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College Physics. Chapter 1 Introduction. Theories and Experiments. The goal of physics is to develop theories based on experiments A theory is a “guess,” expressed mathematically, about how a system works The theory makes predictions about how a system should work
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College Physics Chapter 1 Introduction
Theories and Experiments • The goal of physics is to develop theories based on experiments • A theory is a “guess,” expressed mathematically, about how a system works • The theory makes predictions about how a system should work • Experiments check the theories’ predictions • Every theory is a work in progress
Fundamental Quantities and Their Dimension • Length [L] meters • Mass [M] kilograms • Time [T] seconds • other physical quantities can be constructed from these three • These are called derived units with examples of velocity, m/s, force;kg-m/s2
Units • To communicate the result of a measurement for a quantity, a unit must be defined • What is a gram? How long is a meter? • Defining units allows everyone to relate to the same fundamental amount • How is a second defined?
Systems of Measurement • Standardized systems • agreed upon by some authority, usually a governmental body • SI -- Systéme International • agreed to in 1960 by an international committee • main system used in this text • also called mks(meter, kilogram, second) for the first letters in the units of the fundamental quantities
Systems of Measurements, cont • cgs – Gaussian system • named for the first letters of the units it uses for fundamental quantities • Centimeter, gram, second • US Customary • everyday units • often uses weight, in pounds, instead of mass as a fundamental quantity
Length • Units • SI – meter, m • cgs – centimeter, cm • US Customary – foot, ft • Defined in terms of a meter – the distance traveled by light in a vacuum during a given time of 1/299792458 second. This establishes the speed of light at 299792458 m/sec or its accepted value of 3.00 x 108 m/s.
Mass • Units • SI – kilogram, kg • cgs – gram, g • USC – slug, slug • Defined in terms of kilogram, based on a specific cylinder of platinum and iridium alloy kept at the International Bureau of Weights and Measures located in Sevres, France.
Time • Units • seconds, s in all three systems • 9192631700 times the period of oscillation of radiation from a cesium atom • The current standard cesium clock was built and is housed at the National Institute of Standards and Technology Laboratory in Boulder, Colorado
Approximate Values • Various tables in the text show approximate values for length, mass, and time: See Page 3 • Note the wide range of values • Lengths – Table 1.1 • Masses – Table 1.2 • Time intervals – Table 1.3
Prefixes • Prefixes correspond to powers of 10 • Each prefix has a specific name • Each prefix has a specific abbreviation • See table 1.4 found on page 4 Common prefixes to remember: 109 giga, G 106 mega, M 103 kilo, k 101 deka, da 10-1 deci, d 10-2 centi, c 10-3 milli, m 10-6 micro, 10-9 nano, n
Structure of Matter • Matter is made up of molecules • the smallest division that is identifiable as a substance • Bodies of mass smaller than the molecule will not have characteristics of a unique substance; e.g. protons, neutrons, electrons • Molecules are made up of atoms • correspond to elements
More structure of matter • Atoms are made up of • nucleus, very dense, contains • protons, positively charged, “heavy” • neutrons, no charge, about same mass as protons • protons and neutrons are made up of quarks • orbited by • electrons, negatively charges, “light” • fundamental particle, no structure
Structure of Matter Quarks – up, down, strange, charm, bottom, and top. Up, Charm, and Top have a charge of + ⅔ that of a proton. Down, Strange, and Bottom have a charge of - ⅓ that of a proton. The proton has two up quarks and one down quark. ⅔ + ⅔ - ⅓ = + 1. The neutron has two down quarks and one up quark. - ⅓- ⅓ + ⅔ = 0. The other quarks are indirectly observed and not well understood.
Dimensional Analysis • Technique to check the correctness of an equation • Dimensions (length, mass, time, combinations) can be treated as algebraic quantities • add, subtract, multiply, divide • Both sides of equation must have the same dimensions
Dimensional Analysis, cont. • Cannot give numerical factors: this is its limitation • Dimensions of some common quantities are listed in Table 1.5 on page 5. • Example: [a] = [v]/[t]; L/T = {x}/{t2} T • a = x/t2 • x = at2
Uncertainty in Measurements • There is uncertainty in every measurement, this uncertainty carries over through the calculations • need a technique to account for this uncertainty • We will use rules for significant figures to approximate the uncertainty in results of calculations
Significant Figures • A significant figure is one that is reliably known • All non-zero digits are significant • Zeros are significant when • between other non-zero digits • after the decimal point and another significant figure • can be clarified by using scientific notation
Operations with Significant Figures When multiplying or dividing two or more quantities, the number of significant figures in the final result is the same as the number of significant figures in the least accurate of the factors being combined
Operations with Significant Figures, cont. • When adding or subtracting, round the result to the smallest number of decimal places of any term in the sum • If the last digit to be dropped is less than 5, drop the digit • If the last digit dropped is greater than or equal to 5, raise the last retained digit by 1
Examples of Significant Digits • Addition & Subtraction – The resulting answer is to have the same degree of precision as the least precise number used in the calculation. Ex. 3.6 – 0.57 = 3.0 not 3.03 5.4 + 2.35 = 7.8 not 7.75 Multiplication & Division - The resulting answer is to have the same number of significant digits as the number with the least precise number of significant digits. Ex. 2.0 ÷ 3.0 = 0.67 not 0.6666666666 2.5 x 3.2 = 8.0 not 8
Reading Quiz The quantity 2.67 x 103 m/s has how many significant figures? • 1 • 2 • 3 • 4 • 5
Answer The quantity 2.67 x 103 m/s has how many significant figures? • 1 • 2 • 3 • 4 • 5 Slide 1-8
Importance of Measurements • Data collection and interpretation are essential skills in physics • Repeatability to obtain measurements and the degree of uncertainty of correctness are very important
Importance of Measurements • Precision – The repeatability of the measurement using a given instrument Example: If a board is measured four times and the results are 8.81 cm, 8.78 cm, 8.85 cm, 8.82 cm then the precision of the measurement is ± 0.1 cm. The value beyond the 0.1 cm level is an estimate between 8.8 and 8.9 cm.
Importance of Measurements • Accuracy – The degree of closeness a measurement is to the true and correct value. • Example: • If the ruler used to take the measurements stated previously was manufactured with a 2% error, the accuracy of the 8.8 cm measurement would be 2% or 8.8 x 0.02 = ± 0.176 cm or ± 0.2 cm .
Importance of Measurements • Estimated Uncertainty – Expressing the measurement with the degree of determined precision of the measurement Example: 8.8 ± 0.1 cm is the estimated uncertainty because the measurement was 8.8 cm and the level of precision was determined to be ± 0.1 cm
Importance of Measurements • Percent Uncertainty – The ratio of the uncertainty of the measured value to the measured value multiplied by 100. Example: Uncertainty → 0.1 cm Measured value →8.8 cm % uncertainty = (0.1/8.8) x 100 = 1%
Importance of Measurements • Percent uncertainty Example - A family heirloom diamond is loaned to a friend to show her family. Before giving her the diamond you weigh it and get 8.17 g on a scale of ± 0.05g accuracy. Upon return the diamond weighs 8.09 g. Is this your diamond? Answer – The mass before release was 8.17± 0.05g meaning the weight range was 8.12 – 8.22 g. The mass after return was 8.09 ± 0.05g meaning the weight range was 8.04 – 8.14 g. Since these ranges overlap (8.12 – 8.14) there is no concern about discrepancy.
Percent Error The calculation of % error from the determined data is an integral part of the analysis of physics. % error = |accepted value – measured value|x 100 accepted value Example: Determination of gravity Measured value 9.68 m/s 2 Accepted value 9.80 m/s 2 % error = 9.80 – 9.68 x 100 = 1.22% 9.80
Summary of Formulas • Accuracy – measurement x % error of device • Precision – one power of ten less than the given measurements • Estimated uncertainty – measurement accepted level of precision • Percent Uncertainty – • % uncertainty = uncertainty x 100 measurement • Percent error – % error = |accepted value – measured value| x 100 accepted value
Sample Problems What is the % uncertainty in the measurement 3.76 0.25? Answer: (0.25/3.76) x 100% = 6.6% What is the % uncertainty in the measurement 11.3 0.9? Answer: (0.9/11.3) x 100% = 8%
Answer: (21.1)(27.5) {(21.1)(.2) + (27.5)(.1) + (.1)(.2)} = 580.25 (4.22 + 2.55 + 0.02) = (580 7) cm2 % error = (603 – 580) x 100 = 3.8% 603 Sample Problems In calculating the area of a piece of notebook paper you get measurements of (21.1 .1) cm for the width and (27.5 .2) cm for the length. Determine the area and the uncertainty. If the actual value of the area is 603 cm2, determine the percent error from your calculation. Uncertainty x uncertainty Measurements x uncertainties of other measure measurements
Sample Problems An airplane travels at 950 km/h. How long in seconds does it take to travel 1 km? Answer: (950 km/h)(1h/3600sec) = .26 sec Use dimensional analysis to determine if the equation vf 2 = vo2 + 2as is consistent. Answer: [L]2 = [L]2 + [L][L] [T]2 [T]2 [T]2 [L]2 = [L]2 [T]2 [T]2 The equation is consistent.
Conversions • When units are not consistent, you may need to convert to appropriate ones • Units can be treated like algebraic quantities that can “cancel” each other • See the inside of the front cover for an extensive list of conversion factors • Example:
Order of Magnitude • Approximation based on a number of assumptions • may need to modify assumptions if more precise results are needed • Order of magnitude is the power of 10 that applies • Examples: 27~30, 1006950~1000000
Coordinate Systems • Used to describe the position of a point in space • Coordinate system consists of • a fixed reference point called the origin • specific axes with scales and labels • instructions on how to label a point relative to the origin and the axes
Types of Coordinate Systems • Cartesian – (x,y) or (x,y,z) • Plane polar - (r,) y s r x
Cartesian coordinate system • Also called rectangular coordinate system • x- and y- axes • Points are labeled (x,y)
Plane polar coordinate system • Origin and reference line are noted • Point is distance r from the origin in the direction of angle , ccw from reference line • Points are labeled (r,)
More Trigonometry • Pythagorean Theorem • To find an angle, you need the inverse trig function • for example, • Be sure your calculator is set appropriately for degrees or radians
Problem Solving Strategy • Read the problem • Identify the nature of the problem • Draw a diagram • Some types of problems require very specific types of diagrams
Problem Solving cont. • Label the physical quantities • Can label on the diagram • Use letters that remind you of the quantity • Many quantities have specific letters • Choose a coordinate system and label it • Identify principles and list data • Identify the principle involved • List the data (given information) • Indicate the unknown (what you are looking for)
Problem Solving, cont. • Choose equation(s) • Based on the principle, choose an equation or set of equations to apply to the problem • Substitute into the equation(s) • Solve for the unknown quantity • Substitute the data into the equation • Obtain a result • Include units
Problem Solving, final • Check the answer • Do the units match? • Are the units correct for the quantity being found? • Does the answer seem reasonable? • Check order of magnitude • Are signs appropriate and meaningful?
