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- 1. Science: -The systematic knowledge acquire by systematic knowledge acquire by systematic analysis of nature & natural phenomenon is known as science. Science is a Greek word means ‘to know’ taken from Scintia. OR Knowledge acquired by man through organised attempted is known as science. Branches of science - Chemistry, Zoology, Botany, Geology, Astronomy, Geology, Meteorology, Oceanography, Seismology Physics: -the branch of science in which we study about matter, properties of matter, energy & there inter relation. Scientific Method: -It has following steps- 1) To identify the problem. 2) Collection of the information about the problem. 3) Organisation of the experiments. 4) Recording of the observations. 5) Classification and analysis of observations. 6) Get result and make a hypothesis. 7) Test the hypothesis, if it is true then it is called theory.8. If the theory can explain the entire phenomenon related to the problems then it is called principle. Scientific law. The general statement of happenings in nature after repeated observations and experiments is known as scientific law. It is actually the essence of a large number of observations and experiments. A law has no theoretical or other basis. Theory. The speculation on the basis of determined laws is known as theory. Theory: -Experimentally true hypothesis is called theory. When several laws are determined, then one may make a guess about the way the universe must be constituted in order that such things happen; this guess is called theory. It may be noted that there is a basic difference between a theory and law. Theory is given by man and hence may be wrong but law is made by nature and is always right. Theories are never derived from observations — they are created to explain observations. Hypothesis: -If to know the reason for any phenomenon we make a prediction, this prediction is called hypothesis. A model is a visual picture of the phenomenon under study. Physics in Relation to Science: - 1) Mathematics: -Use of mathematics in physics is more than other branches. Mathematics is a toll of physics and it act as language of physics. Laws and principle of physics are written in the form of mathematical equations. Use of mathematics is most useful in the development of physics without mathematics we cannot understand the principles and laws of physics. 2) Chemistry: -In development of chemistry physics is most useful. Periodic arrangement of elements, nature of valency, chemical structure of large molecule & chemical bond can be explain by atomic physics. X-rays , Diffraction, atomic spectrum, nuclear physics & magnetic resonance are the gift of physics. 3) Biology: -Microscope & electron microscope are gift of physics, which help in the study of cells, virus, bacteria. We can detect fracture or bullet in our body by X-rays. Radiations are helpful in the treatment of various diseases like cancer etc. The entire machines used in biology are from physics. 4) Astronomy: -With the help of telescope we can see & study about stars and planets etc. Spectra of star help in study of the temperature & nature. (a) Medical physics (b) Meteorology (c) Geophysics (d) Environmental physics (e) Communication (f) Material science. Need of Measurement: -In physics we express some phenomenon in terms of various physical quantities. This relation may be expressed in the form of statement or mathematical equation. To prove the equation or statement we need exact measurement of physical quantities. In 18th century magnitude of the physical quantities was measured with the help of our sense or organs. But this measurement was only qualitative estimate not quantitative and not correct. So we require accurate measurement. Measurement: -Measurement is a process of comparison of a physical quantity with the standard amount of the quantity called unit. The reference standard for measurement of a given physical quantity is called unit. The particular amount of the quantity used as a standard of measurement is called the unit of that quantity. Hence measurement means to find out the number of times that unit is contained in the quantity. Magnitude of a physical quantity = numeric value × unit = N × u N1 & N2 are numeric value; u1 & u2 are the units in different system of units. N1 u1 = N2 u2 =Constant. If the size of the unit is small then the magnitude of the physical quantity is large & vice versa. Properties of the Units: -Any standard unit must have following properties- 1) The size of the unit must be comparable to the physical quantity to be measured i.e. neither too small nor too large. 2) It should be well defined. 3) It should not change with time& place. 4) It should be easily reproduced at all the places. 5) It should not change with physical condition such as temperature, pressure etc. 6) It should be easily converted from one system to another.
- 2. Physical quantities: -All the quantities in terms of which laws of physics can be expressed & which can be measured directly or indirectly are called physical quantities. These are of two types- a) Fundamental quantities: -Those quantities which are independent on any other quantities are called fundamental quantities OR A physical quantity made by itself is called an elementary physical quantity. It does not need other quantity or quantities to represent or express itself. Hence they are also called basic or fundamental quantities.e.g. mass, length, time, temperature, electric current, luminous intensity, quantity of matter. b) Derived Quantities: - Those quantities which depend on two or more then two fundamental quantities are called derived quantities. OR A physical quantity, made by combining (compounding) two or more elementary quantities, is called a derived (compounded) physical quantity. e.g. speed, work, power, force, torque etc. 1. Fundamental units The units selected for measuring fundamental quantities are called fundamental units. Those units, which are independent on any other units, are called fundamental units. Meter, kg, sec etc. 2. Derived units. The units of physical quantities which can be expressed in terms of fundamental units (mass, length and time) are called derived units. Those units, which depend on two or more then two fundamental units, are called derived units e.g. Newton, Joule etc. System of Units: - the common systems of units are- 1) Foot-Pound-Second (FPS) system: -It is also known as British system. Foot, Pound and Second are the unit of length, mass & time respectively. 2) Meter-Kilogram-Second (MKS) system: -It is also known as Metric or French system. Meter, Kg and Second are the unit of length, mass & time respectively. 3) Centimetre-gram-Second (CGS) system: -It is derived from MKS system. Centimetre, gram and Second are the unit of length, mass & time respectively. 4) SI Units:- It is modified MKS system. S.I. Units. Systeme internationale d’ units is the new comprehensive and rationalized system of units accepted by the Eleventh Conference of Weights and Measures in 1960. It is used in science and technology all over the world. In SI. there are seven fundamental units or base units and two supplementary units. Fundamental Quantity Fundamental Unit Symbol Length metre m Mass kilogram kg Time second s Current ampere A Temperature kelvin K Luminous intensity candela cd Amount of substance mole mol 1. One Metre is equal to 1,650,76373 wavelengths in vacuum of the radiation (orange red light) emitted by the krpton-86atom in its transition between the states 2p10 and 5d5. 2. One Kilogram is the mass of a platinum-iridium(90 % & 10 %) cylinder kept at the International Bureau of Weights and Measures office Severs near Paris (France). 3. One second is the time interval in which Cesium—l33 atom vibrates 9,192,631,771 times. 4.One Ampere is that constant current which, when flowing conductors of infinite length and negligible area of cross section one metre apart in vacuum, would produce a force of 2 x 10-7 Newton per metre of length. 5. Kelvin is the basic unit of temperature and it is equal to fraction 1/273.16 of the thermodynamic temperature of triple point of water. 6.Cendela is the basic unit of luminous intensity is defined as the luminous intensity, in a perpendicular of area Supplementary quantities Supplementary unit Symbol for the unit Plane angle radian rad Solid angle steradian sr
- 3. 1/600,000 square metre of a black body at a temperature of freezing platinum under a pressure of 101325 Pa. 7. Mole. Mole is the amount of substance of a system, which contain as many elementary entities as there are atoms in 0.012 kg of Cabon C12 . One mole contains 6.0225 x 1023 entities. Radian is the plane angle between two radii of a circle, which cut off on the circumference an arc equal to the length of the radius. Angle (in radian) = Arc/ radius Steradian is the solid angle, which, with its vertex at the centre of sphere, cuts off an area on the surface of the sphere equal to that of a square having sides of length equal to the radius of the sphere. If S is the area cut off on the surface of a sphere of radius r, the solid angle at the centre of the sphere is given by φ(in Steradian) = S/r2 . Coherent system :−A system of units based on certain set of basic or fundamental units from which all the derived units are obtained by multiplication or division without introduction of numerical factors, is called a coherent system of units.Two such coherent system of units are (i) M.K.S. (ii) S.I. (a) Astronomical Units of Length (i) Light year (l.y.) = 9.467 x 1016 m (ii) Astronomical Unit (A.U.) = 1.496 × 1011 m (iii) Parsec = 3.08 ×1016 m (b) Atomic Units of Length (i) Micron (micro-metre) = 10−6 m (n-m) (ii) Milli micron = 10−9 m (iii) Angstrom Unit (A) = 10−10 m (diameter of atom) (iv) Fermi (f) = 10−15 m (diameter of nucleus) ADVANTAGES OF SI UNITS: - (i) Coherent system of units. All the derived SI units can be obtained by dividing and multiplying the base SI units and no numerical factors are involved as used to be the case with units of C.G.S. and M.K.S. systems. For example, the unit of force is obtained when unit of mass is multiplied by unit of acceleration. (ii) Rational system of units. One of the important advantages of SI is that it is a rational system of units. It is evident from the following discussion. (iii) It uses one unit for one physical quantity. For example, all types of energies (e.g. mechanical energy, heat energy, electrical energy etc.) are measured in Joules. However, in C.G.S. and M.K.S. systems, different units are used for different types of energies. Thus in M.K.S. system, mechanical energy is measured in Joules, heat energy in calories and electrical energy in watt-hours. (iv) In C.G.S. and M.K.S. systems, both gravitational and absolute units of a physical quantity are used. For example, in the M.K.S. system, the absolute unit of force is Newton and gravitational unit is kilogram force. This often leads to confusion. However, in SI, only Newton is used as the unit of force. (v) Decimal system. Like C.G.S. and M.K.S. system, SI is also a decimal system. This makes the calculation work a simple affair. (vi) Blend of practical and theoretical work. It can be used by research scientists, the technicians, the practicing engineers and by the layman, thereby blending the theoretical and practical world. Length Order of magnitude (m) Mass Order of magnitude (kg) Radius of proton 10–15 Electron 10–30 Radius of atom 10–10 Proton 10–27 Height of man 100 Ant 10–2 Radius of earth 107 Man 102 Earth-Sun distance 1011 Earth 1024 Distance of nearest star 1010 Sun 1030 Prefix Symbol 1012 tera T 109 giga G 106 mega M 103 kilo k 10–3 milli m 10–6 micro 10–9 nano n 10–12 pico p Time interval Order of magnitude (s) Period of visible light 10–15 Period of microwaves 10–10 Period of human heart 100 Period of earth rotation 105 Life 109
- 4. MEASURING LARGE DISTANCES: - (i) Measurement of size of heavenly bodies. The size of a heavenly body (e.g. diameter of moon) can be found by measuring angle subtended by its diameter AB at a point E on the earth (See Fig. 3.1). For this purpose, we use a telescope. If AB (= d) is the diameter of the moon and D is its distance from earth, then, D d Radius Arc Angle =⇒= θ or d = D θ Since D (distance of the moon from earth) is known and θ is measured, the diameter of the moon can be found. Note that if θ is in radian and D in meters, then diameter d of the moon will also be in meters. (ii) Measurement of distances of heavenly bodies. (a) Parallax method. In this method, a planet P is observed simultaneously from two distant places A and B (See Fig. 3.2). The angle θ ( = ∠ APB) is measured. If AB = d and S (= AP = BP) is the distance of the planet from the surface of earth, then, θ θ d S S d Radius Arc Angle =⇒=⇒= If θ is in radian and b in A.U., then distance S of the planet from earth will be in A.U. 1 A.U. = 1.496 × 1011 m. MEASUREMENT OF MASS: - Mass is the basic property of matter. We normally define the mass of a body as the quantity of matter contained in the body. (i) We can say that mass is a measure of inertia of a body. The more mass a body has the harder is to change its state of motion; it is harder to start it moving from rest or to stop it when it is moving or to change its motion out of straight line. This fact directly follows from Newton’s second law of motion. If a force F acts on a body of mass mi and produces an acceleration `a‘, then according to Newton’s second law of motion, F = mi a a F mi = ...(i) The mass defined in this way is called inertial mass (mi) because it makes use of inertia of the body. Eq (i) tells us that more mass a body has, the larger the force necessary to give it a given acceleration and vice-versa. When a body is in transnational motion, the ratio of resultant force applied on the body to the acceleration produced in it is called inertial mass (mi) of the body. MEASUREMENT OF INERTIAL MASS: - The inertial mass of a body can be determined with the help of inertial balance. Principle. When a long metal strip fixed at one end and carrying an object at the other end is displaced horizontally, it executes vibratory motion. The time period of vibrations depends upon the mass of the object. mT ∝ Theory. It consists of two long metal strips. One end of each strip is clamped to a wooden board in such a way that the flat faces of the strip are vertical; this arrangement permits easy horizontal vibrations. The other ends of the strip support a pan. The object whose inertial mass m1 is to be determined is placed in the pan. When the pan is displaced horizontally, it executes vibratory motion & we measure time period T1. 11 mkT = - - - -(1) The object whose inertial mass m2 is known is placed in the pan. When the pan is displaced horizontally, it executes vibratory motion & we measure time period T2. 22 mkT = - - - -(1) Dividing eq1 & eq2. 2 1 2 1 mk mk T T = Hence 2 1 21 T T mm =
- 5. Therefore, the inertial mass m2 of the object can be determined from the above relation. (ii) Gravitational mass (mG). When the mass of a body which is measure of effect of gravity and the body is not in motion, it is called gravitational mass (mG). However, careful experiments show that they are equal. and can put mG = mi. MEASUREMENT OF GRAVITATIONAL MASS: - A physical balance is used to measure the gravitational mass of a body by comparing it with a known standard mass. Construction:− It consists of a rigid light beam of aluminum or brass. The beam is balanced at its center on a knife edge which rests on a flat piece. The flat piece is fixed on the top of a vertical pillar. Two scale pans of equal masses are suspended at the two ends of the beam. A pointer is attached at the centre of the beam, which moves on a scale fixed at the base of the pillar. When the physical balance is in equilibrium, the pointer remains either at the zero of the scale or it swings equally on both sides of the zero mark. The ends of the pointer are provided with screws for zero mark adjustment. A lever arrangement is provided at the base for raising or lowering the beam. Leveling screws are provided at the base board to make it horizontal. A plumb line is suspended near the vertical pillar to ascertain whether the base board is horizontal or not. Procedure:− In order to find the mass of a body with a physical balance, the following procedure is adopted: (i) The first step is to make the base board horizontal. This can be done by bringing the plumb line just above the pointed mark through the adjustment of leveling screws fixed at the base board. (ii) The next step is to ensure that the beam is horizontal when the pans are empty. For this purpose, the beam is raised gently by the lever and it is ensured that the pointer swings equally on both sides of the zero mark. This can be achieved through the adjustment of screws provided at the ends of the beam. (iii) The beam is now lowered and the body whose mass (m) is to be found out is placed on the left pan of the balance. The known weights are placed on the right pan till the pointer remains at zero of the scale or it swings equally on both sides of the zero mark. Under this condition, the balance is in equilibrium. Consequently, the total mass (M) of weights on the right pan gives the mass of the body. It is because weight of the object is equal to weight on the right pan i.e. m g = M g or m = M Two things are worth noting. First, a physical balance measures the mass of a body at a place and not its weight. Secondly, a physical balance is used for comparing masses of the bodies. Weight. The weight of a body is defined as the gravitational force exerted on it due to earth i.e., The weight of a body is defined as the gravitational force with which earth earth attract the body towards center. Weight, W = gravitational force on the body due to earth W = mass of body × acceleration due to gravity = m × g SPRING BALANCE A spring balance is used to measure the weight of a body. The working of spring balance is based on Hooke’s law stated below : The extension produced in a spring is directly proportional to the applied force provided the spring is not loaded beyond elastic limit i.e. F = k x where F = force applied to the spring k = spring constant x = extension produced in the spring Construction: - It consists of steel helical spring enclosed in a metal case having a slit along its length. The upper end of the spring is attached to a ring provided at the top of the case. The lower end of the spring is attached to a pan on which the body whose weight is to be determined is placed. A pointer P attached to the spring moves in the slit and rests against a scale graduated in newtons. Marks on the scale are equally spaced because of Hooke’s law.
- 6. When there is no load on the pan, the pointer reads zero. When the body to be weighed is placed on the pan, the spring is stretched due to the weight of the body. When the object hangs at rest, the upward force due to the spring balances the downward force due to gravitation attraction of earth (i.e. weight of the body). Therefore, the position of the pointer on the scale gives the weight of the body. MEASUREMENT OF TIME INTERVALS: - How long an event lasts is called time interval. Thus time interval is tagged with two questions : “At what time does an event occur” ? and “how long does the event last ?” The SI unit of time is second. There are 60 seconds in a minute, 60 minutes in an hour and 24 hours in a day. Thus clock is a device that counts the number of repetitions (including the fractions) of some periodic event. For example, we can measure time intervals by counting the number of swings of a pendulum or the number of cycles of an alternating current. Our general concept of clock will not depend on the details of a mechanical or electrical device but on the operational definition of a device that counts the number of repetitions of some periodic event. The instruments commonly used for measuring short time intervals are spring-driven stopwatches or electric stop clocks. Other methods for measuring time intervals are pendulums of suitable lengths, oscillating quartz crystals and oscillating atoms. Dimension Analysis: - When we express a physical quantity in terms of power on fundamental quantity, then it is called dimensional formula of that quantity. For example, the dimensional formula of force is [M L T–2 ] OR Expression which tells how and which of the fundamental quantities are involved in making that physical quantity called dimension formula. The equation relating a derived quantities & fundamental quantities is known as dimensional equation. e.g. Dimension equation of force [F] = [M L T–2 ] The dimensions of a physical quantity may be defined as the powers to which the fundamental quantities (M, L and T) must be raised to represent the physical quantity. Dimension:− In a dimension formula the powers of the fundamental quantities (M, L and T) are called dimension. e.g. Unit of force has one dimension in unit of mass, one dimension in unit of length and minus two dimensions in unit of time. (a) Dimensions of mechanical quantities:−These are expressed in terms of M, L & T (for mass, length & time). (b)Dimensions of electrical quantities:−These are expressed in terms of M. L, T and A (for electric current). (c)Dimensions of thermal quantities:−These are expressed in terms of M, L, T, K (for thermodynamic temperature) and mol (for amount of substance). (d)Dimensions of optical quantities:−These are expressed in terms of M, L, T, K and C (for luminous intensity). Sl. No. Physical Quantity Relation with other physical quantities Dimensional Formula SI Unit 1. Area length × breadth [L] [L] = [L2 ] m2 2. Volume length × breadth × height [L] [L] [L] = [L3 ] m3 3. Density = [ML–3 T0 ] kg m–3 4. Speed or Velocity = [M0 LT –1 ] ms–1 5. Acceleration = [M0 LT–2 ] ms–2 6. Force mass × acceleration [M] [M0 LT –2 ] = [MLT –2 ] N 7. Momentum mass × velocity [M] [M0 LT –1 ] = [MLT –1 ] kg m s–1 8. Work force × distance [MLT –2 ] [L] = [ML2 T –2 ] Nm 9. Power = [ML2 T –3 ] W
- 7. 10. Pressure = [ML–1 T–2 ] Nm–2 11. Kinetic energy × mass × (velocity)2 [M] [M0 LT –1 ]2 = ML2 T –2 Nm 12. Potential energy mass × g × distance [M] [M0 LT –2 ] [L] = ML2 T –2 Nm 13. Impulse force × time [MLT –2 ] [T] = [MLT –1 ] Ns 14. Torque force × distance [MLT–2 ] [L] = ML2 T –2 Nm 15. Stress = [ML–1 T–2 ] Nm–2 16. Strain = [M0 L0 T0 ] Number 17. Elasticity = [ML–1 T –2 ] Nm–2 18. Surface tension = [ML0 T–2 ] Nm–1 19. Force constant of spring = [ML0 T –2 ] Nm–1 20. Gravitational constant = [M–1 L3 T –2 ] Nm2 kg–2 21. Frequency = [M0 L0 T –1 ] s–1 22. Angle = [M0 L0 T 0 ] rad 23. Angular velocity = [M0 L0 T–1 ] rad s–1 24. Angular acceleration = [M0 L0 T –2 ] rad s–2 25. Moment of inertia mass × (distance)2 [M] [L2 ] = [ML2 T 0 ] kgm2 26. Angular momentum moment of inertia × angular velocity [ML2 T 0 ] [M0 L0 T –1 ] = [ML2 T –1 ] kg m2 s–1 27. Heat energy [ML2 T –2 ] J 28. Planck’s constant = [ML2 T –1 ] Js 29. Velocity gradient = [M0 L0 T –1 ] s–1 30. Radius of gyration distance [M0 LT 0 ] m PRINCIPLE OF HOMOGENEITY OF DIMENSIONS:− This principle is based on the fact that only quantities of the same kind (or dimensions) can be added or subtracted. According to this principle, a physical relation is dimensionally correct if the dimensions of fundamental quantities (mass, length and time) are the same in each and every term on either side of the equation. OR In a physical equation, the dimensions of all the terms are similar (homogenous).
- 8. The following are the uses of dimensional equations : (i) To check the correctness of a physical relation. (ii) To recapitulate a forgotten formula. (iii) To derive relationship between different physical quantities. (iv) To convert one system of units to another. (v) To find the dimensions of constants in a given relation. Limitations of Dimensional Analysis: -1) The method does not give any information about the dimensionless constant i.e. a number only. The value of the constant of proportionality has to be determined either by experiment or theoretically.2).It is not applicable to trigonometrical and exponential functions. 3) It cannot be used when a physical quantity depends on more than three physical quantities. For example, using dimensions we cannot derive the relation v2 = u2 + 2as. 4).It is also not applicable if the right hand side of the equation is a sum of two or more expressions. 5).In some cases the constants of proportionality are not dimensionless viz, G, η, Y. K etc. In such eases, the exact nature of forces involved is not known. 6. It does not tell whether the quantity is scalar or vector. QUANTITIES HAVING IDENTICAL DIMENSIONAL FORMULAE (DIMENSIONS):− Quantities Formula 1. Light year, Radius of gyration, Wavelength [M°LT0 ] 2. Rydberg constant, Propagation constant (k = 2π/λ) [M°L−1 T0 ] 3. Momentum, impulse [M1 L1 T −1 ] 4. Angular momentum, Planck's constant [M1 L2 T −1 ] 5. Work, Energy, Moment (Torque), Heat, Enthalpy [M1 L2 T−2 ] 6. Pressure, Stress, Elastic constant, Energy density [M1 L− 1 T −2 ] 7. Surface tension, Surface energy, Spring constant [M1 L° T−2 ] 8. Frequency, Velocity gradient, Angular velocity. [M°L0 T −1 ] 9. Time period, L/R, RC, (RC)1/2 [M°L0 T 1 ] 10. Velocity, E/B, oo/1 ∈µ [M°L1 T−1 ] 11. Gravitational potential, Latent heat [M° L2 T − 2 ] 12. Boltzmann's constant, Entropy. [M1 L 2 T−2 K−1 ] Mistake:− is inaccuracy in measurement due to defective measurement or observation due to the carelessness of the observer viz., recording a wrong reading or due to wrong calculation. Mistakes in measurement can be avoided by taking proper care by the observer. Errors :−When we measure, we may not get the true value, this inaccuracy in measurement is called measurement. The difference between the measured value and the true value is called an error. 1. Constant errors (Instrumental Error) are those, which affect the result by the same amount. For example, if a scale has faulty graduation, the same error is constantly repeated in all observations. To avoid inaccuracy due to constant errors, the same physical quantity is measured by different methods and existence of constant errors in any one of the methods is detected. An instrument may also have zero error. 2. Systematic errors are those, which occur according to some definite rule. For example, in a magnetometer, if the aluminium pointer is not pivoted at the centre of the circular scale, the readings will be inaccurate. Error due to air buoyancy in weighing and radiation loss in calorimetry is systematic errors. These errors can be eliminated by knowing the nature of the error. 3. Accidental errors may occur due to sudden variations in the surroundings viz., variation in atmospheric pressure and temperature. Random errors are not due to any systematic or constant source. This error gives too high or too low results. These errors are beyond the control of the observer. 4. Gross or Personal Error: -The error in reading may be due to the incapability of observer are called personal errors. However, experienced observers will take measurement correctly. 5. Least Count Error: -The error caused due to the least count of the instrument is called the least count error. This error is taken to be half of the smallest division on the scale of the instrument. For example, the least count of a metre scale is in 1 mm. Then the possible error in the length measured using metre scale is 05 mm. Least Count:−The least value of a quantity, which the instrument can measure accurately, is called the least count of the instrument. The maximum error caused by an instrument is equal to its least-count.
- 9. 6. Random Errors:-The errors which occur irregularly’ and at random, in magnitude and direction due to some unknown cause are called random errors. This is also called chance error. It makes to give different results for same measurements taken repeatedly. These errors are not due to any definite cause. 7. Gross (Personal)Errors: -The errors caused due to the carelessness of the person are called gross errors. No correction can be applied. Errors can be minimized by taking mean of the large number of values. The true (mean or absolute) value of a physical quantity is the arithmetic mean of a large number of readings of that quantity. If a1, a2, ... an are the n different readings of a physical quantity in an experiment, then true value of that quantity is True value, ∑ = = +++ = n 1i i _ n321 _ aaOR n a...----aaa a (i) Absolute error. The difference in the magnitudes of mean value and the measured value of a physical quantity is called absolute (i.e. actual) error. The absolute errors in the various measured values are : ∆ a1 = a – a1, ∆ a2 = a – a2 , ........................ ∆ an = a – an (ii) Mean absolute error. The arithmetic mean of the absolute errors of different measurements taken is called mean absolute error. n am a...----aaa n321 _ ∆+∆+∆+∆ =∆ Now the quantity is a = a m ± ∆ a m (iii) Relative The ratio of the mean absolute error to the mean value of the measured quantity is called the relative error or fractional error. m m a a valueTrue errorabsoluteMean errorlative ∆ ==Re (iv)Percentage Error If we multiply the relative error by 100, we get the percentage error. 100× ∆ = m m a a errorPercentage This means that if we make a single measurement, then result of measurement will lie between a – ∆a and a + ∆a. Propagation of Error: - (i) Error in sum. Suppose the result Q is given as the sum of two observed quantities a and b i.e. X = a + b If ∆a and ∆b are the absolute errors in the measurements of quantities a and b, measurements should be recorded as a ±∆a and b ± ∆b. If ∆X is the absolute error in the sum, then X ± ∆X = (a ± ∆a) + (b ± ∆b) or X ± ∆X = (a + b) ± (∆a + ∆b) or ± ∆X = ± (∆a + ∆b) Hence ∆X = + (∆a + ∆b) = – (∆a + ∆b) Maximum absolute error in X = Maximum absolute error in a + Maximum absolute error in b Therefore when two physical quantities are added, the maximum absolute error in the result is equal to the sum of the absolute errors in the physical quantities. (ii) Error in difference Let X = a – b If ∆a and ∆b are the absolute errors in the measurements of quantities a and b, the measurements should be recorded as a ± ∆a and b ± ∆b. If ∆X is the absolute error in the difference, then, X ± ∆X = (a ± ∆a) – (b ± ∆b) or X ± ∆X = (a – b) ± ∆a ± ∆b ± ∆X = ± ∆a ± ∆b Maximum error in X = ∆X = + (∆a + ∆b) = – (∆a + ∆b) Maximum absolute error in X = Maximum absolute error in a + Maximum absolute error in b Therefore when two physical quantities are subtracted, the maximum absolute error in the result is equal to the sum of errors in the physical quantities. (iii) Error in Product Let X = ab If ∆a and ∆b are the absolute errors in the measurement of quantities a and b, then measurements should be recorded as a ± ∆a and b ± ∆b. If ∆X is the absolute error in the result, then,
- 10. X ± ∆ X = (a ± ∆ a) (b ±∆ b) or X ± ∆X = ab ± a ∆b ± b ∆a ±∆a ∆b or ± ∆X = ± b ∆a ± a ∆b± ∆a ∆b Dividing both sides by X (= ab), we get ba b ba ab ab ba a X X ∆∆ ± ∆ ± ∆± = ∆± b b a a X X ∆ ± ∆± = ∆± ⇒ We can ignore the term ba b a∆∆ since it is the product of two small quantities and is therefore small in comparison with other terms. The terms b ,, b a a X X ∆∆±∆± are the relative or fractional errors in X, a and b respectively. . Maximum fractional error in X = Maximum fractional error in a + Maximum fractional error in b Therefore, when two physical quantities are multiplied, the maximum relative error in the result is equal to the sum of relative errors in the physical quantities 100 b 100100 × ∆ ±× ∆± =× ∆± b a a X X Max. Percentage error in X = Max. Percentage error in a + Max. Percentage error in b Therefore, when two physical quantities are multiplied, the maximum percentage error in the result is equal to the sum of percentage errors in the physical quantities. (iv) Error in division Let X = b a If ∆a and ∆b are the absolute errors in the measurements of quantities a and b, the measurement should be recorded as a ± ∆a and b ±∆b. If ∆X is the absolute error in the result, then, X ± ∆X = 2/1 )b/b1)(a/a1( b a )b/b1(b )a/a1(a bb aa − ∆±∆±= ∆± ∆± = ∆± ∆± or X ±∆X = ])b/ba/a(b/ba/a1[ b a ]b/b)2/1(1)[a/a1( b a ∆×∆±∆±∆±=∆−±∆± or X ± ∆X ] ba ba b b a a 1[X ∆∆ ± ∆ ± ∆ ±= We can ignore the term ∆a ∆b/ab since it is small in comparison with other terms. ] ba ba b b a a 1[ X X X X ∆∆ ± ∆ ± ∆ ±= ∆ ± ⇒ b b a a 1 X X 1 ∆ ± ∆ ±= ∆ + b b a a X X ∆ ± ∆ ±= ∆ ± Maximum fractional error in X = Maximum fractional error in a + Maximum fractional error in b Therefore, when two physical quantities are divided, the maximum relative error in the result is equal to the sum of relative errors in the physical quantities. Now b b a a X X ∆ + ∆ = ∆ or 100 b b 100 a a 100 X X × ∆ +× ∆ =× ∆ Max. percentage error in X = Max. Percentage error in a + M ax. Percentage error in b Therefore when two physical quantities are divided, the maximum percentage error in the result is equal to the sum of percentage errors in the physical quantities. (v) Error in power Let X = am If ±∆a is the absolute error in the measurement of the quantity a, then the measurement should be recorded as a ±∆a. If ±∆X is the absolute error in the result, then X ± ∆X = (a ±∆a)n = an n a a 1 ∆ ± ∆ ± ∆ ±=∆± a a ofpowerhigher a a n1aXX n We can ignore the term (∆a/a), since it is small in comparison with other terms. ∆ ±=∆± a a n1XXX Dividing both sides by X we have, ∆ ±= ∆ ± a a n1 X X X X X X
- 11. ∆ ±= ∆ ± a a n1 X X 1 Hence Maximum fractional error in X a a n X X ∆ ±= ∆ ± (iv) General case:− Let n ml c ba X = It can be proved that c c n b b m a a l X X ∆ ± ∆ ± ∆ ±= ∆ ± Significant Figure:- The number of digit up to which the measured value is accurate is call Significant Figure. (i) All nonzero digits are significant e.g. 128.25 g contains five significant figures. (ii) All zeros between two nonzero digits are significant e.g. 107.004 m contains six significant figures. (iii) Unless stated otherwise, all zeros to the left of an understood decimal point but the right of a nonzero digit are not significant e.g. 208,000 m contains three significant figures. (iv) All zeros to the left of an expressed decimal point and to the right of a nonzero digit are significant e.g. 202,000 contains six significant figures. v) All zeros to the right of a decimal point but to the left of a nonzero digit are not significant e.g. 0.000248 kg contains three significant figures. (vi) All zeros to the right of a decimal point and to the right of a nonzero digit are significant e.g. 0.06020 cm and 30.00 cm each contains four significant figures. SIGNIFICANT FIGURES IN CALCULATIONS (i) In addition or subtraction, the number of decimal places in the result should be equal to the smallest number of decimal places of any term in the sum or subtraction. (ii) In multiplication or division, the number of significant figures in the result should be equal to the number of significant figures of the least precise term in multiplication or division. ORDER OF MAGNITUDE:−The nearest power of ten, in which a quantity can be expressed, is called the order of magnitude of the quantity. Rules. For order of magnitude, following two rules are important: Rule 1 If the left: most digit of the quantity is less than 5(1, 2, 3, 4) the decimal is put first to its right. The number of digits right to the decimal, becomes the order of magnitude. Example. 4962 = 4.962 ×103 .The order of magnitude of the quantity is three. Rule 2. If the left most digit of the quantity is 5 or more (5, 6, 7, 8, 9) the decimal is put to its left The number of digits right to the decimal becomes the order of magnitude. Example. 5125 = 0.5125 × 104 . The order of magnitude of the quantity is four. BOUNDING OFF:−(a) Definition. It is the process of reducing the number of decimal places in a decimal number, without much affecting its magnitude.
- 12. (b) Rules. For rounding off, following two rules are important. Rule 1 If the right most digit is less than 5, then it is dropped. Example. 2.454 when rounded off, becomes 2.45. Rule 2. If the right most digit is 5 or more, then digit first to its left is increased by one. Example. 2.455 when rounded off, becomes 2.46. (c) Successive rounding off Let a given number be 23.27354 First round off makes it 23.2735 Second round off makes it 23.274 Third round off makes it 23.27 Fourth round off makes it 23.3.