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Graphing data

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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

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In fact there are several major errors in this graph In fact there are several major errors in this graph How many can you spot? How many can you spot? s t 30 40 50 60 70 80 90 420 409 371 342 325 291 244

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Here are some hints….. Here are some hints….. s t 30 40 50 60 70 80 90 420 409 371 342 325 291 244

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So, it may not be as simple as graphing the exact data that is in the exam question 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 …………..but there are a number of guidelines to help you

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Lets have another quick look at the relevant wording of the question… Lets have another quick look at the relevant wording of the question…

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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 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 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

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

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Graphing data when a body falls freely under gravity u = 0 and a = g => s = gt 2 Here, we have the link between s and t => s = ½ gt 2 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 Note: the t is squared This means we also need to square the t values to ensure we get a straight line graph

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Add a new line to the table and square the t values, using your calculator t 2 /s 2 0.0595 0.085 0.106 0.117 0.138 0.167 0.176 Note: the units for t are also squared

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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)

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Graphing data (i)Ask for graph paper (deducted most marks otherwise!) …..seriously!

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Graphing data (ii)Title the graph You can find a very suitable one in the question

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Graphing data (iii) Decide what data will go on each axis As a rule, their top line is your bottom line – so s will go on the x axis Dont forget to convert to SI units! Dont forget to convert to SI units! t 2 /s 2 0.0595 0.085 0.106 0.117 0.138 0.167 0.176

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To measure g, the acceleration due to gravity, by freefall s / cm So far the graph looks like this…. ….on graph paper….. naturally!

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Graphing data From the formula, we know we needs and t 2, so the middle line of data is not used in the graph. From the formula, we know we needs and t 2, so the middle line of data is not used in the graph. The y axis will hold the t 2 values. Also quote the correct units (s 2 ) t 2 /s 2 0.0595 0.085 0.106 0.117 0.138 0.167 0.176

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To measure g, the acceleration due to gravity, by freefall …. ….on graph paper….. naturally! s /cm t 2 / s 2

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Graphing data The next stage is VERY IMPORTANT Lets have another look at the data we now want to plot….start with s The next stage is VERY IMPORTANT Lets have another look at the data we now want to plot….start with s t 2 /s 2 0.0595 0.085 0.106 0.117 0.138 0.167 0.176

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Graphing data The values go from 30 up to 90…. …..but you MUST start at zero 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 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 You must make equal sized intervals along your x axis

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To measure g, the acceleration due to gravity, by freefall s /cm 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 t 2 / s 2

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Now decide on how your y axis will be divided…. 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!!!!!!!!! t 2 /s 2 0.0595 0.085 0.106 0.117 0.138 0.167 0.176

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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 t 2 / s 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

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You must NEVER…..join-the-dots! You must NEVER…..join-the-dots! Always pick a best-fit line. If the dots dont form an EXACT straight line, make sure there is the same number of dots on each side of the line. Always pick a best-fit line. If the dots dont form an EXACT straight line, make sure there is the same number of dots on each side of the line.

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To measure g, the acceleration due to gravity, by freefall s /m 180000 160000 140000 120000 100000 80000 60000 40000 20000 t 2 / ms 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

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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 t 2 / s 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

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To read a slope from the graph, take two points ON THE LINE, (not from the table) that are far apart 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 Usually we can use the origin as one of these points Then use the formula: slope = Then use the formula: slope =

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s /m (90, 0.176 (0,0) To measure g, the acceleration due to gravity, by freefall t 2 / s 2 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

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From the graph the units are: From the formula

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Other suitable graphs A number of other data experiments will require you to plot a suitable graph A number of other data experiments will require you to plot a suitable graph Make sure to consider carefully what you will plot 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

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Suitable graphs 1 Boyles Law: You will be supplied with P and V measurements. Boyles Law: You will be supplied with P and V measurements. What will you plot against what? What will you plot against what? ANS: P against 1/V P 1/V

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Suitable graphs 2 Measure g, using pendulum You will be supplied with l and T measurements. Measure g, using pendulum You will be supplied with l and T measurements. What will you plot against what?What is slope? What will you plot against what?What is slope? ll ANS: l against T 2 Slope = y/x = l /T 2 = g/4 l T2T2

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Suitable graphs 3 Measure fundamental freq. against length You will be supplied with f and l measurements Measure fundamental freq. against length You will be supplied with f and l measurements What will you plot against what?What is slope? What will you plot against what?What is slope? l l ANS: f against 1/ l Slope = y/x = f/(1/ l ) = ½(T/µ) f l 1/ l

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Suitable graphs 4 Measure fundamental freq. against Tension You will be supplied with f and T measurements. Measure fundamental freq. against Tension You will be supplied with f and T measurements. What will you plot against what?What is slope? What will you plot against what?What is slope? ANS: f against T Slope = y/x = f/T = 1/(2Lµ) f T

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Suitable graphs 5 Measure f of concave mirror or converging lens You will be supplied with u and v measurements. Measure f of concave mirror or converging lens You will be supplied with u and v measurements. What will you plot against what? 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)

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Suitable graphs 6 Verify Snells Law You will be supplied with *
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Suitable graphs 7 Joules Law: You will be supplied with θ and I measurements Joules Law: You will be supplied with θ and I measurements What will you plot against what?What is slope? What will you plot against what?What is slope? ANS: θ against I 2 Slope: y/x = θ/I 2 = Rt/mc θ I2I2

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In summary…. Do it on graph paper Do it on graph paper Title the graph and the axes Include units on the axes Title the graph and the axes Include units on the axes Divide your axes correctly Divide your axes correctly Use slope formula to get required information Use slope formula to get required information And finally……….. And finally……….. Plot your points Plot your points Use a formula to help you decide what goes where (Their top line is your bottom line) Use a formula to help you decide what goes where (Their top line is your bottom line)

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If you make a mistake on your division of axes etc, it is often quicker and neater to start again 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 ….Ask for more graph paper

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Section 5: Graphs in Science

Section 5: Graphs in Science

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