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

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

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

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.

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.

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

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

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.

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.

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

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).

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.

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

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

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.)

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

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.

Example 1 – algebraic solution F 2 = 9F 1

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

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

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”).

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.

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

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