Physics: Physical World (PDF)

PHYSICAL WORLD Chapter-1, PHYSICAL WORLD Science: Science is an organised, systematic and formulated knowledge obtained through observations, experiments and verifications. Note: The word Science originates from the Latin verb “SCIENTIA”, which means “to know”. Scien...

PHYSICAL WORLD Chapter-1, PHYSICAL WORLD Science: Science is an organised, systematic and formulated knowledge obtained through observations, experiments and verifications. Note: The word Science originates from the Latin verb “SCIENTIA”, which means “to know”. Scientific method: Procedure followed in acquiring knowledge in science is called scientific method. It involves the major steps, which are, (i) Systematic observation. (ii) Logical reasoning. (iii) Model making. (iv) Theoretical prediction. (v) Verification OR Rejection of theory. Law: Law is a statement given based on the observation, experimentation and analysis. Ex: Newton’s laws of motion. Theory: The behaviour of the physical system is explained in terms of a set of minimum number of laws is called Theory. Ex: Ptolemy’s geocentric theory. Branches of Science: There are two main branches in science, 1) Physical science 2) Biological science The main branches of Physical science are, 1) Physics 2) Chemistry Physics: The term Physics is derived from Greek word called “FUSIS”, which means “Nature”. Physics is a branch of science which deals with the study of Nature and Natural phenomenon. Two principal thrusts is physics are (i) Unification and (ii) Reduction Unification: Unification is explaining diverse physical phenomena in terms of few concepts and laws. OR It is an effort to see a physical world as manifestation of some universal law in different domains and conditions. Ex: (a) Isaac Newton unified celestial and terrestrial mechanics and showed that same laws of motion and law of gravitation apply to both the domains. (b) Hans Christian Oersted and Michel Faraday showed that electric and magnetic phenomena are inseparable aspects of a unified domain. (c) James Clerk Maxwell unified electricity, magnetism and optics, showed that light is an electromagnetic wave. Reductionism: Explaining the properties of complex system using the properties and interactions of its constituent simpler parts is called reductionism. Ex: Initially thermodynamics was dealing with bulk system in terms of temperature, internal energy etc. Now the kinetic theory and statistical mechanics interpreted these quantities in terms of properties of molecular constituents of the bulk system. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 1  PHYSICAL WORLD Scope of physics: The scope of physics is very broad and covers two domains – macroscopic and microscopic. Macroscopic domain includes phenomenon at the laboratory, terrestrial and astronomical scales. Classical physics deals with macroscopic phenomena and includes the following. (i) Mechanics: The branch concerned with the motion of objects moving at speeds very small compared to the speed of light. (ii) Thermodynamics: The branch dealing with heat, temperature and work. (iii) Electrodynamics: It deals with the electricity, magnetism and electromagnetic fields. (iv) Optics: It is the branch of physics which deals with the nature of light and different properties and phenomena exhibited by light. Microscopic domain includes atomic, molecular and nuclear phenomena and includes the following. (v) Quantum mechanics: The study of motion in micro world of atoms is known as quantum mechanics. Totally physics deals with the macroscopic world like galaxies and universe as well as microscopic world like nucleus of an atom and fundamental particles like electrons, protons, neutrons etc. Excitement of physics: The study of physics is quite interesting and exciting. In physics, we come across wide range of mass, length and time. But in spite of these wide ranges of the order of these physical quantities, it is possible to understand them quite easily. This is the reason for the excitement in physics. Physics, technology and society Technology: Application of scientific knowledge for practical purpose is called Technology. Technologists use the information or knowledge of physics for designing the various applications and other instruments which help to make our material way of life comfortable. Following are a few advancements in technology based on physics. Technology Scientific principle(s) Steam engine Laws of thermodynamics Nuclear reactor Controlled nuclear fission Radio and Television Generation, propagation and detection of electromagnetic waves Lasers Light amplification by stimulated emission of radiation Production of ultra- high magnetic Superconductivity fields Rocket propulsion Newton’s laws of motion Electric generator Faraday’s laws of electromagnetic induction Hydroelectric power Conversion of gravitational potential energy into electrical energy Aeroplane Bernoulli’s principle in fluid dynamics Particle accelerators Motion of charged particles in electromagnetic fields Sonar Reflection of ultrasonic waves Optical fibres Total internal reflection of light U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 2  PHYSICAL WORLD Technology Scientific principle(s) Non-reflecting coatings Thin film optical interference Electron microscope Wave nature of electron Photocell Photoelectric effect Fusion test reactor (Tokamak) Magnetic confinement of plasma Giant Metre wave Radio Telescope Detection of cosmic radio waves (GMRT) Trapping and cooling of atoms by laser beams and Bose-Einstein condensate magnetic fields. Physics has a great impact on the society. In fact discoveries and developments in physics have changed the face of society. Our life has become more comfortable and luxuries than that of our ancestors. Some physicists from different countries of the world and their contributions. Country Name Major contributions/Discoveries of origin Principle of buoyancy Archimedes Greece Principle of lever Galileo Galilei Law of inertia Italy Christiaan Huygens Wave theory of light Holland Universal law of Gravitation Isaac Newton Laws of motion UK Reflecting Telescope Michel Faraday Laws of Electromagnetic induction UK Electromagnetic theory James Clerk Maxwell UK Light – an electromagnetic wave Heinrich Rudolf Hertz Generation of electromagnetic waves Germany J C Bose Ultra short radio waves India W K Roentgen X-Rays Germany J J Thomson Electron UK Discovery of radium and polonium; Studies on Marie Sklodowska Curie Poland natural radioactivity Explanation of photoelectric effect; Albert Einstein Germany Theory of relativity Victor Francis Hess Cosmic radiation Austria R.A. Millikan Measurement of electronic charge USA Ernest Rutherford Nuclear model of atom New Zealand Niels Bohr Quantum model of hydrogen atom Denmark C.V. Raman Inelastic scattering of hydrogen atom India Louis Victor de Borglie Wave nature of matter France M.N. Saha Thermal Ionisation India S.N. Bose Quantum statistics India Wolfgang Pauli Exclusion principle Austria Enrico Fermi Controlled nuclear fission Italy Werner Heisenberg Quantum Mechanics, Uncertainty principle Germany Relativistic theory of electron Paul Dirac UK Quantum statistics U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 3  PHYSICAL WORLD Country Name Major contributions/Discoveries of origin Edwin Hubble Expanding Universe USA Ernest Orlando Lawrence Cyclotron USA James Chadwick Neutron UK Hideki Yukawa Theory of nuclear forces Japan Homi Jehangir Bhabha Cascade process of cosmic radiation India Theory of condensed matter Lev Davidovich Landau Russia Liquid helium Chandrasekhar limit S. Chandrasekhar India Structure and evolution of stars Transistors John Bardeen USA Theory of superconductivity C.H. Townes Maser, Laser USA Unification of weak and electromagnetic Abdus Salam Pakistan interactions Fundamental forces in nature: There are four basic forces in nature. These are (i) Gravitational force (ii) Electromagnetic force (iii) Strong nuclear force (iv) Weak nuclear force (i) Gravitational force * It is the force of attraction between the two bodies due to their masses. * It is always attractive. * It weakest force in nature. * It is a long range force * This force is governed by Newton’s law of gravitation.” * It is also known as “action-at-a-distance” force. (ii) Electromagnetic force * It is the force of attraction or repulsion between two electric charges. * It is both attractive as well as repulsive. * It is charge dependent. * It is long range force. * It is 1037 times stronger than the Gravitational force. * When the charges are at rest the force between them is called electrostatic force. (iii) Strong nuclear force * Forces operating inside the nuclei are called strong nuclear force. * It is short range force (range is ~ 10-15m) * It is charge independent. * It is an attractive force. * It is the strongest force in nature. It is 100 times stronger than electromagnetic force and 1039 times stronger than gravitational force. * It does not obey inverse square law. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 4  PHYSICAL WORLD (iv) Weak nuclear force * It is the force existing between the elementary particles emitted during radio-active decay. * It appears only in certain nuclear processes such as beta decay. * It is not as weak as gravitational force but much weaker than strong nuclear and Electromagnetic force. * It’s range is 10-16m. Relative Name Range Operates among strength Gravitational force 1 Infinite All objects in the universe Very short Some elementary particles. Weak nuclear force sub-nuclear size, Particularly electron and neutrino Electromagnetic Infinite Charged particles force Short Nucleons, heavier Strong nuclear force Nuclear size, elementary particles Nature of physical law: The various phenomena occurring in nature are explained on the basis of certain laws. These laws are expressed in terms of some physical quantities. Several physical quantities may change with time but some physical quantities remain constant in time. The quantities like charge, mass, energy, liner momentum angular momentum etc. of a system are conserved, if no external force acts on the system. Conserved quantities: The physical quantities remain constant during a process are called conserved quantities. Conservation Laws: A law which states the constancy of physical quantity over time within an isolated system is called a conservation law. Ex: Law of conservation of mass. Law of conservation of energy. Law of conservation of charge. Law of conservation of momentum. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 5  PHYSICAL WORLD Suggested Questions. One mark. 1) What is physics? 2) Name the scientist who achieved the unification of electromagnetism and optics. 3) Who introduced the nuclear model of an atom? 4) Who discovered neutron? 5) Who own the Nobel Prize in the field of scattering of light by molecules? 6) Who discovered the law of Inertia? 7) Who proposed the electromagnetic wave theory? 8) Name the scientist who gave the heliocentric theory? 9) Name the scientist who gave the laws of electromagnetic induction? 10) Who discovered radioactivity? 11) Who discovered the X-rays? 12) What is the nature of physical laws? Two marks. 1) Mention the steps involved in scientific method. 2) What is unification? Give one example. 3) Mention two physicists who achieved the unification of electricity and magnetism. 4) What is reductionism? Give an example. 5) Name the two physicists who discovered an electron and electromagnetic wave. 6) Name two Indian scientists who have been awarded Nobel Prize. 7) Mention any two fundamental forces in nature. 8) Name the strongest and weakest fundamental forces in nature. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 6  UNITS AND MEASUREMENTS CHAPTER-2 UNITS AND MEASUREMENTS Physical quantity: A measurable quantity is called a physical quantity. Ex: Length, mass, time, area, volume etc. Fundamental quantities: The physical quantities which are independent of each other are called fundamental quantities. There are SEVEN fundamental quantities. They are, Length, Mass, Time, Electric current, Thermodynamic temperature, Amount of substance and Luminous Intensity Derived quantities: The physical quantities which can be expressed in the form of a product or quotient of the fundamental quantities are called derived units. Ex: Area, Volume, Force, momentum, speed etc. Unit: The basic, arbitrary chosen, internationally accepted standard of reference which is used to express a physical quantity is called a unit. S I System: The system of units which is at present internationally accepted for measurement is the system of International (S I) and it was developed by General conference on weights and measures in 1971. The earlier systems of units are FPS, CGS and MKS system. Fundamental units: The units used to express fundamental quantities are called Fundamental units. The table gives the list of fundamental quantities and their units in SI. Base SI Units quantity Name Symbol Definition The metre is the length of the path travelled by light in vacuum Length metre m during a time interval of 1/299,792,458 of a second. (1983) The kilogram is equal to the mass of the international prototype of the kilogram (a platinum-iridium alloy cylinder) kept at Mass kilogram kg international Bureau of Weights and Measures, at Sevres, near Paris. France. (1889) The second is the duration of the 9,192,631,770 periods of the radiation corresponding to the transition between the two Time second s hyperfine levels of the ground state of the cesium-133 atom.(1967) The ampere is that constant current which, if maintained in two straight parallel conductors of infinite length, of negligible Electric current ampere A circular cross-section, and placed 1 metre apart in vacuum, would produce between these conductors a force equal to newton per metre of length. (1948) Thermodynamic The kelvin, is the fraction 1/273.16 of the thermodynamic kelvin K temperature temperature of the triple point of water. (1967) The mole is the amount of substance of a system, which contains Amount of mole mol as many elementary entities as there are atoms in 0.012 kilogram Substance of carbon - 12. (1971) The candela is the luminous intensity, in a given direction, of a Luminous source that emits monochromatic radiation of frequency candela cd intensity hertz and that has a radiant intensity in that direction of 1/683 watt per steradian. (1979) In addition to the seven fundamental units, two supplimentory units are defined which are given in the table below. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 1  UNITS AND MEASUREMENTS S I Units Quantity Name Symbol Plane angle radian rad Solid angle steradian sr Plane angle: It is the ratio of arc length to the radius of the circle. Solid angle: It is the ratio of spherical area enclosed to the square of the radius of the sphere. Maximum solid angle at the centre of the sphere is, Derived units: The units which can be expressed as combination of base units are called derived units. Ex: ms-1, ms-2, kgms-1, m2, m3 etc. General guidelines for using symbols and units  Symbols for units are written in lower case starting with small letters.  The unit names are never capitalised, however the unit symbols are capitalised only if the symbol for a unit is derived from a proper name of scientist.  Symbols for units do not contain any punctual marks and remain unaltered in the plural. Advantages of SI units  It is a rational system: It uses only one unit for a given quantity.  It is a coherent system: Every unit can be derived from seven fundamental and two supplementary units.  It is a metric system: Multiple and sub multiples of unit can be expressed as the powers of TEN.  It is internationally accepted. Note: () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 2  UNITS AND MEASUREMENTS Common SI prefixes: Multiples and Submultiples Prefixes Symbols 1 000 000 000 000 tera T 1000 000 000 giga G 1 000 000 mega M 1000 kilo k 100 hecto h 10 deka da Base unit: 1 0.1 deci d 0.01 centi c 0.001 milli m 0.000 001 micro 0.000 000 001 nano n 0.000 000 000 001 pico p 0.000 000 000 000 001 femto f 0.000 000 000 000 000 001 atto a Measurement of length: Length of various objects or distances between the objects differ widely ranging from the radius of proton of about 10-15m to the average size of the universe with a radius of about 1026m. Some of the simple measurement of length involves the use of a) A metre scale for lengths from 10-3m to 102m. b) Vernier callipers for lengths to accuracy of about 10-4m. c) A screw gauge or spherometer to measure lengths of the order of 10-5m. In order to measure lengths beyond these ranges some special indirect methods are adopted. One of them is the parallax method. Parallax: It is the change in the position of an object to its background, when the object is seen from two different positions. The distance between the two different points of observation is called the Basis. Measurement of large distance by parallax method: Let and are the two positions of observation of a distant object. Let be the distance between the distant object and the earth. The distance between and be be the angle made by two opposite ends and. ( ) Then, Where is in radian and is called parallactic angle. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 3  UNITS AND MEASUREMENTS Some special units of length  1 fermi = 1f = 10-15m  1 angstrom = 1 ̇ = 10-10m (these are the shorter units of length)  1 astronomical unit: It is the average distance between the earth and sun. 1 AU = 1.496×1011m  1 light year: The distance travelled by the light in one year of time. 1 ly = 9.46 ×1015m  1parsec: It is the distance at which an arc of length equal to one AU subtends an angle of one second at a point. 1pc = 3.08×1016m (parsec is the largest unit of length) Measurement of mass: Mass is the basic property of matter. It is expressed in kg, but for atomic and sub atomic particles, we use unified atomic mass unit (u). unified atomic mass unit (u): One unified atomic mass unit is equal to ( ) of the mass of an atom of carbon-12 isotope including the mass of electrons. 1u = 1.66×10-27kg The mass of various objects differ widely ranging from the mass of an electron about to the mass of universe with about. Masses of commonly available objects are measured using a common balance. Inertial mass of an object is measured using an inertial balance. Masses of microscopic objects are determined by spectroscopic method, using a mass spectroscope. Masses of astronomical objects are estimated using Newton’s law of gravitation. Masses of binary stars are estimated using Kepler’s law of time periods. Measurement of time: Time measurements are done using a clock. Now we use an atomic standard of time which is based on the periodic vibrations produced in a cesium-133 atom. Cesium atomic clocks are very accurate. Measurement of time intervals ranging from 10-16s to 10-24s is estimated using photographic emulsions involved in the decay of elementary particles. Radioactive dating is used to estimate time intervals in the range of several hundred years to millions of years. Note: A Cesium atomic clock is used at the National physical laboratory (NPL), New Delhi to maintain the Indian standard of time. Accuracy, precision of instrument and errors in measurements Accuracy: The accuracy is the measure of how much close the measured value is to the true value of the quantity. Precision: It indicates, to what resolution or limit the quantity is measured. Least count of the instrument: The smallest value that can be measured by the measuring instrument is called least count Ex: least count of meter scale = 0.1 cm = 1 mm least count of vernier callipers = 0.01 cm U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 4  UNITS AND MEASUREMENTS Error: The uncertainty in the measurement is called error. Errors are due to lack of accuracy and insufficient precision. The errors in measurement are classified into two types based on cause, (i) Systematic error (ii) Random error Systematic error: Systematic errors are those errors that tend to be in one direction, either positive or negative and affect each measurement by same amount. these errors are due to known cause. Sources of systematic error (types of systematic errors) a) Instrumental error: These errors occur due to faulty instrument or imperfect design of the measuring instrument. b) Imperfection in experimental procedure: These errors arise due to false procedure or limitations of experimental arrangements. c) Personal errors: These arise due to individual’s bias, lack of attentiveness or bad sights. Methods of reducing systematic errors: Systematic errors can be minimised by, a) Selecting better instruments b) Improving experimental techniques. c) Removing personal bias. Random errors: The random errors are those errors which occur irregularly due to random and unpredictable fluctuations in experimental conditions. Random errors appear due to unknown reasons. Ex: Reading of physical balance may change due to settling of dust, change in temperature, pressure etc. Minimising random errors: Random errors can be minimised by repeating the measurements and taking the arithmetic mean of all measurements. Least count error: This error is associated with the resolution or the precision of the instrument. Minimising least count error: Least count error can be minimised by a) using instruments of higher precision b) improving experimental techniques c) taking mean of all observations. Absolute error, relative error and percentage error: There are three ways to express the magnitude of errors. They are, a) Absolute error b) Relative error c) Percentage error Absolute error: The difference between the individual measured value and true value is called an absolute error. The mean value am of measured values is taken as true value. If a1, a2,….an are the individual measured values in different trails then the mean value is, U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 5  UNITS AND MEASUREMENTS ∑ The absolute error in the measured values are given by,... The absolute error may be either positive or negative. Mean absolute error: The arithmetic mean of the magnitude of the absolute error is called the mean absolute error ( ). | | | | | | ∑| | The final result is expressed as. this means that the true value of lies between the limits and. Relative error: The ratio of mean absolute error to the mean value of the quantity measured is called the relative error. Percentage error: Relative error when expressed in percentage is called percentage error. Combination of errors: In each measurement, there is some error and when we get the final result, these errors are combined to have the net error in the final result. Error due to a sum or difference: Let two physical quantities A and B have measured values respectively where and are their absolute error. The error in the sum is, ( ) ( ) ( ) ( ) The maximum possible error in Z is, For difference, ( ) ( ) ( ) ( ) The maximum error is again When two quantities are added or subtracted, the absolute error in the final result is the sum of the absolute errors in the individual quantities. The absolute errors due to sum or difference always add up. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 6  UNITS AND MEASUREMENTS Error due to a product or a quotient: Let two physical quantities A and B have measured values respectively where and are their absolute error. The error in the product, is, ( ) ( ) ( ) ( since is small) For division, ( )( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( ) When two quantities are multiplied or divided the relative error in the result is the sum of the relative errors in the individual quantities. Errors due to exponentiation If ( ) ( ) ( is very small, ) ( ) U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 7  UNITS AND MEASUREMENTS Generally, ( ) The relative error in a physical quantity raised to the power K is K times the relative error in the individual quantity. Significant figures: In a measured value the reliable digits and the first uncertain digit are known as significant figures. Importance of significant figures: Significant figures indicate the precision of the instrument. Number of significant figure does not change if we measure a physical quantity in different units. Rules for determining the significant figures  All non-zero digits are significant.  Ex: 2341 m → has 4 significant figures.  14.3m → has 3 significant figures  All the zeros between two non-zero digits are significant.  Ex: 308 m → has 3 significant figures.  23.08 m → has 4 significant figures.  In a number without decimal point trailing or terminal zeros are NOT significant.  Ex: 12300 m → has 3 significant figures.  104000 m → has 3 significant figures.  In number with decimal point, trailing or terminal zeros are significant.  Ex: 4.700 m → has 4 significant figures.  23.04000 m → has 7 significant figures.  If the number is less than 1, then zeros on the right of decimal point but to the left of the first non-zero digit are NOT significant.  Ex: 0.067 m → has 2 significant figures.  0.0003080 m → has 4 significant figures. Scientific notation: In this notation, every number is expressed as , where is a number between 1 and 10 is called base number and is any positive or negative exponent of 10. The power of 10 is irrelevant to the determination of significant figures. However all zeros appearing in the base number in the scientific notation are significant. Ex: 4.700 has 4 significant figures. 4.700 has 4 significant figures. Rules for arithmetic operations with significant figures (i) When numbers are added or subtracted, the number of decimal places in the final result should be equal to the smallest number of decimal places of any term. Ex: (a) 436.32 + 227.2 = 663.5 (but not 663.52) (b) 0.3074 – 0.304 = 0.003 (but not 0.0034) U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 8  UNITS AND MEASUREMENTS (ii) In multiplication or division, the number of significant figures in the final result should be equal to the number of significant figures in the quantity having the smallest number of significant figures. Ex: (a) 1.21 ×0.12 = 0.14 (but not 0.1452) (b) (but not 4.7833) Rounding of the uncertain digits (i) If the digit to be dropped in a number is less than 5, then the preceding digit remains unchanged. Ex: 1.344 is rounded as 1.34 (ii) If the digit to be dropped in a number is greater than 5, then the preceding digit is raised by 1. Ex: 1.346 is rounded as 1.35 (iii) If the digit to be dropped in a number is 5, then (a) the preceding digit remains unchanged if it is EVEN. Ex: 1.345 is rounded as 1.34 (b) the preceding digit is raised by 1, if it is ODD. Ex: 1.375 is rounded as 1.38 Dimensions of physical quantities: Dimensions of a physical quantity are the power to which the base quantities are raised to represent the physical quantity. Note: Dimensions of a physical quantity explain its relationship with fundamental quantities. All the derived physical quantities can be expressed in terms of some combination of seven fundamental quantities. Dimensions of a physical quantity are denoted with square bracket. Symbols for dimensions of fundamental quantities Base quantity Symbol for its dimension Length [L] Mass [M] Time [T] Current [A] Thermodynamic temperature [K] Luminous intensity [cd] Amount of substance [mol] -2 Ex: Dimensions of force are M. Hence force has one dimension in mass, one dimension in length and -2 dimensions in time. Dimensional formula: Expression of physical quantity in terms of the base quantities is called dimensional formula. Ex: Dimensional formula of volume is [ ] , Dimensional formula of Speed is [ ] Dimensional equation: Equation obtained by equating a physical quantity with its dimensional formula is called dimensional equation. Ex: [ ] [ ], [ ] [ ] U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 9  UNITS AND MEASUREMENTS Different types of variables and constants:  Dimensional variable: The physical quantities which possess dimensions and have variable values are called dimensional variables. Ex: Area, volume, speed, velocity, acceleration, momentum, force etc.  Dimensionless variables: The physical quantities which have no dimensions but have variable values are called dimensionless variables. Ex: Angle, specific gravity, strain, , , etc.  Dimensional constants: The physical quantities which possess dimensions and have constant values are called dimensional constants. Ex: Planck’s constant, Gravitational constant, speed of light in vacuum etc.  Dimensionless constants: The physical quantities which do not have dimensions but have constant values are called dimensionless constants. Ex: , , pure numbers like 1, 2, 3….etc. Dimensional analysis: The process of examination of dimensions of various physical quantities involved in a relation is called dimensional analysis. Principle of homogeneity: The dimensions of all the terms in an equation must be identical. This principle is called the principle of homogeneity. Uses of Dimensional analysis: By dimensional analysis we can, (i) Check the correctness of an equation. (ii) Deduce relations between physical quantities. (iii) Convert the unit of a physical quantity from one system to another. Check the correctness of the following equation by dimensional analysis (i) [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] The dimensions of each term on both sides of the equation are the same. Thus equation is dimensionally correct. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 10  UNITS AND MEASUREMENTS (ii) [ ] [ ] [ ] [ ] [ ] [ ] ( ) The dimensions of each term on both sides of the equation are the same. Thus equation is dimensionally correct. (iii) ( ) ( ) ( ) [ ] [ ] [ ] [ ] [ ] [ ] ( ) [ ] [ ] ( ) [ ] [ ] ( ) The dimensions of each term on both sides of the equation are the same. Thus equation is dimensionally correct. Deducing the relation 1) The time period of oscillation of a simple pendulum(T) depends on its length(l), mass of the bob(m) and acceleration due to gravity(g). Derive the expression for its time period using the method of dimension. Let (where is constant and dimensionless) [ ] [ ] [ ] U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 11  UNITS AND MEASUREMENTS [ ] [( ) ] [ ] [ ] [( ) ] [ ] [ ] [( ) ] [( ) ] [ ] Then, from the principle of homogeneity, [ ] [ ][ ][ ] [ ] [ ] [ ] [ ] Comparing the exponents on both sides, we have On solving the above equations, and, Now substituting the values of , in the equation ⁄ ⁄ √ 2) The centripetal force(F) acting on a particle moving in a circle depends upon mass(m), velocity(v) and radius of the circle(r). Derive an expression foe centripetal force using the method of dimensions. Given, (where is dimensionless constant) [ ] [ ] [ ] [ ] [( ) ] [ ] [ ] [( ) ] [( ) ] [ ] [ ] [( ) ] [ ] Then, from the principle of homogeneity, [ ] [ ][ ][ ] [ ] [ ] [ ] [ ] On comparing, Solving for b and c, we have Now substituting the values of , in the equation F U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 12  UNITS AND MEASUREMENTS Limitations of dimensional analysis 1) Dimensionally correct equation need not be actually correct. 2) Correctness of the constants appearing in an equation cannot be verified. 3) Equations involving trigonometric and exponential functions cannot be verified. 4) An equation can be derived only if it is of product type. 5) While deriving en equation the value of constant of proportionality cannot be obtained. 6) This method works only if there are as many equations available as there are unknowns. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 13  UNITS AND MEASUREMENTS Quantity symbol formula S I unit DF U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 14  UNITS AND MEASUREMENTS Quantity symbol formula S I unit DF U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 15  UNITS AND MEASUREMENTS Suggested questions. One mark. 1) Define unit. 2) What are derived units? 3) Define unified atomic mass unit. 4) How many metres make one parsec? 5) What are random errors? 6) Define the term relative error. 7) What is dimension of a physical quantity? 8) Write the dimensional formula of work. 9) Write the dimensional formula for linear momentum. 10) Write the dimensional formula for force. 11) State principle of homogeneity of dimensions. Two marks. 1) What are fundamental units? Give an example for fundamental units. 2) With a diagram explain the parallax method of measuring the large distance like a planet or a star from the earth. 3) Distinguish between accuracy and precision of measurement. 4) Mention the types of errors. 5) Define (a) error (b) accuracy. 6) What is systemic error? Mention any one source of systemic error. 7) Write any two methods to minimise the systemic error. 8) The resistance R =V/I, where V=(100±5) volt and I= (10±0.2) A, Find the percentage error in R. 9) Define the term significant figures with example. 10) Write the number of significant figures in the following. a) b) 11) Give the number of significant figures in a) b) 12) Write the SI unit and dimensional formula for acceleration. 13) Write dimensional formula for force and work. 14) Name any two physical quantities, which have same dimensions. 15) Mention the Physical quantity represented by the dimensional formula [ ] 16) What are the advantages of dimensional analysis? Three marks. 1) Name the SI unit of (i) momentum (ii) luminous intensity (iii) solid angle (iv) plane angle (v) Power (vi) Impulse 2) Write the dimensional formula for pressure, wavelength and force. 3) Using the method of dimensions, deduce the relation connecting the time period, mass of the bob, length of the pendulum and acceleration due to gravity. 4) Check the correctness of the equation by the method of dimensions. 5) Check the correctness of the equation by the method of dimensions. 6) Check the correctness of the equation by the method of dimensions. 7) Check the correctness of the equation ⁄ by the method of dimensions. 8) Check the correctness of the equation by the method of dimensions. 9) Check the correctness of the equation ⁄ by the method of dimensions. 10) Write the limitations of dimensional analysis. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 16  MOTION ALONG A STRAIGHT LINE CHAPTER-3 MOTION IN A STRAIGHT LINE Mechanics: Mechanics is the oldest and fundamental branch of physics and it is the study of the state of rest as well as the state of motion of object under the action of force. The study of mechanics is broadly classified in to (i) Statics and (ii) Dynamics Statics: It deals with bodies at rest under the action of system of force. Dynamics: It deals with motion of a body under the action of force. Dynamics is again divided into (a) Kinematics and (b) Kinetics Kinematics: It deals with the description of motion without reference to the cause of motion. Kinetics: It deals with what moves and what causes motion. Some of the terms used in describing motion are given below. Particle: A particle is ideally just a piece or quantity of matter, having no linear dimensions but only position and mass. Event: An event is a physical process that occurs at a point in space and at an instant of time. Observer: A person or equipment which can locate, record, measure and interpret an event is called an observer. Frame of reference: It is the reference in which an observer sits and makes the observations. In order to specify the position, we need to use a reference point and set of axes. The choice of set of axes in a frame of reference depends on the situation. Motion: Motion is change in position of an object with time. Rectilinear motion: Motion of objects along a straight line. Ex: A car moving along a straight road, A freely falling body. Rest: A body is said to be at rest when it does not changes its position with time. Path length: It is the actual distance covered by a body in time. It is also called as distance travelled.  Path length is a scalar quantity.  SI unit of path length is. Dimensions are  Path length depends on the actual path.  Path length is always positive. Displacement: It is the shortest distance between the initial point and final point.  It is vector quantity.  SI unit of displacement is. Dimensions are  Displacement may be positive, negative and zero.  Magnitude of the displacement can never be greater than path length.  When a body moves in straight line displacement is equal to path length.  It is independent of the actual path travelled and it denoted by U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 1  MOTION ALONG A STRAIGHT LINE Difference between path length and displacement Path length Displacement It is the actual distance covered by a body in It is the shortest distance between the initial point time. It is also called as distance travelled and final point Path length is a scalar quantity It is vector quantity Path length is always positive Displacement may be positive, negative and zero Path length is always greater than or equal to Displacement is always less than or equal to path displacement length Speed: Speed is defined as rate of change of position of a particle.  Speed is a scalar quantity.  Its SI unit is. Dimensions are  It is always positive.  Speed gives no indication about the direction of motion of the particle. Average speed: The average speed of a particle in motion is defined as the ratio of the total path length to the total time taken. Instantaneous speed (speed): It is defined as the limit of average speed as the time interval is infinitesimally small. Velocity: Velocity is defined as the rate of change of displacement of a body.  Velocity is a vector quantity.  SI unit is. Dimensions are  Velocity may be positive, negative or zero. Average velocity: The average velocity of a particle in motion is defined as the ratio of total displacement to the total time taken. ̅ ̅ Instantaneous velocity: Velocity is defined as the limit of average velocity as the time interval becomes infinitesimally small.  Instantaneous velocity is also called velocity. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 2  MOTION ALONG A STRAIGHT LINE  In position-time graph, instantaneous velocity at a point is the slope to the tangent drawn to the curve at that point.  Instantaneous speed or speed is the magnitude of velocity. Uniform velocity: If equal changes of displacement take place in equal intervals of time is called uniform velocity. Note: When a body moves with uniform velocity, neither the magnitude nor the direction of the velocity changes. Difference between speed and velocity Speed Velocity It is defined as the ratio of the path length It is defined as the ratio of displacement to to the time taken. the time taken. Speed is a scalar quantity. Velocity is a vector quantity. It is always positive Velocity may be positive, negative or zero. Acceleration: It is defined as rate change of velocity of a particle.  Acceleration is a vector quantity.  SI unit is and dimensions are  Since velocity is a quantity having both magnitude and direction, Acceleration may result from a change in magnitude or a change in direction or changes in both.  Acceleration can be positive, negative or zero.  The negative acceleration is called retardation or deceleration. Average acceleration: It is defined as the total change in velocity divided by the total time taken. ̅ Instantaneous acceleration: It is defined as the limit of the average acceleration as the time interval becomes infinitesimally small. Uniform acceleration: If the velocity of a body changes by equal amount in equal intervals of time, however small these time intervals may be, is called uniform acceleration. Graphical representation of motion: Graph: A diagrammatical representation of variation of one quantity with respect to another quantity is called a graph. Position-time graph: It is a graph obtained by plotting instantaneous positions of a particle versus time.  The slope of the position time graph gives the velocity of the particle. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 3  MOTION ALONG A STRAIGHT LINE Position-time graphs: Slno Type of motion Position-time graph 1 Object at rest 2 Uniform motion along a straight line 3 Motion with positive acceleration 4 Motion with negative acceleration 5 Motion with zero acceleration Velocity time-graph: A graph of velocity versus time is called velocity-time graph.  The area under v-t graph with time axis gives the value of displacement covered in given time.  The slope of tangent drawn on graph gives instantaneous acceleration. Uses of velocity-time (v-t) graph / Significance of velocity-time (v-t) graph:  It is used to study the nature of the motion.  It is used to find the velocity of the particle at any instant of time.  It is used to derive the equations of motion.  It is used to find displacement and acceleration. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 4  MOTION ALONG A STRAIGHT LINE Velocity time-graphs Slno Type of motion v-t graph Motion in positive direction with positive acceleration or uniform 1 acceleration having some initial velocity. Motion in positive direction with 2 negative acceleration having some initial velocity. Motion in negative direction with 3 negative acceleration having some initial velocity. Motion of an object with negative acceleration that changes direction 4 at time having some initial velocity. Kinematic equation for uniformly accelerated motion: For uniformly accelerated motion, we can derive some simple equations that relate displacement ( ), time taken ( ), initial velocity ( ), final velocity ( ), and acceleration ( ). These equations are called Kinematic equations for uniformly accelerated motion. The Equations are, (i) (ii) (iii) Derivation of equation of motion by graphical method (i) Consider a particle in motion with initial velocity and constant acceleration. Let be the final velocity of the body at time. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 5  MOTION ALONG A STRAIGHT LINE From graph, But, and But, slope of v-t graph gives the acceleration. (ii) Consider a particle in motion with initial velocity and constant acceleration. Let be the final velocity of the body at time. From graph, But, (iii) Consider a particle in motion with initial velocity and constant acceleration. Let be the final velocity of the body at time. From graph, ( ) U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 6  MOTION ALONG A STRAIGHT LINE Note: The set above equations were obtained by assuming that at , the position of the Particle is 0 (zero). When at , If the position of the particle is at , then the equations are, Free fall: An object released near the surface of the earth is accelerated downward under the influence of the force of gravity. If the air resistance is neglected, then the motion of the body is known as free fall. Acceleration due to gravity: The acceleration produced in object due to gravity is called acceleration due to gravity, denoted by. Free fall is an example for motion along a straight line under constant acceleration.  Acceleration due to gravity is always a downward vector directed towards the centre of the earth.  The magnitude of is approximately near the surface of the earth.  Acceleration due to gravity is the same for all freely falling bodies irrespective of their size, shape and mass.  The distance traversed by a body falling freely from rest during equal intervals of time are in the ratio this is known as Galileo’s law of ODD numbers. Equations of motion under gravity: The motion of a freely falling body is in Y-direction. If we take vertically upward as positive Y-axis, acceleration is along the negative Y-axis, therefore Then, (i) (iii) For freely falling body the initial velocity, Then, (i) (iii) The graph, graph and graph to a body released from rest at are as shown. 𝑎 𝑡 graph 𝑣 𝑡 graph 𝑦 𝑡 graph Note: (i) Stopping distance: When breaks are applies to a moving vehicle, the distance travelled before stopping is called stopping distance. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 7  MOTION ALONG A STRAIGHT LINE It is an important factor for road safety and it depends on initial velocity and deceleration. (ii) Reaction time: When a situation demands our immediate action, it takes some time before we really respond this time is called reaction time. Relative velocity: The relative velocity of body with respect to body is defined as the time rate of change of displacement of with respect to. Explanation: Consider two bodies A and B moving with constant velocity and respectively, along positive X-axis. Let and be the position of and at any given instant of time , then Separation between and at time is, Here, is the separation between and at and is the time rate of change of relative velocity of with respect to , denoted by. Hence, Similarly velocity of A with respect to B is and it can be shown that Case(1): When two bodies move with the same velocity in same direction, then and and , Then two bodies appear at rest with respect to each other. In this case relative velocity is minimum. Case(2): When two bodies move in the same direction with different velocities, If then and Case(3): When two bodies move in different velocities or same velocities in opposite direction. The magnitude of the relative velocity of either of them with respect to the other is equal to the sum of the magnitude of their velocities. In this case relative velocity is maximum. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 8  MOTION ALONG A STRAIGHT LINE Suggested Questions. One mark. 1) When will the magnitude of displacement equal to the path length? 2) Define average speed. 3) Define instantaneous speed. 4) Define instantaneous velocity. 5) Define average velocity. 6) Define acceleration. 7) What is retardation? 8) What is the acceleration of a body moving with uniform velocity? 9) What does the slope of position-time graph represent? 10) What does the slope of velocity-time graph represent? 11) Draw v-t graph for motion in uniform acceleration. Two marks. 1) Distinguish between distance travelled and displacement of a particle. 2) Distinguish between speed and velocity. 3) Define uniform velocity and uniform acceleration. 4) What is position time graph? Draw graph for an object at rest. 5) Draw position time graph for (a) a particle at rest,(b) a body moving with uniform velocity. 6) Draw the position time graph of a particle moving with a) Positive acceleration. b) Negative acceleration. 7) Draw v-t graph for body moving in uniform acceleration. 8) Define relative velocity. When will the relative velocity of two bodies be zero? 9) Define relative velocity with an example. Three marks. 1) Write the significance of v-t graph. 2) Derive the equation with usual notation by using v-t graph. 3) Define relative velocity. When does the relative velocity become maximum and minimum if two particles are moving along a straight line? Five marks. 1) What is v-t graph? Derive the equation with usual notation by using v-t graph. 2) What is v-t graph? Derive the equation with usual notation by using v-t graph. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 9  MOTION IN A PLANE Chapter 4 MOTION IN A PLANE When a body moves in a plane (a two dimensional motion) or in a space (a three dimensional motion) then the position, displacement, velocity and acceleration of the body have two or three components respectively. Then we need to use Vectors to describe the concept of position, displacement, velocity and acceleration. Scalar quantity: A physical quantity having only magnitude is called a scalar quantity. It is specified completely by a single number along with proper unit. Ex: mass, length, temperature, speed, charge, area etc. Scalars can be added, subtracted, multiplied and divided just as the ordinary numbers. They follow the rules of algebra. Vector quantity: A Physical quantity having both magnitude and direction and obey the triangle law of addition is called Vector quantity. It is represented by a number with an appropriate unit and direction. Ex: Displacement, velocity, acceleration, force, momentum etc. Differences between Scalar quantity and Vector quantity Scalar Quantity Vector Quantity It has only magnitude It has both magnitude and direction They follow the rules of ordinary algebra They follow the rules of vector algebra Ex: Mass, Length, Temperature, Area Ex: Displacement, velocity, Acceleration, Force These changes when magnitude changes or These change when magnitude changes direction changes or both of them changes. Representation of a vector: To represent a vector we use a bold face letters or an arrow placed over a letter. Ex: a= ⃗ ⃗⃗⃗⃗⃗⃗ Here O is called the initial point and P is called the terminal point. Length of the line segment OP represents the magnitude and arrow at the end point indicates the direction. The magnitude of a vector is often called the absolute value and indicated by, | | | ⃗| Classification of vectors: Parallel vectors: Two or more vectors having same direction are called parallel vectors. Anti-parallel vectors (opposite vectors) : Vectors having opposite directions are called anti- parallel vectors (opposite vectors). Equality of vector (Equal vectors): Two (or more) vectors having same magnitude and direction, representing the same physical quantity are called Equal vectors. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 1  MOTION IN A PLANE Negative of a vector: A vector having same magnitude but having opposite direction to that of the given vector is called negative of a given vector. Zero(Null) vector: A vector whose magnitude is zero is called Zero vector. It is represented by ⃗⃗ and the direction is not specified. Properties of Zero vector are, ⃗ ⃗ ⃗⃗ |⃗⃗| ⃗ ⃗⃗ ⃗ ⃗⃗ ⃗ ⃗ ⃗⃗ ⃗⃗ ⃗⃗ ⃗⃗ Unit vector: A vector having unit magnitude is called unit vector. Its purpose is to specify a direction. Unit vector has no dimensions and unit. If ⃗ is a vector, then the unit vector in direction of ⃗ is written as ̂ (read as “a cap”) ⃗ | ⃗| ̂, ⃗ ̂ | ⃗| Note: The unit vectors in the positive directions of x, y and z axes are labelled as ̂ ̂ ̂ respectively. Addition of vectors – Graphical method Two vectors representing the same quantity in the same unit are added using following rules. (i) Triangle method of vector addition: Law of triangle of vectors or Triangular law of vector addition: If two vectors ⃗ and ⃗⃗ are represented by two sides of a triangle in head to tail form, then the closing side of the triangle taken from tail of the first to head of the second represent the vector sum of ⃗ and ⃗⃗. Explanation: Consider two vectors, ⃗ ⃗⃗⃗⃗⃗⃗ and ⃗⃗ ⃗⃗⃗⃗⃗⃗ are of same nature. The triangle ABC is completed by joining A and C. According to triangle law of addition, ⃗⃗⃗⃗⃗⃗ ⃗ represents the sum of ⃗ and ⃗⃗. ⃗ ⃗⃗ ⃗ or ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ Note: In this procedure of vector addition, vectors are arranged head-to-tail. Hence it is called head-to-tail method. Properties vector addition: (a) Vector addition is commutative. (b) Vector addition is Associative. ⃗ ⃗⃗ ⃗ ⃗⃗ ⃗ ( ⃗ ⃗⃗) ⃗ ⃗ ( ⃗⃗ ⃗) U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 2  MOTION IN A PLANE (ii) Parallelogram method of vector addition Law of parallelogram of vectors or Parallelogram law of vector addition: If two vectors are represented by two adjacent sides of a parallelogram, then the diagonal drawn from the common initial point represents their vector sum. Explanation: Vector ⃗ and ⃗⃗ are drawn with a common initial point and parallelogram is constructed using these two vectors as two adjacent sides of a parallelogram. The diagonal originating from the common initial point is vector sum of ⃗ and ⃗⃗. Subtraction vectors – Graphical method: Subtraction of vectors can be defined in terms of addition of vectors. Consider two vectors ⃗ and ⃗⃗ of same nature and another vector ⃗⃗ which is opposite (negative) vector of ⃗⃗ , then ⃗ ( ⃗⃗) ⃗ ⃗⃗ Note: Subtraction vector is neither commutative nor associative. Multiplication of a vector by real(Scalar) number OR Scalar multiplication of a Vector: The product of a vector ⃗ and a positive number (Scalar) gives a vector, whose magnitude is changed by a factor but direction is same as that of ⃗. | ⃗| |⃗ | If is negative, the direction of the vector ⃗ is opposite to the direction of the vector ⃗ and magnitude is – times | ⃗ |. If the multiplying factor is dimensionless then ⃗ have the same dimensions as that of ⃗ and is product of dimensions if has dimensions. Resolution of vectors: Splitting a given vector into a number of components is called resolution of vectors OR The process of finding the components of a given vector is called resolution the vector. Expressions for X and Y components of a Vector: Consider a vector ⃗⃗⃗⃗⃗⃗ ⃗ in X-Y plane, which makes an angle with the positive X-axis. Draw AM and AN perpendicular to X and Y axes respectively. Let ⃗⃗⃗⃗⃗⃗⃗ ⃗ and ⃗⃗⃗⃗⃗⃗⃗ ⃗ From parallelogram law of addition, we have ⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗ ⃗ ⃗ (Here ⃗ is x - component of ⃗ and ⃗ is y - component of ⃗) Vectoricaly ⃗⃗ ⃗⃗ and ⃗⃗ ⃗⃗ Note: ⃗ and ⃗ being perpendicular are called Rectangular components of ⃗ U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 3  MOTION IN A PLANE Magnitude: Magnitude of ⃗ is given by | ⃗| Now, and Taking √ ( ) Note: (i) In terms of unit vectors, ⃗ ̂ ̂ ̂ ̂ where ⃗ ̂ and ⃗ ̂ (ii) If ⃗ is in XYZ plane and makes an angle d with X, Y and Z axes respectively, then , , and ⃗ ̂ ̂ ̂ The magnitude of ⃗ is, √ Find the magnitude and direction of the resultatnt of two vectors ⃗⃗ and ⃗⃗⃗ in terms of their magnitudes and angle between them. Let ⃗⃗⃗⃗⃗⃗ and ⃗⃗⃗⃗⃗⃗⃗ represent the two vectors ⃗ and ⃗⃗ making an angle. Then using the parallelogram method of vector addition ⃗⃗⃗⃗⃗⃗ represents the resultant vector ⃗⃗. ⃗⃗ ⃗ ⃗⃗ Draw SN is normal to OP extended. Magnitude: From geometry, U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 4  MOTION IN A PLANE √ Direction: Let be angle made by the resultant vector ⃗⃗ with the vector ⃗, then ( ) Limitations of Graphical method of adding vectors: (i) It is very difficult method. (ii) It has limited accuracy. To overcome these limitations Analytical method of addition of vectors is preferred. Addition of vectors – Analytical method: In two Dimensions: Consider two vectors ⃗ and ⃗⃗ in X-Y plane. If ⃗ ̂ ̂ and ⃗⃗ ̂ ̂ then, ⃗⃗ ⃗ ⃗⃗ ⃗⃗ ( ̂ ̂) ( ̂ ̂) ⃗⃗ ̂ ̂ ̂ ̂ ⃗⃗ ̂ ( )̂ ⃗⃗⃗ ̂ ̂ Where and In three Dimensions: If ⃗ ̂ ̂ ̂ and ⃗⃗ ̂ ̂ ̂ then, ⃗⃗ ⃗ ⃗⃗ ⃗⃗ ( ̂ ̂ ̂) ( ̂ ̂ ̂) ⃗⃗ ̂ ̂ ̂ ̂ ̂ ̂ ⃗⃗ ̂ ( )̂ ̂ ⃗⃗⃗ ̂ ̂ ̂ This method can be extended to addition and subtraction of any number of vectors. Motion in a plane Position vector: The position vector ⃗ of a particle located in X-Y plane with reference to the origin is given by, ⃗ ̂ ̂ Where and are component of ⃗ along X-axis and Y-axis respectively. Displacement: Consider a particle moves along curve. Initially it is at at time and moves to a new position at time. Then the displacement is given by, ⃗ ⃗ ⃗ ⃗ ̂ ̂ ̂ ̂ ⃗ ̂ ̂ ̂ ̂ ⃗ ̂ ̂ ⃗⃗ ̂ ̂ Where and U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 5  MOTION IN A PLANE Velocity: Average velocity: It is defined as ratio of the displacement to the time taken. ⃗ ̂ ̂ ⃗̅ ⃗̅ ̂ ̂ ⃗̅⃗ ̅ ̂ ̅ ̂ ̅ ̅ Direction of the average velocity is same as that of the displacement. Instantaneous velocity (Velocity): It is given by the limiting value of the average velocity as the time interval approaches to zero. ⃗ ⃗ ⃗⃗ ⃗⃗ The direction of velocity at any point on the path of the object is tangential to the path at that point and in the direction of the motion. The components of the velocity ⃗ are given by, ⃗ ⃗ ⃗ ( ̂ ̂) ⃗ ̂ ̂ ⃗ ̂ ̂ ⃗⃗ ̂ ̂ The magnitude is given by √ and Direction is given by ( ) Acceleration: Average acceleration: It is defined as the change in velocity divided by time interval. ⃗ ( ̂ ̂) ̅⃗ ̅⃗ ̂ ̂ ⃗̅⃗ ̅ ̂ ̅ ̂ Instantaneous acceleration (Acceleration): It is the limiting value of the average acceleration as the time interval approaches zero. ⃗ ⃗ ⃗⃗ ⃗⃗ The Components are given by, ⃗ ⃗ ̂ ̂ ⃗ ⃗ ̂ ̂ U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 6  MOTION IN A PLANE ⃗ ̂ ̂ ⃗⃗ ̂ ̂ In one dimension the direction of velocity and acceleration is same or in opposite direction but in two or three dimensions, velocity and acceleration vectors may have any angle between 0 0 and 1800. Motion in a plane with constant acceleration Consider an object moving in x-y plane and its acceleration ⃗ is constant. Let the velocity of the object be ⃗ at time and ⃗ at time , then (i) ⃗ ⃗ ⃗ In terms of its components, (ii) Displacement is ⃗ ⃗ ⃗ ⃗ In terms of its components, (iii) ⃗ ⃗ ⃗ ⃗ ⃗ In terms of its components, The motion in plane can be treated as two separate simultaneous one-dimensional motions with constant acceleration along two perpendicular directions. Relative velocity in two Dimensions: Suppose two objects and are moving with velocities ⃗ and ⃗ , then the velocity of the object relative to that of is, ⃗ ⃗ ⃗ Similarly, the velocity of the object relative to that of is, ⃗ ⃗ ⃗ Therefore, ⃗ ⃗ and | ⃗ | | ⃗ | Examples for Motion in a plane: (i) Projectile motion (Uniformly accelerated motion) (ii) Circular motion (Non-uniformly accelerated motion) When a particle traces a curve in two dimensional plane, the velocity of the particle changes at least in direction. Hence, a two dimensional motion along a curve is essentially an accelerated motion. Acceleration may be uniform or non-uniform. Projectile: A projectile is any object thrown into air or space. Projectile motion: Motion associated with a projectile in parabolic path is called Projectile motion. Ex: A ball leaving the hand of a bowler, a stone thrown at an angle to the horizontal, an object dropped from an aeroplane in horizontal flight. The motion of projectile may be thought of as the result of two separate, simultaneously occurring components of motion. One component is along a horizontal direction without any acceleration and other along the vertical direction with constant acceleration due to gravity. Note: It was Galileo, who first stated this independency of the horizontal and vertical components of projectile motion. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 7  MOTION IN A PLANE Analysis of Projectile motion: Let a projectile is projected with initial velocity ⃗ that makes an angle with x-axis. The acceleration acting on it is due to gravity and is directed vertically downwards. , hence ⃗ ̂ The components of initial velocity ⃗ are, The components of velocity at time are, The components of displacements at time are, (along X-axis) Path of a projectile: The path described by the projectile is called trajectory. The trajectory is a parabola Expression for path of a projectile (Show that the path of a projectile is Parabola): The displacement of the projectile along X-axis is, The displacement of the projectile along Y-axis is, ( ) ( ) The equation represents a parabola. Hence the trajectory is a parabola. Time of Flight: It is the time during which the projectile is in flight. It is denoted by. Expression for Time of flight: The component of velocity along Y-axis at time t is, At maximum height and time for maximum height,. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 8  MOTION IN A PLANE Time of flight bec use “time of sce t = time of desce t” Maximum height of a projectile: It is the maximum height reached by the projectile in time. It is denoted by. Expression for maximum height of a projectile: ( ) ( ) ( ) Horizontal Range of projectile: It is the horizontal distance covered by the projectile during its flight. It is denoted by Expression for Horizontal Range of projectile: Displacement along X-axis is, Now and ( ) Note: (i) For a given speed of projection, the projectile will have maximum range when is maximum or angle of projection is. Then angle of projection, U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 9  MOTION IN A PLANE (ii) Show that when the angle of projection is. ( ) The maximum range of a projectile is equal to times the maximum height reached. Uniform circular motion: Motion of the object in a circular path at a constant speed is called uniform circular motion. Even though the object moves at a constant speed it has acceleration, because there is a continuous change in its direction of motion. Hence there is a change in its velocity from point to point. Expression for Acceleration: Let ⃗ and ⃗⃗⃗⃗ be the position vectors and ⃗ and ⃗⃗⃗⃗ are the velocities of the object when it is at and as shown. Velocity at a point is along the tangent at that point in the direction of motion. From , ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗ ⃗ ⃗ is change in velocity, which is towards the centre. Since the path is circular, ⃗ and ⃗⃗⃗⃗ are perpendicular to ⃗ and ⃗⃗⃗⃗ respectively. Therefore ⃗ is perpendicular to ⃗. ⃗ Since ⃗ is perpendicular to ⃗, ̅ is along ⃗ and perpendicular to ⃗ and directed towards the centre of the circle. ⃗ ⃗ Its magnitude is given by, | ⃗| | ⃗| Since the velocity vectors ⃗ and ⃗⃗⃗⃗ are always perpendicular to ⃗ and ⃗⃗⃗⃗, the angle between ⃗ and ⃗⃗⃗⃗ is also and are similar. | ⃗| | ⃗| | ⃗| | ⃗| ⃗ | ⃗| U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 10  MOTION IN A PLANE | ⃗| This equation represents the magnitude of acceleration and is directed towards centre. Centripetal acceleration: The acceleration, which is directed towards the centre, is called centripetal acceleration. The term centripetal acceleration was termed by Newton and Centripetal comes from a Greek term which means Centre seeking of towards centre. Note: (i) In uniform circular motion as the object moves from P to Q (in the above figure) in time the line OP turns through an angle , called angular distance. If the distance travelled then, But , where R is the radius of the trajectory. ( ) (ii) Time period (T): Time taken by an object to make one revolution. (iii)Frequency ( ): Number of revolution made in one second. Distance moved in time period Comparing equation (1) and (2), we get Then Acceleration, U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 11  MOTION IN A PLANE Suggested Questions. One mark. 1) What is unit vector? 2) Represent the unit vector in mathematical form. 3) What is Zero (null) vector? 4) Is scalar multiplied by a vector, a vector or a scalar? 5) What is the minimum number of vectors to give zero resultant? 6) When will be the resultant of two given vectors is maximum? 7) What is time of flight of a projectile? 8) At what angle range of a projectile is maximum? or When the range of a projectile does become maximum? 9) What is the relation between maximum height and maximum range of a projectile? 10) For angle of projection , is the range of the projectile. Then write another angle of projection for which the range is same. Two marks. 1) What are scalar and vector? Give example. OR Distinguish between scalar and vector. 2) A unit vector is represented by ̂ ̂ ̂ If the value of and are and respectively, then find the value of. 3) State and explain parallelogram law of vector addition. 4) What is a projectile? Give an example. 5) Write the equation for the trajectory of a projectile motion. What is the nature of its trajectory? Three marks. 1) State and explain the triangle law of vectors addition. 2) What is resolution of vectors? Write expressions for and components (Rectangular) of a vector. or Obtain the equations for rectangular components of a vector in two dimensions. 3) Derive an expression for magnitude of resultant of two concurrent vectors. or Find the magnitude of the resultant of two vectors and in terms of their magnitude and angle between them. 4) Obtain an expression for maximum height reached by a projectile. 5) Obtain an expression for time of flight of a projectile. 6) Obtain the expression for range of a projectile. Five marks. 1) What is projectile motion? Show that trajectory of projectile is a parabola. or What is projectile motion? Derive an expression for trajectory of projectile. or Show that the path of the projectile is a parabola. 2) What is centripetal acceleration? Derive an expression for centripetal acceleration of a particle in uniform circular motion. or What is centripetal acceleration? Derive the expression for radial acceleration. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 12  LAWS OF MOTION Chapter 5 LAWS OF MOTION Motion of the object needs the concept of Velocity and acceleration. In earlier chapter we have studied the motion but not what causes the motion? In this chapter we study about what causes the motion? In early days it was known that some influence was needed to keep the body in motion and it was also known that rest is the natural state of an object. Aristotle’s Law and its Fallacy: Law: An external force is required to keep a body in motion. Fallacy in the law: A moving object comes to rest because; the external force of friction on the object by the floor opposes its motion. If there is no friction no force is required to keep the object in motion. Law of Inertia: If the net external force is zero, a body at rest continues to be at rest and a body in motion continues to be in uniform motion. Note: Aristotle’s view point about the motion of the body was rejected by Galileo and gave the law of Inertia. Inertia: The property of a body to change its state of rest or uniform motion unless some external force acts on it. Mass of a body is measure of inertia. Types of inertia: (i) Inertia of rest: The property of a body to remain at rest. (ii) Inertia of motion: The property of a body to oppose the change in its motion. Newton’s laws of motion: Based on the Galileo’s idea, the intimate relationship between force acting on a body and its motion executed by the body was first understood by Isaac Newton. Newton’s first law of motion: Everybody continues to be in its state of rest or of uniform motion in a straight line unless compelled by some external force to act otherwise. Alternate statement of Newton’s first law: The first law can be stated in terms of acceleration as “If the net external force on a body is zero, its acceleration is zero. Acceleration can non zero only if there is a net external force on the body”. Ex: A passenger in a bus is pushed back when the bus suddenly starts moving. A person in a moving vehicle tends to fall forward when the vehicle suddenly stops. Significance: Newton’s first law of motion gives the definition for force and reveals Inertia, a fundamental property of all matter. Force is a vector quantity and dimensions are [ ] Force: The external agency which changes or tends to change the state of rest or state of uniform motion of a body in a straight line. U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 1  LAWS OF MOTION Momentum: Momentum of a body is defined as the product of its mass and velocity. Momentum = mass velocity ⃗ ⃗ Momentum is a vector quantity and its SI unit is. Dimensions of momentum are [ ] Note: The concept of momentum was introduced by Newton. It is a measure of the ability of a body to impart motion to another. Newton’s Second law of motion: The rate of change of momentum of a body is directly proportional to the applied force and takes place in the direction in which the force acts. ⃗⃗⃗ Derivation of using Newton’s second law of motion: Consider a body of mass , moving with a velocity ⃗⃗⃗⃗ and having momentum ⃗⃗⃗⃗. Let a force acts on it for time. Then velocity changes to ⃗⃗⃗⃗ and momentum to ⃗⃗⃗⃗. Then change momentum is, ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ (⃗⃗⃗⃗ ⃗⃗⃗⃗ ) ⃗⃗⃗ For simplicity, we choose Then, ⃗⃗⃗ ⃗⃗⃗ Significance: Newton’s second law of motion signifies momentum and gives a formula to measure the force. Unit of Force: SI unit of force is or and newton(N): One newton is that force which causes an acceleration of to a body of mass. Some applications of Newton’s second law:  A cricket player lowers his hands while catching a ball  A person falling on a cemented floor gets injured but a person falling on heap of sand is not.  The vehicles are fitted with shockers (springs).  Glass wares and china wares are wrapped with straw pieces before transportation. Components of force: We have, ̂ ̂ ̂ and ̂ ̂ ̂ Then, ̂ ̂ ̂ ( ̂ ̂ ̂) U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 2  LAWS OF MOTION On comparing the co-efficient of ̂ ̂ ̂ We have, , and Impulsive force: Large force acting on a body for a short time is called an impulsive force. Ex: A ball hit by bat, kicking a football, hammering a nail etc. Impulse: It is the product of the force and time interval for which the force acts. It is denoted by. ( ) It is a vector. SI unit of Impulse is ( ). Dimensions are [ ] Note: Impulse and momentum have same dimensions. Impulse-Momentum theorem: Impulse is equal to change in momentum. Proof: Impulse, But we have ( ) ( ) ⃗⃗⃗⃗ Newton’s third law of motion: To every action there is always an equal and opposite reaction. ( ) ⃗ ⃗ Note: The term action and reaction means the force and the force on the object A by the object B and the force on object B by A act at the same instant. Action and reaction forces act on different bodies, not on the same body. So they do not cancel each other. Significance: Newton’s third law signifies that forces never occur singly in nature, but they always occur in pairs. Launching of rocket is based on this law. Illustrations of Newton’s third law:  When a person jumps from a boat, he pushes the boat in the backward direction while the boat pushes him in the forward direction.  A swimmer pushes the water in the backward direction and the water pushes the swimmer in the forward direction.  A person walking on the floor. Law of conservation of momentum: The total momentum of an isolated system of interacting particle is conserved. Ex: Recoil of gun, the motion of rocket is based on this principle. Proof: Consider two bodies A and B, with initial momentum ⃗⃗⃗ and ⃗⃗⃗. Let the bodies collide and get apart with final momentum ⃗⃗⃗⃗ and ⃗⃗⃗⃗ respectively. From Newton’s second law, U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 3  LAWS OF MOTION ⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ But from Newton’s third law, ⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ ( ) ⃗⃗⃗⃗ ⃗⃗⃗ (⃗⃗⃗⃗ ⃗⃗⃗ ) ⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗ Note: Isolated system: It is a system with no external force acts on it. Equilibrium of a particle: Resultant force: Resultant force is that single force which produces the same effect on the body as the net effect of all the forces together. Equilibrium: A set of forces are said to be in equilibrium if their resultant is zero. Equilibrant: The equilibrant is that force which when acts together with those forces keep the body in equilibrium. Equilibrium of a particle: The particle is said to be in equilibrium if the net external force acting on the particle is zero. Equilibrium under two forces: Let two forces, ⃗⃗⃗ and ⃗⃗⃗⃗ act on a particle. The particle will be in equilibrium, if ⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗ ⃗⃗⃗⃗ That is two forces on the particle must be equal and opposite. Equilibrium under three forces: Let three forces, ⃗⃗⃗ , ⃗⃗⃗⃗ and ⃗⃗⃗⃗ act on a particle. The particle will be in equilibrium, if ⃗⃗⃗ ⃗⃗⃗⃗ ⃗⃗⃗⃗ Equilibrium under several forces: A particle is in equilibrium under the action of several forces, if the resultants of the resolved components of these forces in each of the X and Y-directions are independently zero. ∑ and ∑ Note: For a particle to be in equilibrium, minimum number of forces acting on a particle must be two. Common forces in mechanics: Common forces in mechanics are  Gravitational force  Spring force  Tension in the string U N Swamy, Lecturer in Physics, MGGPUC, KUNIGAL Page | 4  LAWS OF MOTION Gravitational force: It is the force of attraction between the two bodies due to their masses. Every object on the earth experiences the force of gravity due to earth and it can act at a distance without need of material medium. Weight: The force exerted by the earth on the object is called the weight of the object. It is given by,. Weight is vector quantity and its unit is. Differences between Mass and Weight Mass Weight It is the amount of matter contained in a body It is the gravitational force of attraction on a body It is a scalar It is vector Mass of the body remains same at all places Weight of the body varies from place to place SI unit is kilogram SI unit is newton Spring force (F): When a spring is compressed or extended by an external force a restoring force i