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Measurement and Units, , {1}, , St. Aloysius H.S.S, Elthuruth, , MEASUREMENT, Measurement is the basis of all experiment and research. The quantities, which can be measured, directly or indirectly, are called physical quantities., Eg: length, mass, time, velocity, volume etc., **NB Units, In order to measure a physical quantity, its value is compared with the value of a standard of the same, kind. This standard is called unit., Eg: metre is the unit of length, kilogram is the unit of mass, second is the unit of time., Thus in the measurement of a physical quantity, two things are involved, – a number and a unit. The unit is the standard quantity with which the measured quantity is compared and the, number shows how many times the measured quantity is greater than the unit., ## A good unit will have the following characteristics., It should be (a) well defined (b) easily accessible (c) invariable (d) easily reproducible., **NB Fundamental and derived units, A set of physical quantities which are independent of each other are known as fundamental quantities., The units of fundamental quantities are called fundamental units., The units of length, mass and time, ie metre, kilogram and second are taken as fundamental units., Physical quantities, which can be expressed in terms of fundamental quantities, are called, derived quantities. The units of derived quantities are called derived units., Eg: 1. Volume is measured in metre3 or m3., 2. Speed can be expressed in meter per second (m/s), **NB Rules for writing units., 1) The symbol of a unit, which is not named after a person, is written in lower letter (small letter)., Eg: metre (m) kilogram (kg), 2) The symbol of a unit, which is named after a person, is written with a capital initial letter., Eg: newton (N), kelvin (K), 3) Full name of a unit, even if it is named after a person is written with a lower initial letter. Eg: newton, kelvin., 4) A compound unit formed by multiplication of two or more units is written after putting a dot or leaving a, space between the two symbols. Eg: newton metre – N.m or N m., 5) A unit in its short form is never written in plural., Systems of units., A complete set of units for all physical quantities with particular basic units is called a system of units., The commonly used systems are:, 1) The C.G.S. System: It is based on three basic units – centimeter, gram and second for length, mass and, time respectively., 2) The F.P.S. System: Here unit of length is foot, mass is pound and time is second., 3) The MKS system: Here metre is unit of length, kilogram for mass and second for time., 4) SI units: In 1960, International Committee for Weights Measures adopted a system of units for all fundamental, physical quantities and is called International system of units or SI units., In SI system, there are seven Fundamental (basic) units and two Supplementary units., Basic units:, 1) Mass, – kilogram – kg, 2) Length, – metre, –m., 3) Time, – second, –s, 4) Temperature – kelvin, –K, 5) Electric current, – ampere, –A

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{2}, , Measurement and Units, , St. Aloysius H.S.S, Elthuruth, , 6) Luminous intensity – candela, – cd., 7) Amount of substance– mole – mol, Supplementary units:, a) Angle – radian – rad, b) Solid angle – steradian – sr., Advantages of SI units:, 1) It is comprehensive., 2) The system is coherent., 3) It is internationally accepted., ** Multiples and Sub multiples of units., To express magnitude of physical quantities, which are very large or small, we use prefixes to the unit., Multiple, 10-1, 10-2, 10-3, 10-6, 10-9, 10-12, 10-15, 103, 106, 109, , Name, deci, centi, milli, micro, nano, pico, fermi, kilo, mega, giga, , Symbol, d, c, m, m, n, p, f, k, M, G, , NOTE: Practical Units., Length :, (i) 1 fermi = 1 fm = 10–15 m, (ii) 1 X-ray unit = 1XU = 10–13 m, (iii) 1 angstrom = 1Å = 10–10 m = 10–8 cm = 10–7 mm = 0.1 m mm, (iv) 1 micron = m m = 10–6 m, (v) 1 astronomical unit = 1 A.U. = 1. 49 × 1011 m  1.5 × 1011 m , 108 km, (vi) 1 Light year = 1 ly = 9.46 × 1015 m, (vii) 1 Parsec = 1pc = 3.26 light year, Mass :, (i) Chandra Shekhar unit : 1 CSU = 1.4 times the mass of sun = 2.8 ×, 1030 kg, (ii) Metric tonne : 1 Metric tonne = 1000 kg, (iii) Quintal : 1 Quintal = 100 kg, (iv) Atomic mass unit (amu) : amu = 1.67 × 10–27 kg mass of, proton or neutron is of the order of 1 amu, , Time :, (i) Year : It is the time taken by earth to complete 1 revolution around the sun in its orbit., (ii) Lunar month : It is the time taken by moon to complete 1 revolution around the earth in its orbit., 1 L.M. = 27.3 days, (iii) Solar day : It is the time taken by earth to complete one rotation about its axis with respect to sun., Since this time varies from day to day, average solar day is calculated by taking average of the duration of, allthe days in a year and this is called Average Solar day., , Measurement of large distances (Parallax method), To measure the distance D of a far away planet S by the parallax method,, we observe itfrom two different positions (observatories) A and B on the Earth,, separated by distance AB = b at the same time as shown in Fig. We measure, the angle between the two directions along which the planet is viewed at these, two points. The angle ASB represented by  , is called the parallax angle, or parallactic angle., arc, As the planet is very far away,  is very small. Since angle , ,, radius, , , i.e,  , , AB, SA, , b, D, , Or distance; D , , b, 

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Measurement and Units, , {3}, , St. Aloysius H.S.S, Elthuruth, , **NB, , DIMENSIONS, All physical quantities can be represented in terms of fundamental quantities – length, mass and time., Dimensions of a physical quantity are the powers to which the fundamental units be raised in, order to represent that quantity., For eg: we know, Area = length x breadth. Here both length and breadth are measured with the help of unit, of length (L). Thus unit of area = L x L = L2. Since unit of mass M and time T are not being used for the, measurement of area, the unit of area can be represented by M0L2T0. Powers 0, 2, 0 of fundamental units are, called the dimensions of area in mass, length and time respectively., ** Dimensional equation, The equation, which indicates the units of a physical quantity in terms of the fundamental units, is, called dimensional equation., Eg: Dimensional equation of velocity is [V] = M0LT–1., 1) Area = length x breadth = L x L = L2 = M0L2T0, 2) Volume = L3 = M0L3T0, 3), 4), 5), , ML–3 T0, mass, Density, , volume, 0, –1, displacement L M LT,  , time 0 1 t, M LT, = M0LT, T–2, v, T, Acceleration  , t, , Velocity , , 6), Force = ma = MLT–2, Note: Quantities such as number, angle and trigonometrical ratio are dimensionless., , **NB Principle of homogeneity, An equation representing a physical quantity will be correct if the dimensions of each term on both, sides of the equation are the same. This is called the principle of homogeneity of dimensions., **NB Uses of dimensional analysis, (1) To check the correctness of an equation., An equation is correct only if the dimensions of each term on either side of the equation are, equal., Eg: Check the accuracy of the equation S  ut , Ans: S  ut , , 1 2, at, 2, , 1 2, at, 2, , Taking dimensions on both sides, L  LT1 T LT2 T2, ie L = L + L,  According to principle of homogeneity, the equation is dimensionally correct., (2) To derive the correct relationship between physical quantities., Dimensions can be used to derive the relation between physical quantities., For Eg: To derive an expression for period of oscillation of a simple pendulum., The period of oscillation (T) of a simple pendulum may depend on (1) length of the pendulum (l);, (2) mass of bob (m) and (3) the acceleration due to gravity (g)., Then T = k l x my gz.

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{4}, , Measurement and Units, , Taking dimensions on both sides,, X, , T   L, ,  M Y  LT 2 , , St. Aloysius H.S.S, Elthuruth, , Z, , T  LX  Z M Y T 2Z, , Equating dimensions of M, L and T,, x+z=0, y=0, –2z = 1, 1, , 1, 2, , l, k, 1, z, g, 2, The value of constant k cannot be found by dimensional method. The value of k is found to be 2 using some, other methods like conducting experiment., , 1, x , 2, ,  T  2, ,  T  kl 2 M 0 g, , , , l, g, , Limitations of dimensional analysis., 1) This method gives us no information about dimensionless constants., 2) We cannot use this method if the physical quantity depends on more than three other physical quantities., 3) This method cannot be used if the left hand side of the equation contains more than one term., 4) Often it is difficult to guess the parameters on which the physical quantity depends., Q1. A force F is given by , 2 F = at + bt where t is time. What are the dimensions of a and b, [MLT-3, MLT-4], Q2. The dimensions of physical quantity X in the equation Force =, , X, is .......... [M2L-2T-2], Density, , Q3. The volume V of water passing through a point of a uniform tube during t seconds is related to the, , V  A u  t  , which one of the, following will be true: (a)      (b)      (c)      (d)     , crosssectional area A of the tube and velocity u of water by the relation, , NB Significant figures:, Significant figures give the number of meaningful digits in a number., The significant figures are the number of digits used to express the measurement of the physical quantity such, that the last digit in it is doubtful and the rest all digits are accurate., The number of significant figures depends on the accuracy of the instrument. More the number of, significant figures in a measurement, more accurate the measurement is., Rules to determine the significant figures, 1) All zeros in between the numerals 1 to 9 are counted., 2) In a measurement involving decimal, the position of decimal is disregarded., 3) All zeros after the last numeral are counted., 4) The zeros preceding the first numeral are not counted., Rules for rounding off to the required number of significant figures., 1) If the digit to be dropped is less than 5, the digit immediately preceding it remains unchanged., 2) If the digit to be dropped is more than 5, the digit immediately preceding it is increased by 1., 3) If the digit to be dropped is 5, then the preceding digit is made even by (a) increasing it by 1 if it is, odd, (b) keeping it unchanged if it is even.

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Measurement and Units, , {5}, , St. Aloysius H.S.S, Elthuruth, , Significant figures in calculations., (1) Significant figures in multiplication and division., The result of multiplying or diving two or more numbers can have no more significant figures than, those present in the number having the least significant figures., Eg: (1) a = 10.43 [sig fig 4], b = 2.8612 [sig fig 5], a x b = 29.842316 =29.84 [sig fig 4], a 16000, ,  3493 .449782  3493, b 4.580, (2) Significant figures in addition and subtraction., In adding or subtracting, the least significant digit of the sum or difference occupies the same relative, position as the least significant digit of the quantities being added or subtracted. Here number of significant, figures is not important; position is important., Eg: (1) 204.9 + [ 9 is least sig.digit. Position – 1st decimal place], 2.10 + [0 is least digit. Position 2nd decimal place], . 0.319 + [ 9 is least sig. Digit. Position 3rd decimal place.], 207.319 = 207.3, In sum, the least sig. Fig should come in the first decimal place., Eg: (2) If a = 10.43 and b=2.8612 then a – b= 10.43 – 2.8612 = 7.5688 = 7.57., , (2) a = 16000, b = 4.580 then, , **NB, , Accuracy and Errors in Measurements., Accuracy of a measurement is the correctness with which the measurement is made. For all, measuring instruments, there is a limit up to which measurements can be taken accurately. This is called least, count of the measuring instrument., Least count of an instrument is the least measurement, which can be made accurately with that, instrument., For Eg: Least count of an ordinary metre scale is 0.1 cm, 0.01 cm is the least count of vernier calipers and, 0.001 cm is that for screw gauge., Deviation between the value obtained in a measurement and the true value of the quantity is called, error in that measurement., The following are the commonly occurring errors:, 1) Constant error: If the error in a series of readings taken with an instrument is same, the error is said to be, constant error., 2) Systematic errors: Errors, which are due to known causes and act according to a definite law are called, systematic errors., (a) Instrumental errors: These errors are due to the defect of the instrument., Eg: 1. Zero error in screw gauge and vernier. 2.Faulty calibration of thermometer, metre scale etc., (b) Personal error: This is due to the mode of observation of the person taking the reading., Eg: Parallax error., (c) Error due to imperfection: This is due to imperfection of the experimental setup., Eg: Whatever precautions are taken, heat is always lost from a calorimeter due to radiation., (d) Error due to external causes: These are errors caused due to change in external conditions like, temp, pressure, humidity etc., Eg: With increase in temperature, a metal tape will expand. Any length measured using this tape, will not give correct reading., 3) Random Error: The errors, which occur irregularly and at random in magnitude and direction, are, called random errors. These errors are not due to any definite cause and so they are also called accidental, errors.

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{6}, , Measurement and Units, , St. Aloysius H.S.S, Elthuruth, , Random errors can be minimized by taking several measurements and then finding the arithmetic, mean. The mean is taken as the true value of the measured quantity., Let a quantity measured n times give values a1, a2, ….an, then the possible value a of the quantity is, a  a  a3  ..... an, a  1 2, n, _, , 1, , n, , n, , a, , i, , i 1, , _, , The arithmetic mean a is taken as the true value of the quantity.., 4) Gross Errors: The errors caused by the carelessness of the person are called gross errors. It may be due, to 1) improper adjustment of apparatus (2) mistakes while taking and recording readings etc., 5) Absolute Errors: The magnitude of the difference between the true value of the quantity measured and the, individual measured value is called the absolute error.,  Absolute error = True value – measured value., __, , If a1, a2,…..,an are the measured values and a is the true value, then absolute error,,, __, , a1  a  a1, __, , a2  a  a2 and so on., 6) Mean absolute error: The arithmetic mean of the absolute error of the different measurements taken is, called mean absolute error., If a1,a2,…….an are the absolute errors in the measurements a1, a2, a3,…an, then, ____, , a , , a1  a2  a3  ..... an, ., n, __, , __, , Here the measured value lies between a  a and a  a, 7) Relative and percentage errors., The ratio of the mean absolute error to the true value of the measured quantity is called relative error., Re lative error , , a, a, ___, , Percentage error =, , a, __, , x 100 %, , a, , **NB Combination of errors, Usually, the value of a physical quantity is obtained by measurement of other quantities. The value will, be obtained on mathematical calculations, using measured values. The error in the final result depends not, only on individual measurements but also on the type of mathematical operations performed to obtain the, result., 1) Error in a sum or difference: —, Let two quantities A and B have errors DA and DB respectively in their measured values. We have, to calculate the error DZ in their sum Z = A + B., ,  Z  Z   A  A   B B

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{7}, , Measurement and Units, , St. Aloysius H.S.S, Elthuruth, , Z  Z  Z  A  A   B  B  A  ,   Z   A  B, ,  Maximum value of  Z   A   B, We will get the same result even if we take the difference., \When two quantities are added or subtracted, the absolute error in the final result is the sum of the, absolute errors in the quantities., , 2) Error in a product or quotient, Let A and B be two quantities and Z be their product. ie Z = AB. If  A  A  and B B are the, measured values of A and B, then Z  Z   A  A  B B, = AB  BA  AB  AB, Dividing LHS by Z and RHS by AB,, , Z  Z, Z, , , , ie, 1 , , B A, AB, , AB, AB, , , , A B, , AB, , Z, A, B,  1 , , Z, A, B, , A B, AB, , neglecting, , A B, being small., AB, , Hence the maximum possible relative error or fractional error in Z is, , Z, , Z, , A, B, , A, B, , The result is true for division also., Therefore, when two quantities are multiplied or divided, the relative error of the result is equal to, the sum of relative errors of the quantities., 3) Error when a quantity is raised to a power., Consider a quantity X  x n Taking logarithm on both sides, log X = log xn = n log x., X, x,  n, X, x, Thus if a quantity has to be raised to a power n, then the relative error of the result is n times the, relative error of that quantity., , Differentiating, we get, , Q4. The pressure on a square plate is measured by measuring the force on the plate and the length of the, sides of the plate. If the maximum error in the measurement of force and length are respectively 4%, and 2%, Find the maximum error in the measurement of pressure., [8%], Q5.The resistance, , R  V I where V= 100 ± 5 volts and I = 10 ± 0.2 amperes. What is the total error in R?[7%], , Q6. The length of a cylinder is measured with a meter rod having least count 0.1 cm. Its diameter is, measured with venier calipers having least count 0.01 cm. Given that length is 5.0 cm. and radius is 2.0 cm., The percentage error in the calculated value of the volume will be ........., [3%], Q7. In an experiment, the following observation’s were recorded : L = 2.820 m, M = 3.00 kg, l = 0.087, cm,Diameter D = 0.041 cm Taking g = 9.81 2 m / s using the formula , Y , error in Y?, , 4Mg, , the maximum permissible, D 2 l, [6.5%], , ********************************************************************************

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{8}, , Measurement and Units, , St. Aloysius H.S.S, Elthuruth, , Questions Set I, 1. If x = at + bt2 where x is in meters and t is in seconds. What are the units of a and b?, [1], 2. Fill ups., (i) 3.0m/s2 = —————————— km/hr2, (ii) 6.67 10-11Nm2/kg2 = ———————————— g-1cm3s-2, [1], 3. Write S.I unit of luminous intensity and temperature?, [1], -16, 4. Calculate the time taken by the light to pass through a nucleus of diameter 1.5610 m., (speed of light is 3108 m/s), [2], 5. If force (F) acceleration (A) and time (T) are taken as fundamental units, then find the dimension of energy. [2], 6. Two resistances R1 = 100 ± 3 and R2 = 200 ± 4 are connected in series. Then what is the equivalent, resistance?, [2], 7. If velocity, time and force were chosen the basic quantities, find the dimensions of mass?, [2], 8. The velocity v of water waves may depend on their wavelength  density of water  and the acceleration, due to gravity g. Find relation between these quantities by the method of dimension?, [3], 9. The force acting on an object of mass m traveling at velocity u in a circle of radius r is giving by, The measurements recorded as m = 3.5kg ± 0.1kg u = 20m / s ± 1m / s r =12.5m ± 0.5m. Find the, maximum possible (1) fractional error (2) % error in the measurement of force. How will you record the, reading?, [3], Questions Set II, 1. What is the difference between Ao and A.U.?, [1], 2. Define S.I. unit of solid angle?, [1], 3. Name physical quantities whose units are electron volt and pascal?, [1], 4. When a planet X is at a distance of 824.7 million kilometers from earth its angular diameter is measured to, be 35.721 of arc. Calculate the diameter of ‘X’., [2], 5. A radar signal is beamed towards a planet from the earth and its echo is received seven minutes later., Calculate the velocity of the signal, if the distance between the planet and the earth is 6.3 x 1010m?, [2], 6. Give two methods for measuring time intervals?, [2], 7. Find the dimensions of latent heat and specific heat?, [2], a , , 8. In Vandar Waals’ equation, P  2  (V  b)  R , find the dim ensions of a and b ?, [2], V , , 9. E, m, L and G denote energy, mass, angular momentum and gravitational constant respectively. Determine, the dimensions of EL2 / m5G2, [2], 10. (a) State which of the following are dimensionally current, (i) Pressure = Energy per unit volume, (ii) Pressure = Momentum volume time, 4m, (b) The density of cylindrical rod was measured by the formula:-    D 2 l . The percentage in m, D and l, , are 1%, 1.5% and 0.5%. Calculate the % error in the calculated value of density?, Answers Set I., 1., , a, , x,  m/s ;, t, , (i) 3. 0 m / s 2 , , b , , 3 x 103,  1, , h, ,  3600 , , 2, , x,  m / s2, t, , km / hr 2  3.9 x 104 km / hr 2, , [3]

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{9}, , Measurement and Units, , (ii), , 6 . 67 x 10, ,  11, 8, ,  6 . 67 x 10, , Nm, , 2, , 2, , / kg, ,  6 . 67 x 10, ,  11, , kg, , St. Aloysius H.S.S, Elthuruth, , 1, , m 3 s  2  6 . 67 x 10, ,  11, , x 10, , 3, , x (10 2 ) 3, , g  1 cm 3 s  2, , 3. candela (cd) ; kelvin (K)., 4. time  S / v , , 1.56 x 10 16,  5.2 x 10  25 s., 3 x 108, , 5. F  M L T 2 ; A  LT 2 .......(1), ,  M  F A 1 .......(2), , F  MA, , From (1), L  A T 2, ,  Dimensions of energy  M L2 T  2  F A 1 A 2 T 4 T  2  F A T 2, 6, , ., , Here R 1  100  3  ;, , R 2  200  4  In Series, R S  R 1  R 2  (100  3 )  ( 200  4 ), ,  300  7 , 7. force, f  m a , 8, , mv, t, , m , , tf, FT, ,  F T V 1 ., v, V, , Let v  a p b g c . v  ka p b g c ..........(1); k  cons tan t., , , , , , , , Taking dim ensions on both sides , L T 1  L a M L3, , i.e., , L T   L, 1, , a  3b  c, ,  L T , b, , 2, , c, , M b T  2c, , Equating dim ensions of M , L and T on both sides , a  1 / 2; b  0; c  1 / 2., Substuting in (1), v  k  g, , F 0.1, 1, 0.5, m v2, F m, v r, ,  2, ,  0.17, ;, , 2, , 9. (i) F , i.e., F, 3.5, 20 12.5, r, F, m, v, r, (ii ) % error in F  0.17  100  17%, Also, , (iii ) F , , m v2, 3.5  20 2, ,  112 N ., r, 12.5, , F  0.17 x 112  19 N . Therefore measurement of force, F = 112  19 N, , Answers Set II, 1 : Ao and A.U. both are the units of distances but 1Ao = 10-10m and 1A.U. = 1.496 1011m., 2: One steradian is defined as the angle made by a spherical plane of area 1 square meter at the centre of a, sphere of radius 1m., 3: Energy and pressure., 4: r  824.7 x 106 km. ;   35.72l , , 35.72, , x, radian, 60 x 60 180, , Now diameter l  r  824.7 x 10 6 x, , 5: x  c , , 35.72, , x,  1.429 x 105 km., 60 x 60 180, , t, 2x, 2  6.31010,  c , ,  3  108 m / s., 2, t, 7  60, , 6: (1) Radioactive dating – to know age of fossil fuels, rocks etc. (2) Atomic clocks – used to note periodic, vibrations taking place within two atoms., 7. (1) Latent Heat , , Q (heat energy), M L2 T 2, ,  M 0 L2 T  2, m (mass), M

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{10}, , Measurement and Units, , (2) Specific heat  S  , , St. Aloysius H.S.S, Elthuruth, , Q, M L2 T 2, ,  M 0 L2 T  2 K 1., mxQ, MK, , 8, , ., , P a, , V2, ,  a  P V2 , , F 2 MLT 2, V , x L3, 2, A, L, ,  , , 2, , , , M L T 2 L6,  M L5 T 2 Also b  V  M 0 L3 T 0, 2, L, , 2, 2, L  M L2 T 1 m  M G  M 1L3T 2, 9. E  M L T, 2, M L2 T  2 (M L2T 1 ) 2 M 3 L6 T  4, Dimensions of E L 5 2 ,  3 6  4  1 {Dimensionl ess}, mG, M 5 (M 1L3T  2 ) 2, M L T, , 1  2, 10.(a) (i) Pr essure  F / A  M L T , , M L2 T 2,  M L1 T  2 {Dimensiona lly correct}, 3, L, , (ii) M L1 T 2  M L T 1  L3  T  M L4 T 0 {Dimensiona lly not correct}, (b), , , m, D, l, 100 , 100  2, 100 , 100  1  2 1.5  0.5  4.5%, , m, D, l, , Additional Questions And Answers:, Q1. State one law that holds good in all natural processes., Ans. One such laws is the Newton’s gravitation law, According to this law everybody in this nature are, attracts with other body with a force of attraction which is directly proportional to the product of their masses, and inversely proportionally To the square of the distance between them., Q2: Among which type of elementary particles does the electromagnetic force act?, Ans : Electromagnetic force acts between on all electrically charged particles., Q3. Name the forces having the longest and shortest range of operation., Ans : longest range force is gravitational force and nuclear force is shortest range force., Q4. If ‘slap’ times speed equals power, what will be the dimensional equation for ‘slap’?, Ans . Slap x speed = power Or slap = power/speed = [MLT-2], Q5. If the units of force and length each are doubled, then how many times the unit of energy would be, affected?, Ans : Energy = Work done = Force x length So when the units are doubled, then the unit of energy will, increase four times., Q6. Can a quantity has dimensions but still has no units?, Ans : No, a quantity having dimension must have some units of its measurement., Q7. Justify L +L = L and L – L =L., Ans: When we add or subtract a length from length we get length, So L +L = L AND L – L =L,, justified., Q8. Can there be a physical quantity that has no unit and no dimensions?, Ans : yes, like strain., Q9. Given relative error in the measurement of length is 0.02, what is the percentage error?, Ans: percentage error = 2 %, Q10. If g is the acceleration due to gravity and  is wavelength, then which physical quantity does, represented by g  ., Ans. Speed or velocity., <><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>, , “Try Try till you succeed with a determined mind set and focussed vision,, Success is never far from reach”.