Short Notes on Units and Measurements PDF

Units and Measurements SHORT NOTES Fundamental Quantity Derived Quantity Significant Figure or Digits The physical quantities which Those quantities which can Rules to find out the number of signific...

Units and Measurements SHORT NOTES Fundamental Quantity Derived Quantity Significant Figure or Digits The physical quantities which Those quantities which can Rules to find out the number of significant figures: do not depend on any other be expressed in terms of 1. Rule: All the non-zero digits are significant e.g., 1984 has physical quantities for their fundamental/base quantities. 4 SF. measurements. e.g., Angle, speed or velocity 2. Rule: All the zeros between two non-zero digits are significant. e.g., Mass, Length, Time Acceleration, force etc. e.g., 10806 has 5 SF. Temperature, current, luminous 3. Rule: All the zeros to the left of first non-zero digit are not Intensity & mole significant. e.g., 00108 has 3 SF. 4. Rule: If the number is less than 1, zeros on the right of the System of Units decimal point but to the left of the first non-zero digit are not (a) FPS System: Here length is measured in foot, mass in significant. e.g., 0.002308 has 4 SF. pounds and time in second. 5. Rule: The trailing zeros (zeros to the right of the last non-zero digit) in a number with a decimal point are significant. e.g., (b) CGS System: In this system, L is measured in cm, M is 01.080 has 4 SF. measured in g and T is measured in sec. 6. Rule: The trailing zeros in a number without a decimal point (c) MKS System: In this system, L is measured in metre, M is are not significant e.g., 010100 has 3 SF. But if the number measured in kg and T is measured in sec. comes from some actual measurement then the trailing zeros become significant. e.g., m = 100 kg has 3 SF. Principle of Homogeneity 7. Rule: When the number is expressed in exponential form, According to this, the physical quantities having same dimension the exponential term does not affect the number of S.F. For can be added or subtracted with each other and for a given equation, example in x = 12.3 = 1.23 × 101 =.123 × 102 = 0.0123 × 103 dimensions of both sides must be same. = 123 × 10–1, each term has 3 SF only. B Rules for arithmetical operations with significant figures: For eg, in equation F = A m+ +C , v 1. Rule: In addition or subtraction the number of decimal places all the three parts of R.H.S have same dimension as force on L.H.S. in the result should be equal to the number of decimal places of that term in the operation which contain lesser number of Dimensions decimal places. e.g., 12.587 – 12.5 = 0.087 = 0.1 ( second term contain lesser i.e., one decimal place) The fundamental or base quantities along with their powers needed to express a physical quantity is called dimensions 2. Rule: In multiplication or division, the number of SF in the product or quotient is same as the smallest number of SF in e.g., [F] = [MLT–2] is dimension of force. any of the factors. e.g., 5.0 × 0.125 = 0.625 = 0.62. Usage of Dimensional Analysis Rounding Off (i) To check the correctness of a given formula. Rules for rounding off the numbers: (ii) To establish relation between quantities dimensionally. 1. Rule: If the digit to be rounded off is more than 5, then the (iii) To convert the value of a quantity from one system of preceding digit is increased by one. e.g., 6.87≈ 6.9 units to other system. 2. Rule: If the digit to be rounded off is less than 5, than the Limitations of Dimensional Analysis preceding digit is unaffected and is left unchanged. e.g., 3.94 (i) It does not predict the numerical value or number ≈ 3.9 associated with a physical quantity in a relation 3. Rule: If the digit to be rounded off is 5 then the preceding digit is increased by one if it is odd and is left unchanged if it e.g., v= u + 1 at & v = u + at is even. e.g., 14.35 ≈ 14.4 and 14.45 ≈ 14.4 3 5 Both are dimensionally valid. Representation of Errors (ii) It does not derive any relations involving trigonometric, 1. Mean absolute error is defined as logarithmic and exponential functions ∆a1 + ∆a2 +... + ∆an n ∆a e.g., P = P0e–bt cannot be derived dimensionally.= 2 ∆a = ∑ i n i =1 n (iii) It does not give any information about dimensionally Final result of measurement may be written as: constants or nature of a quantity (vector/scalar) associated with a relation. a = am ± ∆a 1 2. Relative Error or Fractional Error: It is given by To Find Smaller Measurements ∆a Mean absolute Error Vernier Calliper = a m Mean value of measurement (i) Least count: Suppose movable Jaw is slided till the zero of vernier scale coincides with any of the mark of the main ∆a scale. 3. Percentage Error = × 100% am  n −1  Let, n V.S.D = (n – 1) MSD ⇒ 1VSD =   M.S.D  n  Combination of Errors \ Vernier constant = 1 M.S.D – 1 V.S.D (i) In Sum: If Z = A + B, then ∆Z = ∆A + ∆B.  n − 1 1 Maximum fractional error in this case is = 1 − n  MSD = n MSD   ∆Z ∆A ∆B = + (ii) Total reading = MSR + VSR Z A+ B A+ B = MSR + n ×VC (ii) In Difference: If Z = A – B, then maximum absolute error where MSR = Main scale reading is ∆Z = ∆A + ∆B and maximum fractional error in this case VC = Vernier constant i.e. least count ∆Z ∆A ∆B n = nth division of vernier scale coinciding with main scale. = + Z A− B A− B Screw Gauge (iii) In Product: If Z = AB, then the maximum fractional error, This instrument works on the principle of micro-meter screw. It is used to measure very small (mm) measurements. It is provided ∆Z ∆A ∆B = + with linear scale and a circular scale. Z A B (i) Pitch of the screw gauge (iv) In Division: If Z = A/B, then maximum fractional error is Distance moved in n-rotation of cir-scale = ∆Z ∆A ∆B No.of full -rotation = + Z A B Pitch ∆Z ∆A (ii) L.C = (v) In Power: If Z = A then n =n Total number of division on the circular scale Z A (iii) Total Reading (T.R) = L.S.R + C.S.R Ax B y L.S.R = Linear scale Reading = N where In more general form if Z = Cq C.S.R = Circular Scale Reading = n × L.C then the maximum fractional error in Z is If nth division of circular scale coincides with the linear scale ∆Z ∆A ∆B ∆C line, then =x +y +q \ Total reading = N + n × (L.C) Z A B C 2