PHY 111 Lecture 1 Measurement and Units Jupeb ADDITION PDF

Lecture Overview 1. Space and time; physical quantities, units, dimension and dimensional analysis 2. Kinematics; Fundamental laws of mechanics, 3. Statics and dynamics; work and energy. Conservation laws. 4. Moments and energy of rotation; simple harmonic motion; motion of simple systems;...

Lecture Overview 1. Space and time; physical quantities, units, dimension and dimensional analysis 2. Kinematics; Fundamental laws of mechanics, 3. Statics and dynamics; work and energy. Conservation laws. 4. Moments and energy of rotation; simple harmonic motion; motion of simple systems; 5. Elasticity; Hooke's law, Young's shear and bulk moduli, 6. Hydrostatics; Pressure, buoyancy,Archimedes' principles. 7. Surface tension; adhesion, cohesion, capillarity, drops and bubbles. 8. Temperature; heat, gas laws; Laws of thermodynamics; kinetic theory of gases. 9. Sound; Types and properties of waves as applied to sound and light energies. 10. Superposition of waves. Propagation of sound in gases, solids and liquids and their properties. 11. The unified spectra analysis of waves. Applications.  LECTURE 1 SPACEAND TIME; UNITS AND DIMENSION.  Contents of Chapter 1 The Nature of Science Physics and Its Relation to Other Fields Models, Theories, and Laws Measurement and Uncertainty; Significant Figures Units, Standards, and the SI System Converting Units Order of Magnitude: Rapid Estimating Dimensions and Dimensional Analysis © 2014 Pearson Education, Inc.  Introduction Physics as a branch of science, deals with nature and the relationship between matter and energy. It is the natural science that is based on experiments, measurements and mathematical analysis with the purpose of finding physical laws that govern everything in the universe. © 2014 Pearson Education, Inc.  Nature of Science The process involved before a law in physics is accepted: Hypothesis Theories Law Model Principles © 2014 Pearson Education, Inc.  Nature of Science Hypothesis: A tentative explanation that can be tested by further investigation. Experiment is a controlled method of testing a hypothesis Example: Avogadro's hypothesis  Nature of Science Theory A comprehensive explanation of an important feature of nature supported by facts that have been repeatedly confirmed through observation and experimentation over time. Quantitative description of experimental observations. Not just WHAT is observed but WHY it is observed as it is and HOW it works the way it does. Example: Brownian theory Tests of theories: – Experimental observation: More experiments, more observation. – Predictions: Made before observations & experiments.  Nature of Science Law: they are similar to scientific theories in that they are principles that can be used to predict the behavior of natural world. Both law and theories are well supported by observation and/or experimental evidence. It summarizes the relationship between variables. E.g Newton’s law. The difference between them are: Law are frequently written as an equation and describes what happens. Theories are more of explanation. It explains why we observe what we do. © 2014 Pearson Education, Inc.  Nature of Science Model: Models are very useful during the process of understanding phenomena. An analogy of a physical phenomenon to something we are familiar with. A model creates mental pictures; care must be taken to understand the limits of the model and not take it too seriously. A description, simulation, graphic, or 3-D representation of theory used to help enhance understanding. Example: The Bohr model of the atom, for example, depicts electrons circling the atomic nucleus in a fashion similar to planets in the solar system, but it is by no means represents the true nature of the atom. © 2014 Pearson Education, Inc.  Nature of Science Principles: Principles are approaches to analyzing a problem and serve as templates for setting up equations specific to your problem. A principle is similar to a law, but applies to a narrower range of phenomena. Example: Archimedes Principles © 2014 Pearson Education, Inc.  Physical Quantities Physical quantities are physical properties in physics that can be measured. They have two parts: eg. 20 kg 1. Numerical 2. Alphabetical Numerical part : size/magnitude Alphabetical part: it is the quantity that specifies the specific physical quantity-SI unit. © 2014 Pearson Education, Inc.  Physical Quantities Unit Definition: A unit is a physical quantity that can be counted or measured using standard size defined by custom or law. unit as used in physics refers to the standard measure of a quantity. Every measurement or quantitative statement requires a unit. Example: 20 sugar does not mean anything, rather 20 kg of sugar. There are two categories of units 1. fundamental/base units 2. Derives units  Physical Quantities Units If a unit becomes officially accepted, it is called Standard Unit. Group of Unit and Combination is called SYSTEM OF UNITS. Example: SI Units, British Units SI = International Systems of Units  UNIT SYSTEMS Full forms of different standard systems: 1.The full form of CGS system is Centimeter Gram Second system. In the CGS system, fundamental units are Centimeter, Gram, and second 2.The full form of FPS system is Foot Pound Second system. In the FPS system, fundamental units are Foot, Pound, and second. 3.The full form of MKS system is Meter Kilogram Second system. In the MKS system, fundamental units are meter, kilogram, and second. 4.The full form of SI is System International(International System of Units). In the SI system, the fundamental units are Meter, Kilogram, and Second.  Physical Quantities Units ❑ British Units: SI (Système International) Units also called Metric System L = inches, feet, MKS system CGS System miles, L = meters (m) centimeter (cm) M = slugs (pounds), T = seconds M = kilograms (kg) gram (g) T = seconds (s) seconds (s) All things in classical mechanics can be expressed in terms of base quantities: Length (L),MASS(M), TIME (T)  Units, Standards, and the SI System Quantity Unit Standard Length Meter Length of the path traveled by light in 1/299,792,458 second. Time Second Time required for 9,192,631,770 periods of radiation emitted by cesium atoms Mass Kilogram Platinum cylinder in International Bureau of Weights and Measures, Paris © 2014 Pearson Education, Inc.  Units, Standards, and the SI System These are the standard SI prefixes for indicating powers of 10. Many are familiar; Y, Z, E, h, da, a, z, and y are rarely used. © 2014 Pearson Education, Inc.  Converting Units Converting between metric units, for example from kg to g, is easy, as all it involves is powers of 10. Converting to and from British units is considerably more work. For example, given that 1 m = 3.28084 ft, this 8611-m mountain is 28251 feet high. © 2014 Pearson Education, Inc. Unit Conversion  The advantages of S.I. system (i) S.I. has broader base. It has seven base units and two supplementary units. (ii) S.I. is rational, i.e., This system makes use of only one unit for one physical quantity, e.g., for energy of any type, i.e., mechanical or heat or electrical. There is only one unit, Joule (J) but in M.K.S. system unit for mechanical energy is Joule. (iii) S.I. is coherent. In this system, all the derived units can be easily obtained from basic and supplementary units (iv) S.I. is metric system which means that multiples and submultiples can be expressed as powers of 10. (v) S.I. is the modernised and improved form of M.K.S. system so it is readily accepted in almost all the countries of the world. (vi) The symbols fixed by S.I. are not subject to change whatever may be the language, e.g., unit of mass kilogram having symbol ‘kg’ has to be written like this only in French or German or Hindi or Punjabi etc.  Disadvantage of SI Unit It has disadvantages such as it mainly focuses on only one unit so the importance of other units is diluted. Also the SI unit cannot always accurately define a quantity.  Physical Quantities Categories of physical quantities: 1. Fundamental/base 2. Derived 3. Supplementary Fundamental/base They are quantities which are distinct in nature that cannot be defined by other quantities but upon which other physical quantities are derived. There are 7 base quantities © 2014 Pearson Education, Inc.  Fundamental/base quantities Definitions: Mass - is the amount of substance or material in an object. Length - is the distance between two points. Time - is the duration between two instants or events. Current - is the flow of electrons from a high potential to a low potential. Temperature - is the measure of how hot or cold an object or substance is based on a comparative scale. Amount of Substance - is a measure of the amount of elementary entities such as atoms, molecules and other particles. Luminous Intensity - is a measure of the amount of light that a point source radiates in a given direction.  Physical Quantities Fundamental/base quantities  Physical Quantities Derive quantities They are quantities whose definitions are based on other physical quantities (fundamental quantities) by multiplication or division or both. A derived quantity is produced when two (2) or more fundamental quantities are combined, either by multiplication or division.  Derive quantities Definitions: Displacement - is how far off an object is from its rest position, usually a straight line distance. Speed - is how fast an object is moving. Velocity - is how fast an object is moving in a specific direction. Acceleration - is the change in velocity of a period of time. Acceleration could be positive, negative (retardation) or zero (no change in velocity). Force - is a push or a pull which may speed up, slow down, change direction or shape of an object. Energy - is the ability to do work. Density - is the amount of matter in a given volume of an object or substance. Moment - is the turning effect of a Force. Power - is the rate at which work is done. Work - is a force of 1 N displacing an object by 1 m in the direction of the force. Momentum - is the quantity of motion of a moving body, measured as a product of an object's mass and velocity. Pressure - is the force acting on a surface per unit area. Derive quantities  Supplementary Quantities They are geometrical quantities of circle and sphere such as plane angle and solid angle. Similarly, their units not classified as base or derived units. Measurement and Uncertainty; Significant Figures Measurement is a process of detecting an unknown physical quantity by using standard quantity. No measurement is exact; there is always some uncertainty due to limited instrument accuracy (systematic error) difficulty reading results (random/human error). Experiments are NEVER 100% accurate. – Always have uncertainty in final result. There are two types of measurement errors: Random Errors Systematic Errors Measurement and Uncertainty; Significant Figures Random Errors Random errors are statistical fluctuations (in either direction) in the measured data due to the precision limitations of the measurement device. Random errors usually result from the experimenter's inability to take the same measurement in exactly the same way to get exact the same number. For Example :You measure the mass of a ring three times using the same balance and get slightly different values: 17.46 g, 17.42 g, 17.44 g. How to minimize it Take more data. Random errors can be evaluated through statistical analysis and can be reduced by averaging over a large number of observations. Measurement and Uncertainty; Significant Figures Systematic error systematic error is consistent, repeatable error associated with faulty equipment or a flawed experiment design. These errors are usually caused by measuring instruments that are incorrectly calibrated or are used incorrectly. A worn out instrument. For example, a plastic tape measure becomes slightly stretched over the years, resulting in measurements that are slightly too high, An incorrectly calibrated or tared instrument, like a scale that doesn’t read zero when nothing is on it, A person consistently takes an incorrect measurement. For example, they might think the 3/4″ mark on a ruler is the 2/3″ mark. How to minimize it Systematic errors are difficult to detect and cannot be analyzed statistically, because all of the data is off in the same direction(either to high or too low). Spotting and correcting for systematic error takes a lot of care. Add or subtract known error. Measurement and Uncertainty; Significant Figures Preventing Errors Random error can be reduced by: Using an average measurement from a set of measurements, or Increasing sample size. systematic error is difficult to detect — and therefore prevent. In order to avoid these types of error, know the limitations of your equipment and understand how the experiment works. This can help you identify areas that may be prone to systematic errors.  Some terminology True value – standard or reference of known value or a theoretical value. Accuracy: This is the closeness of the measured values to the true value. Precision: reproducibility or agreement with each other for multiple trials. It is the closeness of the measured values to each other: the closer they are to each other, the more precise they are. Uncertainty: The interval in which the true value lies is called the uncertainty in the measurement. Absolute Uncertainty or ± value The absolute uncertainty in the mean value of measurements is half the range of the measurements.  Error calculation of Uncertainty E.g. Suppose the measurements of the diameter of a pin by a Vernier Caliper are as follows: 0.25mm; 0.24mm;0.26mm; 0.23mm;0.27mm; Solution The mean = (0.25 + 0.24 + 0.26 + 0.23 + 0.27)/5 =125/5 = 0.25mm The range = 0.27 - 0.23 = 0.04mm Absolute Uncertainty = ± 0.04/2 = ± 0.02 So, the mean value = mean ± range/2 = 25 ± 0.04/2 = 25 ± 0.02  Error calculation of Uncertainty Combining uncertainties + and - Adding or subtracting quantities then sum all individual absolute uncertainties eg 2.1 ± 0.1 + 2.0 ± 0.2 = 4.1 ± 0.3 eg 2.1 ± 0.1 - 2.0 ± 0.2 = 0.1 ± 0.3 this method overestimates the final uncertainty  Error calculation of Uncertainty When Dividing or multiplying quantities, then sum all of the individual relative uncertainties eg (2.5 ± 0.1) x (5.0 ± 0.1) = (2.5 ± 4%) x (5.1 ± 2%) =12.5 ± 6% (or 0.75 or 0.7) eg (5.2 ±0.1) / (0.84 ± 0.05) 5.2/0.84 =6.19 = 6.2 Find % uncertainty: (0.1/5.2)x100 = 1.92 = 2 (0.05/0.84)x100 = 5.95 = 6 Sum of % uncertainty = 2%+6% = 8% Absolute uncertainty =(8/100)x6.2=0.496 =0.5 = 6.2 ± 0.5 or 6.2 ± 8%  Error calculation of Uncertainty Rule: percentage uncertainty are added Estimated uncertainty is written with a ± sign; for example: 8.8 ± 0.1 cm Percent uncertainty is the ratio of the uncertainty to the measured value, multiplied by 100: © 2014 Pearson Education, Inc.  Significant Figures The rules of significant figures: 1. Any figures that is non-zero, are considered as a significant figure. 2.Zeros at the beginning of a number are not significant Example: 0.254 ----------------- 3 s.f 3.Zeros within a number are significant. Example: 104.6 m ---------------- 4 s.f 4. Zeros at the end of a number after the decimal point are significant. Example: 27050.0 ------------------- 6 s.f  Significant Figures … 5. Zeros at the end of a whole number without a decimal point may or may not be significant. It depends on how that particular number was obtained, using what kind of instrument, and the uncertainty involved. Example:500m ------------------- could be 1 or 3 sf. Convert the unit: 500m = 0.5km (would you say it has 1 sf ? ) 500m = 50 000cm (would you say it has 1 or 5 sf ?) How to solve this problem ?  Significant figures Addition and Subtraction processes The rule: The final result of an addition and/or subtraction should have the same number of significant figures as the quantity with the least number of decimal places used in the calculation. Example: 23.1 + 45 + 0.68 + 100 = 169 Example: 23.5 + 0.567 + 0.85 = 24.9  Significant figures Multiplication and division processes The rule: The final result of an multiplication and/or division should have the same number of significant figures as the quantity with the least number of significant figures used in the calculation. Example: 0.586 x 3.4 = 1.9924 = 2.0 Example: 13.90 / 0.580 = 23.9655 = 24.0  Percentage Error This is the formula for "Percentage Error": Example: The report said the carpark held 240 cars, but we counted only 200 parking spaces. What is the percentage error in the report.  Exercise 1) You measure a block with a ruler that has a precision of 1mm. You obtain a length of 2.4cm. a) What is the absolute uncertainty in this measurement? b) What is the percentage uncertainty in this measurement? 2) The following range of results is obtained for the mass of a bouncy ball: 13.2g, 13.4g, 13.3g, 13.4g, 13.2g, 13.3g a) Calculate a value for the mass of the bouncy ball. b) What is the absolute uncertainty in these measurements? c) What is the percentage uncertainty in these measurements?  Exercise 3. How many Significant figures in each term? a. 34.6209 b. 0.003048 c. 5010.0 d. 4032.090 4.Solve the following equations using the correct number of significant figures. a. 34.683 + 58.930 + 68.35112 b. 45001 - 56.355 - 78.44 c. 0.003 + 3.5198 + 0.0118 d. 36.01 - 0.4 - 15 5.Solve the following equations using the correct number of significant figures. a. 98.1 x 0.03 b. 57 x 7.368 c. 8.578 / 4.33821 d. 6.90 / 2.8952  Exercise 6. A builder wants to calculate the area of a square tile. He uses a rule to measure the two adjacent sides of a square tile and obtains the following results: Length of one side = 84 mm ± 0.5 mm Length of perpendicular side = 84 mm ± 0.5 mm. 7. A metallurgist is determining the purity of a sample of an alloy that is in the shape of a cube by determining the density of the material. The following readings are taken: Length of each side of the cube= 24.0 mm ± 0.5 mm Mass of cube = 48.23 g ± 0.05 g She calculates (i) the density of the material and (ii) the percentage uncertainty in the density of the material.  Exercise 8. A bullet travels a distance of l=154±0.5 m in the time t=0.4±0.05 s. a) Calculate the fractional uncertainty for the speed of the bullet. b) Calculate the percentage uncertainty for the speed of the bullet. c) Write down the speed of the bullet using the absolute uncertainty.  Exercise 9. A ball is thrown in the air and 5 different students are individually measuring the time it takes to fall back down using stopwatches. The times obtained by each student are the following: 6.2 s, 6.0 s, 6.4 s, 6.1 s, 5.8 s a) What is the uncertainty of the results? b) How should the resulting time be expressed?  Exercise 10. The current passing through a resistor is I=3±0.1 A and the resistance of the resistor is R=13±0.5 Ω. The electrical power, measured in watt (W), supplied to the resistor is given by P=RI2. a) Write down the value of the supplied power correct to one significant figure. b) Find the percentage uncertainty for the current passing through the resistor and its resistance. c) Find the absolute uncertainty for the electrical power.  MCQ Which of the following is the standard unit of measurement of mass? a) Gram b) Kilogram c) Pound d) Dozen Which of the following is the correct relation? a) 1 Kg = 100 g b) 1 Kg = 1 g c) 1 g = 0.001 Kg d) 1 g = 0.01 Kg 1. Identify the primary quantity from the following. a) Mass b) Density c) Speed d) Volume 2. Which of the following is a use of dimensional analysis? a) To check the dimensional correctness of an equation b) To solve the equation dimensionally c) To get the number of dimensional constants d) To understand the dimensional equation 3. The dimension whose unit does not depend on any other dimension’s unit is known as ________ a) Fundamental dimension b) Dependent dimension c) Independent dimension d) Absolute dimension 4. Which one of the following is a dimensionless quantity? a) Mass b) Weight c) Specific weight d) Reynold’s number 5. [MLT-2] matches with the dimensional formula of ____ a) Force b) Modulus of elasticity c) Displacement d) Strain 6. What is the maximum number of unknowns that can be found through a simple dimensional equation? a) 6 b) 5 c) 3 d) 2 7. If, v stands for velocity, L for length, T for time and M for mass, what is the value of x in the equation – L = (vT/M)x? a) 3 b) 2 c) 1 d) 0 8. Assuming standard notations, which of the following quantities is dimensionless? a) v/a b) P/Fv c) FE/L d) V2/g  9.Which of the following statements are true for dimensional analysis? The functional relationship between dependent and (a) non-dependent variables can be expressed into dimensionless terms by dimensional analysis. It is used to change the theoretical equation into a (b) dimensionless form. It helps to convert the units of quantities from one (c) system to another system. (d) All of the above. 10. If the units of length and time are doubled, what will be the factor by which the unit of acceleration will change? a) 1 b) 0.25 c) 0.5 d) 2 11. a. T −1 b. T 2 c. L1 d. L−2  mv 12. A physical quantity is given byD= qB where m is the mass, vis the speed, q is the charge, and B is the magnetic field. The dimension of D will be? a) L b) M c) T d) Dimensionless 13  14. The principle of dimensional homogeneity implies that (a) only the dimensional quantities may be added or subtracted only the variables with the same dimensions may be multiplied (b) or divided only the variables with the same dimensions may be added or (c) subtracted (d) dimensions of the two sides of the equation may not be the same 15 ANSWER 1A 2A 3A 4D 5A 6 C (A simple dimensional equation uses three basic parameters – mass, length and time. Hence the maximum number of unknowns that can be found is 3.) 7D 8B 9D 10 C 11 A 12 A 13 A 14 C 15 D  DIMENSIONS AND DIMENSIONAL ANALYSIS