Units and Measurements

Units and measurements Measurement The comparison of any physical quantity with its standard unit is called measurement. Physical Quantities The term "physical quantity" refers to those quantities that can be used to describe the rules of physics. A measureable quantity is one that is physical. The physical quantities are therefore: length, mass, time, pressure, temperature, current, and resistance. The physical quantities are classified into : (i) Fundamental quantities or base quantities (ii) Derived quantities The physical quantities that are independent of each other are called fundamental quantities. All the other quantities which can be expressed in terms of the fundamental quantities are called the derived quantities. Units The reference standard used to measure the physical quantities is called the unit. Properties of Unit 1. The unit must be a reasonable size. 2. The unit has to be well defined. 3. It should be simple to replicate the unit anywhere. 4. The component cannot alter over time. 5. Physical parameters like pressure, temperature, etc. shouldn't cause the unit to alter. 6. The unit must be simple to compare experimentally to physical quantities of a similar nature. Types of Units (i) Fundamental Units Those physical independent to quantities each other which are are called fundamental quantities and their units are called fundamental units. (ii) Derived Units Those physical quantities which are derived from fundamental quantities are called derived quantities and their units are called derived units. e.g., velocity, acceleration, force, work etc. Systems of Units A system of units is the complete set of units, both fundamental and derived, for all kinds of physical quantities. The common system of units which is used in mechanics are given below: 1. CGS System In this system, the unit of length is centimetre, the unit of mass is gram and the unit of time is second. 2. FPS System In this system, the unit of length is foot, the unit of mass is pound and the unit of time is second. 3. MKS System In this system, the unit of length is metre, the unit of mass is kilogram and the unit of time is second. 4. SI System This system contain seven fundamental units and two supplementary fundamental units. Basic Units S.No. Quantities Units Symbol 1 Length metre m 2 Mass kilogram kg 3 Time second S 4 Temperature kelvin kg 5 Electric current ampere A 6 Luminous candela cd mole mol intensity 7 Amount of substance Supplementary Fundamental Units Radian and steradian are two supplementary fundamental units. It measures plane angle and solid angle respectively. S.No. Supplementary Supplementary Fundamental Units Symbol Quantities 1 Plane angle radian rad 2 Solid angle steradian Sr Definitions of Fundamental Units The seven fundamental units of SI have been defined as under. 1. 1 kilogram A cylindrical prototype mass made of platinum and iridium alloys of height 39 mm and diameter 39 mm. It is mass of 5.0188 x 1025 atoms of carbon-12. 2. 1 metre 1 metre is the distance that contains 1650763.73 wavelength of orange-red light of Kr-86. 3. 1 second 1 second is the time in which cesium atom vibrates 9192631770 times in an atomic clock. 4. 1 kelvin 1 kelvin is the (1/273.16) part of the thermodynamics temperature of the triple point of water. 5. 1 candela 1 candela is (1/60) luminous intensity of an ideal source by an area of cm' when source is at melting point of platinum (1773 ℃). 6. 1 ampere 1 ampere is the electric current which it maintained in two straight parallel conductor of infinite length and of negligible cross-section area placed one metre apart in vacuum will produce between them a force 2 x 10-7N per metre length. 7. 1 mole 1 mole is the amount of substance of a system which contains a many elementary entities (may be atoms, molecules, ions, electrons or group of particles, as this and atoms in 0.012 kg of carbon isotope 6C12. Some Practical Units 1. 1 fermi =10-15 m 2. 1 X-ray unit = 10-13 m 3. 1 astronomical unit = 1.49 x 1011 m (average distance between sun and earth) 4. 1 light year = 9.46 x 1015 m 5. 1 parsec = 3.08 x 1016 m = 3.26 light year Some Approximate Masses Object Kilogram Our galaxy 2 x 1041 Sun 2 x 1030 Moon 7 x 1022 Asteroid Eros 5 x 1015 Measurement of Mass 1. The unit used to measure mass of atoms or molecules is unified atomic mass unit(u). 1u = 1.66 * 10-27kg 2. Mass of commonly available objects can be determinded by a common balance. 3. Large masses in the universe like planets, stars, etc., can be measured by using gravitational method. 4. For measurement of small masses of atomic/subatomic particles etc., we make use of mass spectrograph. Measurement of Time 1. To measure any time interval we need a clock. 2. We now use an atomic standard of time, which is based on the periodic vibrations produced in a cesium atom -cesium clock or in general atomic clock. 3. The cesium atomic clocks are very accurate. Accuracy and precision of a Measurement 1. Accuracy of a measurement is a measure of how close the measured value is to the true value of the quantity. 2. Precison is the resolution or limit with which the quantity os measured. 3. The accuracy may depend on the limit or resolution of the measuring instrument. Error in Measurement 1. Error is the uncertainty in a measurement. 2. It is the difference between measured value and true value of the quantity. Types of Errors 1. Systematics Errors 2. Random Errors 3. Least count Errors Systematics Errors Errors that frequently go in one way, either positively or negatively, are referred to as systemic errors. Some of the Source of systematic errors are : 1. Instrumental errors Instrumental mistakes that result from issues with the design or calibration of the measurement device. 2. Imperfection in experimental technique or procedureA thermometer placed under the armpit will always read lower than the real body temperature when used to measure a person's temperature. 3. Personal errors Individual biases, improper instrument settings, carelessness when taking observations without following the necessary safeguards, etc. are examples of personal mistakes. Minimizing Systematic Errors By enhancing experimental methods, using better tools, and minimising human bias, systematic errors can be minimized. Random Errors The errors that occur irregularly and are therefore random in terms of sign and size are known as random errors. These might appear as a result of unpredictable and random variations in the experimental environment. Least count error The smallest value that can be measured by the measuring instrument is called its least count. The instrument's resolution error is the one with the least count error. Error The lack in accuracy in the measurement due to the limit of accuracy of the instrument or due to any other cause is called an error. 1. Absolute Error The difference between the true value and the measured value of a quantity is called absolute error. If a1 , a2, a3, ..., an are the measured values of any quantity a in an experiment performed n times, then the arithmetic mean of these values is called the true value (am) of the quantity. am = (a1 +a2 + a3+ .....+an)/n The absolute error in measured values is given by Δa1 = am - a1 Δa2 = am - a1 ............. Δam = Δam - Δan 2. Mean Absolute Error Δa = (∣Δa1∣ +∣Δa2∣ + .....+ ∣Δan∣)/n The arithmetic mean of the magnitude of absolute errors in all the measurement is called mean absolute error. 3. Relative Error The ratio of mean absolute error to the true value is called relative Relative Error = Mean absolute error / True value = Δa/am 4. Percentage Error The relative error expressed in percentage is called percentage error. Pecentage Error = (Δa/am)*100% Propagation of Error i) Error in Addition or Subtraction Let x = a + b or x = a - b If the measured values of two quantities a and b are (a ± Δa and (b ± Δb), then maximum absolute error in their addition or subtraction. Δx = ±(Δa + Δb) (ii) Error in Multiplication or Division Let x = a x b or x = (a/b). If the measured values of a and b are (a ± Δa) and (b ± Δb), then maximum relative error Δx/x =± (Δa/a + Δb/b) Significant Figures In the measured value of a physical quantity, the number of digits about the correctness of which we are sure plus the next doubtful digit, are called the significant figures. Rules for Finding Significant Figures 1. All non-zeros digits are significant figures, e.g., 4362 m has 4 significant figures. 2. All zeros occuring between non-zero digits are significant figures, e.g., 1005 has 4 significant figures. 3. All zeros to the right of the last non-zero digit are not significant, e.g., 6250 has only 3 significant figures. 4. In a digit less than one, all zeros to the right of the decimal point and to the left of a non-zero digit are not significant, e.g., 0.00325 has only 3 significant figures. 5. All zeros to the right of a non-zero digit in the decimal part are significant, e.g., 1.4750 has 5 significant figures. Significant Figures in Algebric Operations (i) In Addition or Subtraction In addition or subtraction of the numerical values the final result should retain the least decimal place as in the various numerical values. e.g., If l1= 4.326 m and l2 = 1.50 m Then, l1 + l2 = (4.326 + 1.50) m = 5.826 m As l2 has measured upto two decimal places, therefore l1 + l2 = 5.83 m (ii) In Multiplication or Division In multiplication or division of the numerical values, the final result should retain the least significant figures as the various numerical values. e.g., If length 1= 12.5 m and breadth b = 4.125 m. Then, area A = l x b = 12.5 x 4.125 = 51.5625 m2 As l has only 3 significant figures, therefore A= 51.6 m2 Rules of Rounding Off Significant Figures 1. If the digit to be dropped is less than 5, then the preceding digit is left unchanged. e.g., 1.54 is rounded off to 1.5. 2. If the digit to be dropped is greater than 5, then the preceding digit is raised by one. e.g., 2.49 is rounded off to 2.5. 3. If the digit to be dropped is 5 followed by digit other than zero, then the preceding digit is raised by one. e.g., 3.55 is rounded off to 3.6. 4. If the digit to be dropped is 5 or 5 followed by zeros, then the preceding digit is raised by one, if it is odd and left unchanged if it is even. e.g., 3.750 is rounded off to 3.8 and 4.650 is rounded off to 4.6. Dimensions Dimensions of any physical quantity are those powers which are raised on fundamental units to express its unit. The expression which shows how and which of the base quantities represent the dimensions of a physical quantity, is called the dimensional formula. Dimensional Formula of Some Physical Quantities S.No. Physical Dimensional MKS Quantity Formula Unit 1 Area [L2] metre2 2 Volume [L3] metre3 3 Velocity [LT-1] ms-1 4 Acceleration [LT-2] ms-2 5 Force [MLT-2] newton (N) 6 Work or energy [ML2T-2] joule (J) 7 Power [ML2T-3] Js-1 or watt 8 Pressure or stress [ML-1T-2] Nm-2 9 Linear momentum [MLT-1] kg ms-1 or Impulse 10 Density [ML-3] kg m-3 11 Strain Dimensionless Unitless 12 Modulus of [ML-1T-2] Nm-2 elasticity 13 Surface tension [ML-2] Nm-1 14 Velocity gradient [T-1] Second-1 15 Coefficient of [ML-1T-1] kg m-1s-1 [M-1L3T-2] Nm2/kg2 velocity 16 Gravitational constant 17 Moment of inertia [ML2] Kg m2 18 Angular velocity [T-1] rad/s 19 Angular [T-2] rad/S2 [ML2T-1] Kg m2S-1 acceleration 20 Angular momentum 21 Specific heat [L2T-2Theta-1] Kcal kg-1K-1 22 Latent heat [L2T-2] kcal/kg 23 Planck's constant [ML2T-1] J-s 24 Universal gas [ML2T-2Theta-1] J/mol-k constant Homogeneity Principle If the dimensions of left hand side of an equation are equal to the dimensions of right hand side of the equation, then the equation is dimensionally correct. This is known as homogeneity principle. Mathematically [LHS] = [RHS] Applications of Dimensions 1. To check the accuracy of physical equations. 2. To change a physical quantity from one system of units to another system of units. 3. To obtain a relation between different physical quantities. Limitation of dimensional analysis 1. The dimensional consistency does not guarantee correct equations. 2. The arguments of special functions, such as the trigonomrtric, logarithmic and exponential functions are dimensionless. 3. A pure number, ratio of similar physical quantities. 4. Dimensionless constants cannot be obtained by this method. 5. If an equation fails consistency test, it is proved wrong, but if it passes, it is not proved right. 6. It does not distinguish between the physical quantities having same dimensions.