Order of Magnitude Estimations: estimations to the nearest power of ten The ranges of magnitude that occur in the universe: Sizes: 10-15 m to 10 +25 m (subnuclear particles to etent of the visible universe). Masses: 10-30 kg to 10 +50 kg (electron to mass of the universe). Times: 10-23 s to 10 +18 s (passage of light across a nucleus to the age of the universe). Size of a proton 10-15 m Size of an atom 10-10 m Phsical Measurement Significant Figures and Decimal Places Measurement Significant Decimal Figures Places 4003 m 4 0 160 N 2 0 160. N 3 0 30.00 kg 4 2 0.00610 m 3 5 Fundamental Units Quantit Units Smbol Length meter m Mass kilogram kg Time second s Electric current ampere A Temperature Kelvin K Amount mole mol Luminous intensit candela cd Derived units are a combination of fundamental units (that ma have an alternate name). Eample: 1 kg m/s 2 = 1 N Sig Fig Rules Addition and Subtraction Rule The sum or difference of measurements ma have no more decimal places than the least number of decimal places in an measurement. Eample: Add 2.34 m, 35.7 m and 24 m = 62 m Multiplication and Division Rule When multipling or dividing, the number of significant figures retained ma not eceed the least number of significant figures in an of the factors. Eample: 0.304 cm 73.84168 cm. = 22.4 cm 2 Accurac: An indication of how close a measurement is to the accepted value (a measure of correctness) Precision: An indication of the agreement among a number of measurements made in the same wa (a measure of eactness) Sstematic Error: An error associated with a particular instrument or eperimental technique that causes the measured value to be off b the same amount each time. (Affects the accurac of results - Can be eliminated b fiing source of error shows up as non-zero -intercept on a graph) Random Uncertaint: An uncertaint produced b unknown and unpredictable variations in the eperimental situation. (Affects the precision of results - Can be reduced b taking repeated trials but not eliminated shows up as error bars on a graph) 1. Multiple trials can reduce the random uncertaint but not a sstematic error. (Repeated measurements can make our answer more precise but not more accurate.) 2. An accurate eperiment has low sstematic error. 3. A precise eperiment has low random uncertaint. 1
Metric Prefies and Conversions 3. Convert 0.0340 pm into kilometers..0340 pm 1 10-12 m 1 km 1 pm 1 10 3 m = 3.40 10-17 km Prefi to prefi --- must make pit stop at base unit first 4. Convert 12.8 cm 2 into m 2. 12.8 cm 2 1 10-2 m 1 10-2 m 1 cm 1 cm = 1.28 10-3 m 2 Squared units --- two conversions Cubed units --- three conversions Factor-Label Method for Converting Units 1. Convert 45 centimeters into meters. 45 cm 1 10-2 m = 0.45 m or 4.5 10-1 m 1 cm 2. Convert 1.9 A into microamps. 1.9 A 1 µa 1 10-6 A = 1.9 10 6 µa 5. Convert 4700 kg/m 3 into g/cm 3 4700 kg 1 10 3 g 1 10-2 m 1 10-2 m 1 10-2 m 1 m 3 1 kg 1 cm 1 cm 1 cm = 4700 10-3 = 4.7 g/cm 3 6. Convert 55 mph into m/s. (1.0 mile 1.6 km) 55 miles 1.61 km 1 10 3 m 1 hr 1 hr 1 mile 1 km 3600 s = 27 m/s 2
Ever measured value has uncertaint. Uncertainties on Raw Data: a) b) c) Uncertainties on Processed Data: a) b) c) A child swings back and forth on a swing 10 times in 36.27s ± 0.01 s. How long did one swing take? Data Processing Measurements of time are taken as: 14.23 s, 13.91 s, 14.76 s, 15.31 s. 13.84 s, 14.18 s. What value should be reported? Voltage ± uncertaint Absolute Uncertaint Fractional Uncertaint Percentage Uncertaint V ± ΔV 11.6 V ± 0.2 V Calculations with Uncertainties 1. Addition/Subtraction Rule: When two or more quantities are added or subtracted, the overall uncertaint is equal to the sum of the absolute uncertainties. Eample: The sides of a rectangle are measured to be (4.4 ± 0.2) cm and (8.5 ± 0.3) cm. Find the perimeter of the rectangle. ON REFERENCE TABLE: 3
2. Multiplication/Division Rule: When two or more quantities are multiplied or divided, the overall uncertaint is equal to the sum of the percentage uncertainties. Eample: The sides of a rectangle are measured to be (4.4 ± 0.2) cm and (8.5 ± 0.3) cm. Find the area of the rectangle. ON REFERENCE TABLE: 3. Power Rule: When the calculation involves raising to a power, multipl the percentage uncertaint b the power. 1 (Don t forget that = 2 ) Eample: The radius of a circle is measured to be 3.5 cm ± 0.2 cm. What is the area of the circle with its uncertaint? ON REFERENCE TABLE: 4
Eercises 1. Five people measure the mass of an object. The results are 0.56 g, 0.58 g, 0.58 g, 0.55 g, 0.59 g. How would ou report the measured value for the object s mass? 2. Juan Deroff measured 8 floor tiles to be 2.67 m ±0.03 m long. What is the length of one floor tile? 3. The first part of a trip took 25 ± 3 s, and the second part of the trip took 17 ± 2s. a. How long did the whole trip take? b. How much longer was the first part of the trip than the second part? 4. A car traveled 600. m ± 12 m in 32 ± 3 s. What was the speed of the car? 5. The time t it takes an object to fall freel from rest a distance d is given b the formula: where g is the acceleration due to gravit. A ball fell 12.5 m ± 0.3 m. How long did this take? t = 2d g 5
Analzing Data Graphicall The masses of different volumes of alcohol were measured and then plotted (using Logger Pro). Note there are three lines drawn on the graph the best-fit line, the line of maimum slope, and the line of minimum slope. The slope and -intercept of the best-fit line can be used to write the specific equation and the slopes and -intercepts of the ma/min lines can be used to find the uncertainties in the specific equation. The specific equation is then compared to a mathematical model in order to make conclusions. General Equation: = m + b Specific Equation: M = (0.66 g/cm 3 )V + 0.65 g Uncertainties: slope: 0.66 g/cm 3 ± 0.11 g/cm 3 -intercept: 0.65 g ± 3.05 g Mathematical Model: D = M/V so M = DV Conclusion Paragraph: 1. The purpose of the investigation was to determine the relationship between volume and mass for a sample of alcohol. 2. Our hpothesis was that the relationship is linear. The graph of our data supports our hpothesis since a best-fit line falls within the error bars of each data point. 3. The specific equation of the relationship is M = (0.66 g/cm 3 )V + 0.65 g. 4. We believe that enough data points were taken over a wide enough range of values to establish this relationship. This relationship should hold true for ver small volumes, although if it becomes too small for us to measure with our present equipment we won t be able to tell, and for ver large volumes, unless the mass becomes so large that the liquid will be compressed and change the densit. 5. Zero falls within uncertaint range for -intercept (0.65 g ± 3.05 g) so our results agree with math model and no sstematic error is apparent 6. B comparison to the mathematical model we conclude that the slope of the graph represents the densit. Therefore the densit of the sample is 0.66 g/cm 3 ± 0.11 g/cm 3. 7. The literature value for the densit of this tpe of alcohol is 0.72 g/cm 3. Our results agree with the literature value since the literature value falls within the eperimental uncertaint range of 0.66 g/cm 3 ± 0.11 g/cm 3. 6
Graphical Representations of Relationships between Variables Constant = c Direct (linear) = m + b Quadratic = c 2 Inverse = c/ Inverse quadratic (inverse square) = c/ 2 Square root = c Independent Variable: the measurable quantit that ou var intentionall Dependent Variable: the measurable quantit that changes as a result Control Variables: other important measurable quantities (not pieces of equipment) that could be independent variables and so must be held constant (controlled). Graph Straightening Linearizing a graph: Purpose: Transforming a non-linear graph into a linear one b an appropriate transformation of the variables and a re-plotting of the data points. To be able to find the constant of proportionalit and write the specific equation so the relationship can be compared to a mathematical model. To linearize this... To linearize this... To linearize this... To linearize this... X 2 1/ 1/X 2... graph this.... graph this.... graph this.... graph this. 7
Eample of Graphicall Analzing Non-Linear Data Research Question: What is the relationship between kinetic energ and speed for a uniforml accelerating object? Data Collection General Equation: Specific Equation: Mathematical Model: Data Processing Comparison: The graph can be straightened b plotting KE vs. the square of the speed. Conclusion Paragraph: Note that no error bars are required on the straightened graph, nor are ma/min lines, but these could be found if desired. The purpose of the investigation was to determine the relationship between the kinetic energ and the speed of a uniforml accelerating object. Our hpothesis was that the relationship is quadratic. The graph of our data supports our hpothesis. The specific equation of the relationship is KE = (4.9 J/(m 2 /s 2 )) v 2. B comparison to the mathematical model of KE = ½ mv 2 we conclude that the slope of the graph represents one-half the mass of the object. Therefore the mass of the object is 9.8 kg. The measured mass of the object was 10.0 kg. This is consistent with our eperimental value and represents a percent difference of 2.0%. 8
Linearizing Graphs Using Logarithms This is a special linearizing (straightening) technique that works with general equations that are power functions. Power Function: Method of straightening: Derivation: Eamples 1. 2. 3. 4. 9
Eercises Linearizing Data with Logarithms In each eample below, straighten each graph b logarithms. Then, write the specific equation for each relationship. What is the most probable tpe of relationship in each case? 1. Time (s) ± 0.2 s Displacement (m) ± 2 m 0 0 General Equation: Specific equation: 1.0 3 Slope: 2.0 13 3.0 30 4.0 41 Y-intercept: Tpe of relationship: 5.0 72 2. Mass (kg) ± 0.1 kg Acceleration (m/s 2 ) ±0.1 m/s 2 1.1 12.0 2.1 5.9 General Equation: Slope: Specific equation: 3.0 4.1 3.8 3.0 Y-intercept: Tpe of relationship: 5.0 2.5 6.2 2.0 3. Distance (m) Force (N) ± 0.1 m ± 0.2 N 1.5 6.7 General Equation: Specific equation: 2.0 3.8 2.4 2.4 Slope: 3.1 1.7 3.6 1.2 Tpe of relationship: 3.9 0.9 4.6 0.7 Y-intercept: 5.2 0.6 10