force by dimensional analysis (Take the value of k = 1). 1.34 Check the correctness of the following equation by dimensinal analysis mv 2 (i) F= where F is force, m is mass, v is velocity and r is radius r2 1 g (ii) n = where n is frequency, g is acceleration due to gravity 2π l and l is length. 35  1 (iii) mv 2 = mgh 2 where m is mass, v is velocity, g is acceleration 2 due to gravity and h is height. 1.35 Convert using dimensional analysis 18 (i) kmph into m s–1 5 5 (ii) m s–1 into kmph 18 (iii) 13.6 g cm–3 into kg m–3 Answers 1.1 (a) 1.2 (a) 1.3 (b) 1.4 (a) 1.5 (c) 1.6 (c) 1.7 (b) 1.8 (b) 1.9 (d) 1.10 (d) 1.24 6.68 × 10–12 AU 1.25 1.097 × 1030 1.26 52.925 km 1.27 4, 5, 4, 3, 3 1.28 9.86 1.29 4.8 g cm–3 1.30 4.255 m2, 0.0855 m3 1.31 0.4 % Pr mv 2 1.32 T = 1.33 F = 2 r 1.34 wrong, correct, wrong 1.35 1 m s–1, 1 kmph, 1.36 × 104 kg m–3 36  2. Kinematics Mechanics is one of the oldest branches of physics. It deals with the study of particles or bodies when they are at rest or in motion. Modern research and development in the spacecraft design, its automatic control, engine performance, electrical machines are highly dependent upon the basic principles of mechanics. Mechanics can be divided into statics and dynamics. Statics is the study of objects at rest; this requires the idea of forces in equilibrium. Dynamics is the study of moving objects. It comes from the Greek word dynamis which means power. Dynamics is further subdivided into kinematics and kinetics. Kinematics is the study of the relationship between displacement, velocity, acceleration and time of a given motion, without considering the forces that cause the motion. Kinetics deals with the relationship between the motion of bodies and forces acting on them. We shall now discuss the various fundamental definitions in kinematics. Particle A particle is ideally just a piece or a quantity of matter, having practically no linear dimensions but only a position. Rest and Motion When a body does not change its position with respect to time, then it is said to be at rest. Motion is the change of position of an object with respect to time. To study the motion of the object, one has to study the change in position (x,y,z coordinates) of the object with respect to the surroundings. It may be noted that the position of the object changes even due to the change in one, two or all the three coordinates of the position of the 37 objects with respect to time. Thus motion can be classified into three types : (i) Motion in one dimension Motion of an object is said to be one dimensional, if only one of the three coordinates specifying the position of the object changes with respect to time. Example : An ant moving in a straight line, running athlete, etc. (ii) Motion in two dimensions In this type, the motion is represented by any two of the three coordinates. Example : a body moving in a plane. (iii) Motion in three dimensions Motion of a body is said to be three dimensional, if all the three coordinates of the position of the body change with respect to time. Examples : motion of a flying bird, motion of a kite in the sky, motion of a molecule, etc. 2.1 Motion in one dimension (rectilinear motion) The motion along a straight line is known as rectilinear motion. The important parameters required to study the motion along a straight line are position, displacement, velocity, and acceleration. 2.1.1 Position, displacement and distance travelled by the particle The motion of a particle can be described if its position is known continuously with respect to time. The total length of the path is the distance travelled by the particle and the shortest distance between the initial and final position of the particle is the displacement. The distance travelled by a particle, however, is different from its displacement from the origin. For example, if the particle moves from a Fig 2.1 Distance and displacement point O to position P1 and then to 38 position P2, its displacement at the position P2 is – x2 from the origin but, the distance travelled by the particle is x1+x1+x2 = (2x1+x2) (Fig 2.1). The distance travelled is a scalar quantity and the displacement is a vector quantity. 2.1.2 Speed and velocity Speed It is the distance travelled in unit time. It is a scalar quantity. Velocity The velocity of a particle is defined as the rate of change of displacement of the particle. It is also defined as the speed of the particle in a given direction. The velocity is a vector quantity. It has both magnitude and direction. displacement Velocity = time taken Its unit is m s−1 and its dimensional formula is LT−1. Uniform velocity A particle is said to move with uniform velocity if it moves along a fixed direction and covers equal displacements in equal intervals of time, however small these intervals of time may be. In a displacement - time graph, t (Fig. 2.2) the slope is constant at all the points, when the particle moves with uniform velocity. Fig. 2.2 Uniform velocity Non uniform or variable velocity The velocity is variable (non-uniform), if it covers unequal displacements in equal intervals of time or if the direction of motion changes or if both the rate of motion and the direction change. 39 Average velocity Let s1 be the displacement of a body in time t1 and s2 be its displacement in time t2 (Fig. 2.3). ∆s The average velocity during the time interval (t2 – t1) is defined as vaverage = change in displacement ∆t change in time s -s ∆s = t - t = ∆t 2 1 2 1 O From the graph, it is found Fig. 2.3 Average velocity that the slope of the curve varies. Instantaneous velocity It is the velocity at any given instant of time or at any given point of its path. The instantaneous velocity v is given by ∆s ds v = Lt = ∆t → 0 ∆ t dt 2.1.3 Acceleration If the magnitude or the direction or both of the velocity changes with respect to time, the particle is said to be under acceleration. Acceleration of a particle is defined as the rate of change of velocity. Acceleration is a vector quantity. change in velocity Acceleration = time taken If u is the initial velocity and v, the final velocity of the particle after a time t, then the acceleration, v −u a= t Its unit is m s−2 and its dimensional formula is LT−2. dv d ⎛ ds ⎞ d 2s The instantaneous acceleration is, a = = ⎜ ⎟= dt dt ⎝ dt ⎠ dt 2 Uniform acceleration If the velocity changes by an equal amount in equal intervals of time, however small these intervals of time may be, the acceleration is said to be uniform. 40 Retardation or deceleration If the velocity decreases with time, the acceleration is negative. The negative acceleration is called retardation or deceleration. Uniform motion A particle is in uniform motion when it moves with constant velocity (i.e) zero acceleration. 2.1.4 Graphical representations The graphs provide a convenient method to present pictorially, the basic informations about a variety of events. Line graphs are used to show the relation of one quantity say displacement or velocity with another quantity such as time. If the displacement, velocity and acceleration of a particle are plotted with respect to time, they are known as, (i) displacement – time graph (s - t graph) (ii) velocity – time graph (v - t graph) (iii) acceleration – time graph (a - t graph) Displacement – time graph When the displacement of the 2 particle is plotted as a function of time, it is displacement - time graph. ds As v = , the slope of the s - t 3 dt 1 graph at any instant gives the velocity of the particle at that instant. In Fig. 2.4 the particle at time t1, has a O t1 t2 t3 positive velocity, at time t2, has zero Fig. 2.4 Displacement - velocity and at time t3, has negative time graph velocity. Velocity – time graph When the velocity of the particle is plotted as a function of time, it is velocity-time graph. dv As a = dt , the slope of the v – t curve at any instant gives the 41 acceleration of the particle (Fig. 2.5). B ds A But, v = or ds = v.dt dt If the displacements are s1 and v dt s2 in times t1 and t2, then s2 t2 dt ∫ ds = ∫ v dt s1 t1 D C t2 O s2 – s1 = ∫ v dt = t1 area ABCD Fig. 2.5 Velocity - time graph The area under the v – t curve, between the given intervals of time, gives the change in displacement or the distance travelled by the particle during the same interval. Q P Acceleration – time graph When the acceleration is plotted as a a dt function of time, it is acceleration - time graph (Fig. 2.6). dt dv a = (or) dv = a dt S R dt O If the velocities are v1 and v2 at times Fig. 2.6 Acceleration t1 and t2 respectively, then – time graph v2 t2 t2 ∫ dv = ∫ a dt (or) v2 – v1 = ∫ a.dt = area PQRS v1 t1 t1 The area under the a – t curve, between the given intervals of time, gives the change in velocity of the particle during the same interval. If the graph is parallel to the time axis, the body moves with constant acceleration. 2.1.5 Equations of motion For uniformly accelerated motion, some simple equations that relate displacement s, time t, initial velocity u, final velocity v and acceleration a are obtained. (i) As acceleration of the body at any instant is given by the first derivative of the velocity with respect to time, 42  dv a = (or) dv = a.dt dt If the velocity of the body changes from u to v in time t then from the above equation, v t t ∫ dv = ∫ a dt = a ∫ dt ⇒ [v ]vu = a [t ]0 t u 0 0 ∴ v – u = at (or) v = u + at...(1) (ii) The velocity of the body is given by the first derivative of the displacement with respect to time. ds (i.e) v = (or) ds = v dt dt Since v = u + at, ds = (u + at) dt The distance s covered in time t is, s t t 1 2 ∫ ds = ∫ u dt + ∫ at dt (or) s = ut + 2 at...(2) 0 0 0 (iii) The acceleration is given by the first derivative of velocity with respect to time. (i.e) dv dv ds dv ⎡ ds ⎤ 1 a = dt = ds ⋅ dt = ds ⋅v ⎢⎣∵ v = dt ⎥⎦ (or) ds = a v dv Therefore, s v v dv 1 ⎡v 2 u 2 ⎤ ∫ ds = ∫ a (i.e) s = a ⎢2 − 2⎥ 0 u ⎣ ⎦ s = 1 2a (v 2 − u2 ) (or) 2as = (v2 – u2) ∴ v2 = u2 + 2 as...(3) The equations (1), (2) and (3) are called equations of motion. Expression for the distance travelled in nth second Let a body move with an initial velocity u and travel along a straight line with uniform acceleration a. Distance travelled in the nth second of motion is, sn = distance travelled during first n seconds – distance travelled during (n –1) seconds 43  Distance travelled during n seconds 1 2 Dn = un + an 2 Distance travelled during (n -1) seconds 1 D (n –1) = u(n-1) + a(n-1)2 2 ∴ the distance travelled in the nth second = Dn− D(n –1) ⎛ 1 2⎞ ⎡ 1 2⎤ (i.e) sn = ⎜ un + an ⎟ - ⎢u(n - 1) + a(n - 1) ⎥ ⎝ 2 ⎠ ⎣ 2 ⎦ ⎛ 1⎞ sn = u + a ⎜ n - ⎟ ⎝ 2⎠ 1 sn = u + a(2n - 1) 2 Special Cases Case (i) : For downward motion For a particle moving downwards, a = g, since the particle moves in the direction of gravity. Case (ii) : For a freely falling body For a freely falling body, a = g and u = 0, since it starts from rest. Case (iii) : For upward motion For a particle moving upwards, a = − g, since the particle moves against the gravity. 2.2 Scalar and vector quantities A study of motion will involve the introduction of a variety of quantities, which are used to describe the physical world. Examples of such quantities are distance, displacement, speed, velocity, acceleration, mass, momentum, energy, work, power etc. All these quantities can be divided into two categories – scalars and vectors. The scalar quantities have magnitude only. It is denoted by a number and unit. Examples : length, mass, time, speed, work, energy, 44 temperature etc. Scalars of the same kind can be added, subtracted, multiplied or divided by ordinary laws. The vector quantities have both magnitude and direction. Examples: displacement, velocity, acceleration, force, weight, momentum, etc. 2.2.1 Representation of a vector Vector quantities are often represented by a scaled vector diagrams. Vector diagrams represent a vector by the use of an arrow drawn to scale in a specific direction. An example of a scaled vector diagram is shown in Fig 2.7. From the figure, it is clear that (i) The scale is listed. (ii) A line with an arrow is drawn in a specified direction. (iii) The magnitude and direction of the vector are clearly labelled. In Y the above case, the diagram shows that Scale : 1cm=1N the magnitude is 4 N and direction is 30° to x-axis. The length of the line OA=4N gives the magnitude and arrow head A gives the direction. In notation, the Head vector is denoted in bold face letter m 4c such as A or with an arrow above the → letter as A, read as vector 30º X A or A vector while its magnitude O Tail is denoted by A or by A. Fig 2.7 Representation of a vector 2.2.2 Different types of vectors (i) Equal vectors A Two vectors are said to be equal if they have the same magnitude and same direction, wherever be their → → B initial positions. In Fig. 2.8 the vectors A and B have Fig. 2.8 → → the same magnitude and direction. Therefore A and B Equal vectors are equal vectors. 45  A A B A B B Fig. 2.9 Fig. 2.10 Fig. 2.11 Like vectors Opposite vectors Unlike Vectors (ii) Like vectors Two vectors are said to be like vectors, if they have same direction but different magnitudes as shown in Fig. 2.9. (iii) Opposite vectors The vectors of same magnitude but opposite in direction, are called opposite vectors (Fig. 2.10). (iv) Unlike vectors The vectors of different magnitude acting in opposite directions → → are called unlike vectors. In Fig. 2.11 the vectors A and B are unlike vectors. (v) Unit vector A vector having unit magnitude is called a unit vector. It is also defined as a vector divided by its own magnitude. A unit vector in the → ^ direction of a vector A is written as A and is read as ‘A cap’ or ‘A caret’ or ‘A hat’. Therefore, ^ A → ^ → A= (or) A = A |A| | A| Thus, a vector can be written as the product of its magnitude and unit vector along its direction. Orthogonal unit vectors There are three most common unit vectors in the positive directions of X,Y and Z axes of Cartesian coordinate system, denoted by i, j and k respectively. Since they are along the mutually perpendicular directions, they are called orthogonal unit vectors. (vi) Null vector or zero vector A vector whose magnitude is zero, is called a null vector or zero → vector. It is represented by 0 and its starting and end points are the same. The direction of null vector is not known. 46 (vii) Proper vector All the non-zero vectors are called proper vectors. B (viii) Co-initial vectors Vectors having the same starting point are called O → → A co-initial vectors. In Fig. 2.12, A and B start from the Fig 2.12 same origin O. Hence, they are called as co-initial Co-initial vectors vectors. (ix) Coplanar vectors Vectors lying in the same plane are called coplanar vectors and the plane in which the vectors lie are called plane of vectors. 2.2.3 Addition of vectors As vectors have both magnitude and direction they cannot be added by the method of ordinary algebra. Vectors can be added graphically or geometrically. We shall now discuss the addition of two vectors graphically using head to tail method. → → Consider two vectors P and Q which are acting along the same → → line. To add these two vectors, join the tail of Q with the head of P (Fig. 2.13). → → → → → The resultant of P and Q is R = P + Q. The length of the line → → AD gives the magnitude of R. R acts in the same direction as that of → → P and Q. In order to find the sum of two vectors, which P Q are inclined to each other, triangle law of vectors A BC D or parallelogram law of vectors, can be used. P C Q D (i) Triangle law of vectors A B If two vectors are represented in magnitude and direction by the two adjacent sides of a triangle R A D taken in order, then their resultant is the closing Fig. 2.13 side of the triangle taken in the reverse order. Addition of vectors 47  To find the resultant of → → two vectors P and Q which are acting at an angle θ, the following procedure is adopted. → First draw O A = P (Fig. 2.14) Then starting from → the arrow head of P, draw the vector AB = Q. Finally, draw Fig. 2.14 Triangle law of vectors → a vector OB = R from the → → tail of vector P to the head of vector Q. Vector OB = R is the sum → → → → → of the vectors P and Q. Thus R = P + Q. → → The magnitude of P + Q is determined by measuring the length → → → of R and direction by measuring the angle between P and R. → The magnitude and direction of R, can be obtained by using the sine law and cosine law of triangles. Let α be the angle made by the → → → resultant R with P. The magnitude of R is, R 2 = P 2 + Q 2 – 2PQ cos (180 o – θ) R = P 2 + Q 2 + 2PQ cos θ The direction of R can be obtained by, P Q R = = sin β sin α sin (180o -θ ) (ii) Parallelogram law of vectors If two vectors acting at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram, then their resultant is represented in magnitude and direction by the diagonal passing through the common tail of the two vectors. → → Let us consider two vectors P and Q which are inclined → to → each other at an angle θ as shown in Fig. 2.15. Let the vectors P and Q be represented in magnitude and direction by the two sides OA and OB of a parallelogram OACB. The diagonal OC passing through the → common tail O, gives the magnitude and direction of the resultant R. CD is drawn perpendicular to the extended OA, from C. Let → → COD made by R with P be α. 48 From right angled triangle OCD, OC2 = OD2 + CD2 = (OA + AD)2 + CD2 = OA2 + AD2 + 2.OA.AD + CD2...(1) B C In Fig. 2.15 BOA = θ = CAD Q R From right angled ∆ CAD, AC2 = AD2 + CD2...(2) Substituting (2) in (1) O A D P OC2 = OA2 + AC2 + 2OA.AD...(3) Fig 2.15 Parallelogram law of vectors From ∆ACD, CD = AC sin θ...(4) AD = AC cos θ...(5) Substituting (5) in (3) OC2 = OA2 + AC2 + 2 OA.AC cos θ Substituting OC = R, OA = P, OB = AC = Q in the above equation R2 = P2 + Q2 + 2PQ cos θ (or) R = P 2 + Q 2 + 2PQ cos θ...(6) Equation (6) gives the magnitude of the resultant. From ∆ OCD, CD CD tan α = = OD OA + AD Substituting (4) and (5) in the above equation, AC sin θ Q sin θ tan α = = OA + AC cos θ P + Q cos θ −1 ⎡ Q sin θ ⎤ (or) α = tan ⎢ ⎥...(7) ⎣ P + Q cos θ ⎦ Equation (7) gives the direction of the resultant. Special Cases (i) When two vectors act in the same direction In this case, the angle between the two vectors θ = 0 o , cos 0o = 1, sin 0o= 0 49 From (6) R = P 2 + Q 2 + 2PQ = (P + Q ) −1 ⎡ Q sin 0o ⎤ From (7) α = tan ⎢ o ⎥ ⎣ P + Q cos 0 ⎦ (i.e) α = 0 Thus, the resultant vector acts in the same direction as the individual vectors and is equal to the sum of the magnitude of the two vectors. (ii) When two vectors act in the opposite direction In this case, the angle between the two vectors θ = 180°, cos 180° = -1, sin 180o = 0. From (6) R = P 2 + Q 2 - 2PQ = (P − Q ) ⎡ 0 ⎤ −1 From (7) α = tan-1 ⎢ ⎥ = tan (0) = 0 ⎣P −Q ⎦ Thus, the resultant vector has a magnitude equal to the difference in magnitude of the two vectors and acts in the direction of the bigger of the two vectors (iii) When two vectors are at right angles to each other In this case, θ = 90°, cos 90o = 0, sin 90o = 1 From (6) R = P 2 + Q2 ⎛Q ⎞ From (7) α = tan−1 ⎜ ⎟ ⎝P ⎠ → → The resultant R vector acts at an angle α with vector P. 2.2.4 Subtraction of vectors The subtraction of a vector from another is equivalent to the addition of one vector to the negative of the other. For example Q − P = Q + (− P ). → → → → Thus to subtract P from Q, one has to add – P with Q → → → (Fig 2.16a). Therefore, to subtract P from Q, reversed P is added to the 50 → → Q. For this, first draw AB = Q and then starting from the arrow head → → → of Q, draw BC = (− P ) and finally join the head of – P. Vector R is the → → → → sum of Q and – P. (i.e) difference Q – P. P P Q A BC D Q A Q B C D A B Q+[-P] -P R C A C (a) (b) Fig 2.16 Subtraction of vectors The resultant of two vectors which are antiparallel to each other is obtained by subtracting the smaller vector from the bigger vector as shown in Fig 2.16b. The direction of the resultant vector is in the direction of the bigger vector. 2.2.5 Product of a vector and a scalar Multiplication of a scalar and a vector gives a vector quantity which acts along the direction of the vector. Examples → (i) If a is the acceleration produced by a particle of mass m under → → the influence of the force, then F = ma → → (ii) momentum = mass × velocity (i.e) P = mv. 2.2.6 Resolution of vectors and rectangular components A vector directed at an angle with the co-ordinate axis, can be resolved into its components along the axes. This process of splitting a vector into its components is known as resolution of a vector. Consider a vector R = O A making an angle θ with X - axis. The vector R can be resolved into two components along X - axis and Y-axis respectively. Draw two perpendiculars from A to X and Y axes respectively. The intercepts on these axes are called the scalar components Rx and Ry. 51  → Then, OP is Rx, which is the magnitude of x component of R and → OQ is Ry, which is the magnitude of y component of R Y From ∆ OPA, O P Rx A cos θ = = (or) Rx = R cos θ Q OA R O Q Ry Ry R sin θ = = (or) Ry = R sin θ OA R and R 2 = Rx2 + Ry2 X O Rx P Also, R can be expressed as Fig. 2.17 Rectangular → → → components of a vector R = Rxi + Ry j where i and j are unit vectors. ⎡R ⎤ In terms of Rx and Ry , θ can be expressed as θ = tan−1 ⎢ R ⎥ y ⎣⎢ x ⎦⎥ 2.2.7 Multiplication of two vectors Multiplication of a vector by another vector does not follow the laws of ordinary algebra. There are two types of vector multiplication (i) Scalar product and (ii) Vector product. (i) Scalar product or Dot product of two vectors A If the product of two vectors is a scalar, → → then it is called scalar product. If A and B are O two vectors, then their scalar product is written B →→ → → as A.B and read as A dot B. Hence scalar product Fig 2.18 Scalar product of two vectors is also called dot product. This is also referred as inner or direct product. The scalar product of two vectors is a scalar, which is equal to the product of magnitudes of the two vectors and the cosine of the → → angle between them. The scalar product of two vectors A and B may → → → → → → be expressed as A. B = |A| |B| cos θ where |A| and |B| are the → → → magnitudes of A and B respectively and θ is the angle between A and → B as shown in Fig 2.18. 52 (ii) Vector product or Cross product of two vectors If the product of two vectors is a vector, then it is called vector → → product. If A and B are two vectors then their vector product is written → → → → as A × B and read as A cross B. This is also referred as outer product. The vector product or cross product of two vectors is a vector whose magnitude is equal to the product of their magnitudes and the sine of the smaller angle between them and the direction is perpendicular to a plane containing the two vectors. C If θ is the smaller angle through which → → A should be rotated to reach B, then the cross → → product of A and B (Fig. 2.19) is expressed A xB B as, → → → → ^ → O A × B = |A| |B| sin θ n = C A → → → where |A| and |B| are the magnitudes of A → → B xA and B respectively. C is perpendicular to the → → → plane containing A and B. The direction of C is along the direction in which the tip of a → → D screw moves when it is rotated from A to B. Fig 2.19 Vector product → Hence C acts along OC. By the same of two vectors → → argument, B × A acts along OD. 2.3 Projectile motion A body thrown with some initial velocity and then allowed to move under the action of gravity alone, is known as a projectile. If we observe the path of the projectile, we find that the projectile moves in a path, which can be considered as a part of parabola. Such a motion is known as projectile motion. A few examples of projectiles are (i) a bomb thrown from an aeroplane (ii) a javelin or a shot-put thrown by an athlete (iii) motion of a ball hit by a cricket bat etc. The different types of projectiles are shown in Fig. 2.20. A body can be projected in two ways: 53  Fig 2.20 Different types of projectiles (i) It can be projected horizontally from a certain height. (ii) It can be thrown from the ground in a direction inclined to it. The projectiles undergo a vertical motion as well as horizontal motion. The two components of the projectile motion are (i) vertical component and (ii) horizontal component. These two perpendicular components of motion are independent of each other. A body projected with an initial velocity making an angle with the horizontal direction possess uniform horizontal velocity and variable vertical velocity, due to force of gravity. The object therefore has horizontal and vertical motions simultaneously. The resultant motion would be the vector sum of these two motions and the path following would be curvilinear. The above discussion can be summarised as in the Table 2.1 Table 2.1 Two independent motions of a projectile Motion Forces Velocity Acceleration Horizontal No force acts Constant Zero Vertical The force of Changes Downwards gravity acts (∼10 m s–1) (∼10 m s-2) downwards In the study of projectile motion, it is assumed that the air resistance is negligible and the acceleration due to gravity remains constant. 54 Angle of projection The angle between the initial direction of projection and the horizontal direction through the point of projection is called the angle of projection. Velocity of projection The velocity with which the body is projected is known as velocity of projection. Range Range of a projectile is the horizontal distance between the point of projection and the point where the projectile hits the ground. Trajectory The path described by the projectile is called the trajectory. Time of flight Time of flight is the total time taken by the projectile from the instant of projection till it strikes the ground. 2.3.1 Motion of a projectile thrown horizontally Let us consider an object thrown horizontally with a velocity u u1=0 from a point A, which is at a height A h from the horizontal plane OX u (Fig 2.21). The object acquires the C u following motions simultaneously : u2 (i) Uniform velocity with which h it is projected in the horizontal D u direction OX u3 (ii) Vertical velocity, which is non-uniform due to acceleration due B X to gravity. O R The two velocities are Fig 2.21 Projectile projected horizontally from the top of a tower independent of each other. The horizontal velocity of the object shall remain constant as no acceleration is acting in the horizontal direction. The velocity in the vertical direction shall go on changing because of acceleration due to gravity. 55 Path of a projectile Let the time taken by the object to reach C from A = t Vertical distance travelled by the object in time t = s = y 1 From equation of motion, s = u1t + at 2...(1) 2 Substituting the known values in equation (1), 1 1 2 y = (0) t + gt 2 = gt...(2) 2 2 At A, the initial velocity in the horizontal direction is u. Horizontal distance travelled by the object in time t is x. x ∴ x = horizontal velocity × time = u t (or) t =...(3) u 2 1 ⎛x⎞ 1 x2 Substituting t in equation (2), y = g⎜ ⎟ = g 2...(4) 2 ⎝u⎠ 2 u (or) y = kx2 g where k = is a constant. 2u 2 The above equation is the equation of a parabola. Thus the path taken by the projectile is a parabola. u C Resultant velocity at C At an instant of time t, let the body be at C. At A, initial vertical velocity (u1) = 0 u2 v At C, the horizontal velocity (ux) = u Fig 2.22 At C, the vertical velocity = u2 Resultant velocity From equation of motion, u2 = u1 + g t at any point Substituting all the known values, u2 = 0 + g t...(5) The resultant velocity at C is v = u x2 + u22 = u 2 + g 2 t 2...(6) u2 gt The direction of v is given by tan θ = =...(7) ux u where θ is the angle made by v with X axis. 56 Time of flight and range The distance OB = R, is called as range of the projectile. Range = horizontal velocity × time taken to reach the ground R = u tf...(8) where tf is the time of flight At A, initial vertical velocity (u1) = 0 The vertical distance travelled by the object in time tf = sy = h 1 Sy = u1t f + g t f 2 From the equations of motion...(9) 2 Substituting the known values in equation (9), 1 2h h = (0) tf + g t 2f (or) tf =...(10) 2 g 2h Substituting tf in equation (8), Range R = u...(11) g 2.3.2 Motion of a projectile projected at an angle with the horizontal (oblique projection) Consider a body projected from a point O on the surface of the Earth with an initial velocity u at an angle θ with the horizontal as shown in Fig. 2.23. The velocity u can be resolved into two components A (u3=0) ux u2