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SPH3UIB DAY 3/4/5 NOTES METRIC SYSTEM,GRAPHING RULES, LOG-LOG DATA ANALYSIS

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Metric System Metric prefix word Metric prefix symbol Power of ten nanon10 -9 microµ10 -6 millim10 -3 centic10 -2 kilok10 3 megaM10 6 gigaG10 9

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Metric system The standard units in Physics are kilograms (kg), seconds (s) and metres (m). The Newton (N) is 1 kgm/s 2 and The Joule (J) is 1 kgm 2 /s 2. To use standard units, students must be able to convert units to standard units for effective communication of data in labs.

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Examples 12.0 cm is converted into metres by shifting the decimal place left two spaces. 12.0 cm = 0.120 m 12.0 g is converted into kg by dividing by 1000 or shifting the decimal left three times. 12.0 g = 0.0120 kg If a mass is given as 12.0 mg, then the decimal shifts left 3 times to grams and then 3 more to kg. Once we get to really small/large numbers, scientific notation is needed. 12.0 mg = 0.0000120 kg = 1.20 x 10 -5 kg.

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Metric practice Convert to standard units: kg, m, s: 1) 12.0 µm 2) 33.45 mm 3) 12.0 µg 4) 12.67 ns 5) 123.4 Gm 6) 7654 Mg 7) 45.258 Ms 8) 0.000458 km 9) 0.025478 cs

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Metric practice Convert to standard units: kg, m, s: 1) 12.0 µm 1.20 x 10 -5 m 2) 33.45 mm 0.03345 m 3) 12.0 µg 1.20 x 10 -8 kg 4) 12.67 ns 1.267 x 10 -8 s 5) 123.4 Gm 1.234 x 10 11 m 6) 7654 Mg 7.654 x 10 6 kg 7) 45.258 Ms 4.5258 x 10 7 s 8) 0.000458 km 0.458 m 9) 0.025478 cs 2.5478 x 10 -4 s

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Graphing Expectations Use a full page of graph paper for graphs in lab reports or assignments. Include the variable and units on each axis (distance (m), time (s), etc.) The title of the graph is in the form of y vs. x (distance versus time, no units needed in title). Calculations are NOT done on the graph, but on a separate page. Errors are indicated by circling dots if no absolute error is known, or error bars for labs.

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Graphing Expectations A best fit line or curve is usually expected for all graphs. Slope calculations include units and are rounded based on sig digs (determined by precision of measuring devices in the lab (absolute error)). Use +/- half the smallest division of measuring devices in labs, for precision and to determine how many decimals you must measure to.

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Graph data from error worksheet Plot displacement versus time and velocity versus time.

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Determining data relationships Once data is plotted, several general shapes may arise: linear, power or inverse (possibly a root curve). If a curve results, we wish to determine the relationship between the variables. Once we get a linear graph, we can determine an equation for the data (and a formula may ensue).

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Ratios and Proportions The statement of how one quantity varies in relation to another is called a proportionality expression. The goal in physics is to correlate observational data and determine the relationship between the two variables: dependent and independent. We need to take data and determine a relationship and find how the change in one quantity affects the other.

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Example 1 Notice that as time doubles, so does distance. As time triples, distance triples. This is a direct relationship (or direct variation). This object is undergoing uniform motion d t time (s) distance (m) 128 256 384 4112 5140

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Example 2 Notice that as frequency goes from 5 to 50, (a factor of 10), the period changed from 0.2 to 0.02 (a factor of 1/10). ƒ 1/T Frequency (Hz) Period (s) 50.2 100.1 200.05 500.02

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Ratios and Proportions Any proportionality can be expressed as an equation by adding a proportionality constant (use the letter “k”, typically). From Ex. 2: ƒ = k (1/T) The “k” constant can be calculated with known values. (We could take numbers from Example #2 and sub in all the values and find the average “k” value.)

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Ratios and Proportions Chart ws Chart 1 will be done as an example in class.

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Example 1 – algebraic solution You are given that F v 2. If the speed triples, how many times greater is the force? F = kv 2 means that F 1 = k v 1 2 and a new force F 2 = k v 2 2. Taking a ratio of these two expressions eliminates the “k” constant.

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Example 1 – algebraic solution F 2 = 9F 1

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Ratios and Proportions ws Answers are on the bottom of the page

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Log-Log Data Analysis One method is to re-plot data by changing the manipulated (x-axis) data to see its effects on the responding (y-axis) variable. This can be tedious and prone to human error. Using the rules of logarithms, a quicker and more efficient method arises to find an equation. Any curved graph can be expressed as: y = kx n

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Log-Log Data Analysis y = kx n Taking the log of both sides and using log rules: log y = logkx n log y = log k + nlog x log y = nlog x + log k This is now in a form like y = mx + b, where the slope is “n”, the exponent of the relationship. The y-intercept will also give the value for k (log k, which can be converted to k), which is the constant (but if axes are broken, a re-plot will yield “k”).

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Log-Log Data Analysis Step 1: Take the log of all data and plot these numbers as log y versus log x. (log d vs. log t) Step 2: Find the slope. The slope of this graph yields “n” and has no units. (n uses s.d. from data) Step 3: Re-plot the graph as y versus x n, which will yield a straight line verifying your value of “n”. Step 4: Find the slope. The slope of the second graph yields the value, with units, of “k”. Step 5: The final equation can be stated in its final form as y = kx n.

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Log Log Assignment Do practice first, then assignment due Friday.

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