SPH4U/IB DAY 3/4/5 NOTES METRIC SYSTEM,GRAPHING RULES, RATIOS AND PROPORTIONS, LOG-LOG DATA ANALYSIS

Metric System Metric prefix word Metric prefix symbol Power of ten nano n 10-9 micro µ 10-6 milli m 10-3 centi c 10-2 kilo k 103 mega M 106 giga G 109

Metric system The standard units in Physics are kilograms (kg), seconds (s) and metres (m). The Newton (N) is 1 kgm/s2 and The Joule (J) is 1 kgm2/s2. 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 12.0 µm 33.45 mm 12.0 µg 12.67 ns 123.4 Gm 7654 Mg Convert to standard units: kg, m, s: 12.0 µm 33.45 mm 12.0 µg 12.67 ns 123.4 Gm 7654 Mg 45.258 Ms 0.000458 km 0.025478 cs

Metric practice 12.0 µm  1.20 x 10-5 m 33.45 mm  0.03345 m Convert to standard units: kg, m, s: 12.0 µm  1.20 x 10-5 m 33.45 mm  0.03345 m 12.0 µg  1.20 x 10-8 kg 12.67 ns  1.267 x 10-8 s 123.4 Gm 1.234 x 1011 m 7654 Mg  7.654 x 106 kg 45.258 Ms  4.5258 x 107 s 0.000458 km  0.458 m 0.025478 cs  2.5478 x 10-4 s

Unit analysis Useful to catch simple algebra mistakes! Carry units all the way through your calculations. Multipliers used when it’s not just a simple decimal shift, or when two units are changed at the same time. Ex) Convert 232 km/h into standard units. 232 km (1000 m) (1h) = h km 3600 s 64.4 m/s

Metric System and Unit Analysis ws’s Metric ws and Unit Analysis ws

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.

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 d = (28m/s)t time (s) distance (m) 1 28 2 56 3 84 4 112 5 140

Sub in values from table and avg Example 2 Frequency (Hz) Period (s) 5 0.2 10 0.1 20 0.05 50 0.02 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 ƒ = k/T Sub in values from table and avg ƒ = 1/T

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  v2. If the speed triples, how many times greater is the force? F = kv2 means that F1 = k v12 and a new force F2 = k v22. Taking a ratio of these two expressions eliminates the “k” constant.

Example 1 – algebraic solution F2 = 9F1

Typical MC questions Using your formula sheet, you can quickly solve proportion questions without showing all the algebra. Ex) F = ma means that Force is proportional to acceleration, for a constant mass. If Force doubles, acceleration doubles. Rearranging: a = F/m shows that acceleration is inversely proportional to mass if the Force is constant. If mass doubles, acceleration is halved. You can do this for any formula – just rearrange the formula for the variable you are solving for to determine whether it’s direct or inverse (and may have a power!!).

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 = kxn

Log-Log Data Analysis (This is not tested) y = kxn Taking the log of both sides and using log rules: log y = logkxn 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 xn, 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 = kxn.

Log Log Assignment Do practice first, then assignment.