Graphing data

Sometimes students think it is a straightforward matter of graphing one line of data against the other…….

s

t

30 40 50 60 70 80 90

420

409

371

342

325

291

244

In fact there are several major errors in this graph

How many can you spot?

s

t

30 40 50 60 70 80 90

420

409

371

342

325

291

244

Here are some hints…..

s

t

30 40 50 60 70 80 90

420

409

371

342

325

291

244

So, it may not be as simple as graphing the exact data that is in the exam question

…………..but there are a number of guidelines to help you

Let’s have another quick look at the relevant wording of the question…

The word “suitable” is important. This is usually a strong hint that the data in the table needs to be manipulated a bit before you graph it

That means that you may have to square the values of one line of data…or maybe halve it or double it etc, before you try to graph it

To decide how to manipulate the data, you must refer back to the formula that is relevant to that experiment

In the example above, the relevant formula is:

212

s ut at

Graphing data

212

s ut at when a body falls freely under gravity u = 0 and a = g

=> s = ½ gt2

Here, we have the link between ‘s’ and ‘t’

Note: the ‘t’ is squared

This means we also need to square the ‘t’ values to ensure we get a straight line graph

Add a new line to the table and square the t values, using your calculator

t2/s2 0.0595 0.085 0.106 0.117 0.138 0.167 0.176

Note: the units for ‘t’ are also squared

Now you are ready to draw the graph.There are a few easy things you can do straight away:

(i) Ask for graph paper (no marks otherwise!)

(ii) Title the graph

(iii) Decide what data will go on each axis

(iv) Title the axes (include units)

Graphing data

(i) Ask for graph paper (deducted

most marks otherwise!)

…..seriously!

Graphing data

(ii) Title the graphYou can find a very suitable one in the question

Graphing data

(iii) Decide what data will go on each axisAs a rule, “their top line is your bottom line” – so ‘s’ will go on the x axis

Don’t forget to convert to SI units!

t2/s2 0.0595 0.085 0.106 0.117 0.138 0.167 0.176

To measure g, the acceleration due to gravity, by freefall

s / cm

So far the graph looks like this….….on graph paper…..

naturally!

Graphing data From the formula, we know we need

‘s’ and ‘t2’, so the middle line of data is not used in the graph.

The y axis will hold the ‘t2’ values. Also quote the correct units (s2)

t2/s2 0.0595 0.085 0.106 0.117 0.138 0.167 0.176

To measure g, the acceleration due to gravity, by freefall

….….on graph paper…..

naturally!

s /cm

t2 / s2

Graphing data

The next stage is VERY IMPORTANTLet’s have another look at the data we now want to plot….start with ‘s’

t2/s2 0.0595 0.085 0.106 0.117 0.138 0.167 0.176

Graphing data

The values go from 30 up to 90….…..but you MUST start at zero

Use as much of your graph sheet as possible……but make sure you go at least as far as 90…ideally up to 100

You must make equal sized intervals along your x axis

To measure g, the acceleration due to gravity, by freefall

s /cm0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

t2 / s2

Now decide on how your y axis will be divided….

The values go from approx 0.06 to 0.17…..but you MUST start at zero

You must go AT LEAST AS FAR as 0.176 Try to use as much of the page as possible, using

EQUAL sized divisions

Do NOT write the above readings on your graph!!!!!!!!!

t2/s2 0.0595 0.085 0.106 0.117 0.138 0.167 0.176

To measure g, the acceleration due to gravity, by freefall

s /m

180000

160000

140000

120000

100000

80000

60000

40000

20000

0

Now, start plotting your points

Identify a point by placing a dot exactly at the point, and draw a small circle around it to highlight it

t2 / s2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

You must NEVER…..join-the-dots!

Always pick a “best-fit” line. If the dots don’t form an EXACT straight line, make sure there is the same number of dots on each side of the line.

To measure g, the acceleration due to gravity, by freefall

s /m

180000

160000

140000

120000

100000

80000

60000

40000

20000

t2 / ms2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

s /cm

0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

To measure g, the acceleration due to gravity, by freefall

t2 / s2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

To read a slope from the graph, take two points ON THE LINE, (not from the table) that are far apart

Usually we can use the origin as one of these points

Then use the formula: slope = 2 1

2 1

y yx x

s /m

(90, 0.176

(0,0)

To measure g, the acceleration due to gravity, by freefall

t2 / s2 0.18

0.16

0.14

0.12

0.10

0.08

0.06

0.04

0.02

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

2 1

2 1

0.176 00.9 00.196

y ySlope

x x

2y sx m

2

2

2

121

02

12

s ut at

s gt

s gt

From the graph the units are:

From the formula

2

2

2slope

2 210.2 m/s

slope 0.196

y tx s g

g

Other “suitable” graphs

A number of other data experiments will require you to plot a “suitable” graph

Make sure to consider carefully what you will plot

Study the next few examples and decide what should be plotted on y and x axes. Also determine how to get slope

Study how many graphs are straight lines

“Suitable graphs 1” Boyles Law:

You will be supplied with P and V measurements.

What will you plot against what?

ANS: P against 1/V

P

1/V

“Suitable graphs 2” Measure g, using pendulum

You will be supplied with l and T measurements.

What will you plot against what? What is slope?

ANS: l against T2 Slope = y/x = l /T2 = g/4∏

l

T2

“Suitable graphs 3” Measure fundamental freq. against length

You will be supplied with f and l measurements

What will you plot against what? What is slope?

ANS: f against 1/ l Slope = y/x = f/(1/l) = ½√(T/µ)

f

1/ l

“Suitable graphs 4” Measure fundamental freq. against Tension

You will be supplied with f and T measurements.

What will you plot against what? What is slope?

ANS: f against √T Slope = y/x = f/√T = 1/(2L√µ)

f

√T

“Suitable graphs 5” Measure f of concave mirror or converging lens

You will be supplied with u and v measurements.

What will you plot against what?

ANS: 1/v against 1/u To get f: Rather than use slope, take any point on line (1/u,1/v) = (x,y), then 1/f = 1/u + 1/v = x+y

1/v

1/u

(1/u, 1/v)

“Suitable graphs 6” Verify Snell’s Law

You will be supplied with <i and <r measurements.(or possibly Real depth and Apparent depth)

What will you plot against what? What is slope?

ANS: Sin i against Sin r (or real against apparent) Slope = y/x = Sin i/Sin r = n = refractive index

Sin i

Sin r

“Suitable graphs 7” Joules Law:

You will be supplied with ∆θ and I measurements

What will you plot against what? What is slope?

ANS: ∆θ against I2 Slope: y/x = ∆θ/I2 = Rt/mc

∆θ

I2

In summary…. Do it on graph paper

Title the graph and the axes Include units on the axes

Divide your axes correctly

Use slope formula to get required information

And finally………..

Plot your points

Use a formula to help you decide what goes where (Their top line is your bottom line)

If you make a mistake on your division of axes etc, it is often quicker and neater to start again

….Ask for more graph paper