Lesson 7: Graphing Graphing is an essential skill for both Physics 20 and 30. You You MUST be able to follo all of the rules of properly draing a graph! and also be able to do basic interpretation of graphs. "hen you are presented ith a chart of nu#bers that you are going to graph! you should start by identifying hich $ariable is the #anipulated $ariable and hich is the dependent $ariable. %n a lab! you are usually atching to see one thing change on its on! hile letting so#ething ● &ust plod along in an e'pected ay. The one that you(re atching for changes in is called the or $ariable! ● dran on the . The one that is changing in a regular e'pected pattern is the or ● $ariable! dran on the . *s a rule of thu#b! ti#e al#ost alays goes on the ')a'is. ●
Graphing Rules Graphs you dra #ust ha$e the folloing #arks.
basic characteristics. %f you #iss any part you ill lose
Title ● ●
Your Your title should be short! but still clearly tell hat you ha$e graphed. The #ost co##on and reco##ended ay to na#e your graph is to say hat your y)a'is and ')a'is are. +ne ay to say it is , ● The other co##on ay is to say , ●
Labeled Axis ●
●
Make sure to rite out the of hat you ha$e graphed on each a'is! along ith the you used. %f you are using any sort of scientific notation for the nu#bers! #ake sure you sho it here also.
A Well Well Chosen Scale ●
The infor#ation you plot should alays co$er at least /1 of the area on your graph. Start by looking at the #a'i#u# $alues you ha$e for both the ' and y a'is. ● Then check out ho #any #a&or ,ticks you ha$e on each a'is of the actual graph. ● i$ide your #a'i#u#s by the ticks to find out roughly hat to label each tick as. ●
Data Plotted Correctly ●
●
●
%ts too bad hen a person does all this ork! and then does a sloppy &ob of plotting their infor#ation. Make sure you are as careful ca reful as possible hen #arking your points on your graph! otherise e$erything else is a aste of ti#e. You You should alays put around each dot! since they #ight be hard to see on the graph paper. %t also shos that each e ach data point po int is a bit ,iffy. ,iffy.
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Best Fit Line This step is so#eti#es optional! since your data #ight not gi$e you a graph that has a straight line linear relationship. %f your graphed data looks like a cur$ed e'ponential relationship! dra a s#ooth cur$ed line ● through your data points instead. "hen draing a best fit line do ,connect the dots. %nstead! you should try to dra a co#pletely straight line that pases through as #any of your ● data points as possible. Try to get as #any points abo$e the line as belo. ● This is the line that you Use the for#ula... ● slope =
● ●
calculate your slope fro#.
rise Δ y ( y 2 − y 6 ) = = run Δ ' ( ' 2− ' 6)
You #ust kno this for#ula and kno ho to use it9 You are alloed to use the data points you plotted9 You fit line. ;
the slope of the folloing best fit line. d as a function of t
>0
?otice that on the graph % ha$e #arked off to points on the best fit line... <'6 ! y6= @ <2! 65= <'2 ! y2= @ <8.2! 80=
0
% use these to calculate the slope.
80
d <#=
read to points fro# the best
y 2− y 6 slope = ' 2−' 6
30
slope =
80 −65 8.2 −2
slope =60 # / s
20
% a# careful to gi$e #y final anser as a single $alue < = ● shoing units based on the a'is
60
0 0
6
2
3
8
t
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Line Straightening * special skill in$ol$ing graphs is a techniAue called ! so#eti#es called an . Do not make the mistake of :ssentially! you ill so#eti#es ha$e data to graph that ould gi$e calculating some sort of ● average based on the you a cur$ed line. numbers given if you are *nalyBing this line is $ery difficult... calculating a single slope ○ asked to use an “averaging or area under the line is i#possible. technique” for a graphing "hat e need to do is co#e up ith a ay to #anipulate the info problem. ● you ha$e so that! hen plotted! it ill gi$e you a of best fit
20.0
6.0
50.0
2.0
650
3.0
320
8.0
00
.0
irst e need to identify the #ain for#ula <hich e ha$en(t studied yet9=. on(t orry! Auestions you(ll do ill be based on for#ulas you($e seen before. ● 2 # $ G c = r ?e't get rid of the unneeded parts of the for#ula. The only $ariables e are concerned about according to the chart of $alues gi$en are c and $. ● 2
G c α $
?o e$aluate hat this relationship is telling us. ● ?othing has been done to c ! but $ is being sAuared. 2454206
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●
●
●
Do#e up ith a ne colu#n of $alues ith $ sAuared. ?o! really! &ust sAuare $. He sure to #ake the units sAuared as ell.
20.0
6.0
6.0
50.0
2.0
8.0
650
3.0
I.0
320
8.0
6>
00
.0
2
The graph ould be c $s $2. Try graphing it yourself and then calculate the slope... you should get about 20 ?4#24s2 ○ To figure out hat the slope actually #eans in this case! you ould look at your graph this ay... 2 G c # #$ y G c Δ slope = = 2 and fro# the original for#ula G c = 2 = r r Δ ' $ $ slope =
G c 2
$
slope = ●
=
# r
# r
So if e ant to deter#ine the #ass of the ob&ect... # slope = r # =slope ( r ) # =20 ( 6.6) # = 23 kg
There are se$eral ays help yourself do these sorts of Auestions. Dalculators and spreadsheets can gi$e you #athe#atically perfect $alues for slope and y)intercept. Jead Cinear Jegression on a T% 53 K T% 58 or Cinear Jegression on a T% ?spire to learn ho to • do it on a calculator. You can atch a $ideo for T% 53 K 58 or T% ?spire also. • There are also $ideos for doing it in a regular spreadsheet
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