sph3uib day 3/4/5 notes metric system,graphing rules, log-log data analysis

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

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.

Examples12.0 cm is converted into metres by shifting the

decimal place left two spaces. 12.0 cm = 0.120 m12.0 g is converted into kg by dividing by 1000 or

shifting the decimal left three times. 12.0 g = 0.0120 kgIf 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 1011 m

6) 7654 Mg 7.654 x 106 kg

7) 45.258 Ms 4.5258 x 107 s

8) 0.000458 km 0.458 m

9) 0.025478 cs 2.5478 x 10-4 s

Graphing ExpectationsUse 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 ExpectationsA 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 relationshipsOnce 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 motiond t

time (s)

distance (m)

1 282 563 844 1125 140

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)

5 0.2

10 0.1

20 0.05

50 0.02

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 v2. If the speed triples, how many times greater is the force?

F = kv2 means thatF1 = k v1

2 and a new force F2 = k v22.

Taking a ratio of these two expressions eliminates the “k” constant.F

F

v

v1

2

12

22

Example 1 – algebraic solution

F

F1

2

1

9

F

F

v

v1

2

12

22

F

F

v

v

1

2

12

1

23

F2 = 9F1

Ratios and Proportions ws

Answers are on the bottom of the page

Log-Log Data AnalysisOne 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 Analysisy = 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 kThis 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 AnalysisStep 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 due Friday.