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1. Interpreting Information – Graphing

Equipment needed in Investigation 1:

Wooden Disks, Ruler, and Tape Measure

Nuts, Bolts and Balance

The purpose of this lab is to develop skill in interpreting data through an alternative means –

creating an appropriate graph or graphs and examining them to determine what the data can tell us.

Introduction

In laboratory investigations, you generally control one quantity (variable) and measure its

effect on a second quantity (variable) while holding all other factors constant. After the data is

collected, it is helpful to visualize the relationship between these variables by making a graph of

the second variable versus the first variable using proper graphing techniques. The resulting graph

will give you a better understanding of the relationship between the two variables and enable you

to interpret the physical system it represents.

When making a graph, use the following steps.

n Identify the independent variable (the quantity you control) and the dependent variable (the

quantity affected by the independent variable).

n Use the horizontal axis for the independent variable and the vertical axis for the dependent

variable. Label each axis with the name of the variable and the unit.

n Choose a scale. Make the graph as large as possible by spreading out the data on each of the

axes to cover more than half the available grid. Let each grid division stand for a convenient

amount. Choosing three grid divisions to equal 10 is not good because then each grid division is a

fraction. Choosing five grid divisions to equal 10 would be better. To avoid clutter, do not

number every grid division.

n Plot each data point as a dark dot with a small circle around it.

n If the data points appear to lie in a straight line, draw the best straight line through them with

a ruler and pencil. The line should go through as many points as possible with the same number of

points above or below the line. Do not just “connect the dots.” If the points do not form a straight

line, draw the best smooth curve possible keeping in mind the curve should go through as many

points as possible with the same number of points above or below the curve.

n Graphs do not necessarily go through the origin (0,0). Think about your experiment and

decide if the data would logically include a (0,0) point.

n Each graph should have a brief, yet meaningful, title. Examples are “Cart Moving with

Constant Velocity” or “Position versus Time” or “Variation of Pressure with Change in Volume.”

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The following is an example of a graph prepared using good graphing techniques.

There are several relationships that occur frequently in physical processes, as illustrated by the

four graphs below. If the dependent variable varies linearly with (and is directly proportional to

)

the independent variable, the graph will be a straight line passing through the origin. If the

dependent variable varies linearly with (but is not directly proportional to) the independent

variable, the graph will be a straight line that does not pass through the origin. If the dependent

variable varies with the square of the independent variable, the graph will be a parabola. If the

dependent variable varies inversely with the independent variable, the graph will be a hyperbola.

Linear Graphs: A linear graph that passes through the origin is mathematically represented by

the equation where the constant m is called the slope of the straight line. In this case, the

dependent variable y is directly proportional to the independent variable x. A linear graph that

does not pass through the origin is mathematically represented by the equation where

the constant b is the intercept along the y-axis and the constant m is the slope.

Parabolic Graphs: An example of an object moving along a parabolic path is projectile motion

under the influence of gravity. The dependent variable y does not vary linearly with the

independent variable x. Parabolic curves are mathematically represented by the equation

where k is a constant.

Reciprocal, Inverse, or Hyperbolic Graphs: These names collectively refer to graphs

represented mathematically by the equation where k is a constant. As an example,

y = mx y = mx + b y = kx2 y = k/x

y mx=

y mx b= +

y kx= 2

y k x=

Force

(dyne)

Elongation

(cm)

0.0

5

0.7

0

0.10 1.20

0.15 1.30

0.20 2.20

0.25 2.50

0.30 2.80

0.35 3.60

Force (dyne)

0

0.1 0.2 0.3 0.4 0.5

Elongation versus Force

0

1

2

2

3

4

5

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pressure is inversely proportional to volume for an ideal gas. The dependent variable y decreases

nonlinearly as the independent variable x increases. It is important to note that a straight-line

graph will result if the variable y is plotted versus the quantity (1/x).

Graphs are useful because they can give you information that you have not directly deter-

mined experimentally. This information can be obtained from the graph very easily. Reading the

pair of values for the dependent and independent variables at a point on the graph that lies between

two of the data points used to construct the graph is called interpolation. Reading a pair of values

from a point on the graph falling outside the range of data points used to construct the graph is

called extrapolation. In addition to the coordinates, the slope of a graph also contains information.

For many experiments the slope may be all that you really want to find from the graph. The slope

of a straight-line graph usually has an important interpretation in terms of the physical system

represented on the graph. As an example, the slope of a position-versus-time graph is Dx/Dt and is

the velocity, whereas the slope of a velocity-versus-time graph is Dv/Dt and is the acceleration.

Finding the slope of a graph.

A linear graph has a constant slope (or steepness) associated with it. In the general equation

for a straight line (y = mx + b), the slope is m and the y-intercept is b. The slope m can be either a

positive or negative number. If m is positive, the line slopes uphill because y increases as x

increases. If m is negative, the line slopes downhill because y decreases as x increases. The slope

can be calculated from the graph by choosing two points along the straight line (do not use actual

data points or (0,0)), as illustrated in the graph below. These points are separated on the x axis by

an amount Dx and on the y axis by an amount Dy. For good accuracy, always measure the slope

over as wide a range of Dx and

Dy

as possible. The slope is defined as the ratio of the change in the

vertical to the change in the horizontal, that is, slope = Dy/Dx = (y2 – y1)/(x2 – x1).

This is an example of finding the slope

of a graph. First, choose a point (mark-

ed x) near each end of the graph. Then

draw the legs of the triangle representing

Dx and Dy. In this case,

Dx = x2 – x1

= 0.45 dyne – 0.05 dyne

= 0.40 dyne

and

Dy = y2 – y1

= 4.5 cm – 0.5 cm

= 4.0 cm

Slope = Dy/Dx = 4.0 cm/0.40 dyne

Finally, the slope = 10 cm/dyne.

0 0.1 0.2 0.3 0.4

0.5

Elongation versus Force

Force (dyne)

0

1

2

2

3

4

5

x

x

Dx

Dy

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During the course of the semester, you will often conclude your laboratory investigations

by comparing your results in either of two ways. In some of the investigations, the true value of

the quantity being measured is well known. In these labs, the accuracy of your experiment will be

determined by comparing the experimental value you obtained with the known or actual value.

This is done by calculating the percentage error.

Percentage Error = -Actual – Experimental

Actual

– × 100% (1)

In other investigations, a given quantity will be measured or calculated by two different

methods. There will then be two different experimental values E1 and E2, but there may not be a

true value to use for comparison. In these labs, the percentage difference between the two

experimental values will be calculated. Note that this does not tell you anything about the

accuracy of the experiment, but it is a measure of the precision of your experiment.

Percentage Difference = 5 1!21″(1!41″)

67

5 × 100% (2)

The average is chosen as the basis for comparison when there is no reason to think one of the

values is more reliable than the other. Note that both of these quantities are often referred to as the

uncertainty.

Investigation 1-1. Circumference and Diameter

(1) You will determine the relationship between the diameter and circumference of circular

objects by graphing. There are four wooden disks at your lab station. Using the best available

method, measure (DO NOT CALCULATE) the diameter and circumference of each disk.

Record these values in Part B of your Lab Report.

(2) Make a graph of your data on the grid provided by plotting the diameter, D, on the horizontal

axis and the circumference, C, on the vertical axis. Choose the scales for the graph so that

your data cover a majority of the grid. Draw a best-fit straight line that passes through most

data points. If it is reasonable to extend the line through the origin, do so. Your graph

represents the relationship between the diameter and the circumference.

(3) In Part C of your Lab Report, determine the slope of the graph.

Investigation 1-2. Determining Relationships from Graphs

In Part D of your Lab Report, you will use graphing techniques to determine a physical property of

an object. The goal is to measure the mass of a bolt indirectly by making some measurements and

interpreting them using a graph.

(1) Place one nut on the bolt.

(2) Measure and record the mass of the bolt + nut combination.

(3) Add another nut on the bolt/nut combination.

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(4) Measure and record the mass of the bolt + 2 nuts.

(5) Repeat with 3 and 4 nuts threaded on the bolt.

(6) Graph your data with the y-axis as the mass values and the x-axis as the number of nuts

threaded on the bolt.

(7) Calculate the mass of the bolt as the y-intercept.

(8) Calculate the mass of the nut as the slope.

(9) Check your calculation by measuring the mass of a nut by itself and the mass of the bolt by

itself. Calculate the uncertainty in your value.

Investigation 1-3. Area of a

Circle

Area and volume of regular-shaped objects can be calculated by using the appropriate

formula. But this is not possible for irregularly shaped objects and we must resort to an

operational definition. An operational definition of the area of any shaped figure, no matter the

shape, is to find the number of standard squares that fit inside the figure. You will use the

operational definition of area to find the areas of the four circles below. A grid of standard

squares, with each square 0.5 cm x 0.5 cm = 0.25 cm2, has been imposed over the four circles. By

counting the number of standard squares and estimating the partial squares (those squares which

are more than half inside the circle count as one, while those which are less than half inside the

circle are not counted), you can determine approximately the unknown area.

0 1 2 3 4 5 6 7 8 9 10

Length (cm)

L

en

gt

h

(c

m

)

10

9

8

7

6

5

4

3

2

1

0

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(1) Find the area of each circle below by counting the number of squares (according to the

operational definition above) and multiplying by the area of one square. Record the area in the

second column of the table provided in Part E of your Lab Report.

(2) Determine, to the nearest 0.1 cm, the radius of each of the four circles and record the values in

Part E of your Lab Report.

(3) Make a graph of your data on the grid provided in Part F of your Lab Report. Plot the radius

on the horizontal axis and the area on the vertical axis. If the points do not appear to lie on a

straight line, draw a smooth curve that passes as close to as many data points as possible.

(4) When a graph is not a straight line you need to identify the type of curve. Often it will be one

of the graphs discussed earlier. To verify which of these curves you have, the data are

systematically manipulated until a graph is found that yields a straight line. For example, you

can square one of the variables, or take the square root of a variable, or find the reciprocal of a

variable.

To help determine the type of curve, square the radius and record the values in the fourth column

of the table in Part E of your Lab Report.

(5) Make a second graph of your data on the grid provided. Plot the square of the radius on the

horizontal axis and the area on the vertical axis. Draw a line that best fits the data.

(6) In Part H of your Lab Report, determine the slope of this last graph.

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LAB REPORT FOR INVESTIGATION 1

Name ___________________________________ Date ___________________________

Lab Period (Day & Time) ___________________ Station_________________________

All measurements must have units. Calculations not involving ratios of like quantities must have

units. Either put the unit with each number or, if the numbers are contained in columns,

label the column with the unit. Include all steps in your calculations. Follow the format and

general procedures described in the Grading Policy.

Investigation 1-1

Part A: Measurements of Diameter and Circumference

Disk Diameter Cicumference

(cm) (cm)

1 4.2 14.5

2 6.8 20.7

3 7.5 24.7

4 11.2 35.5

5 14.8 43.7

Part B: Using the graph paper provided, plot this information.

Part C: Determination of the slope of the circumference-versus-diameter graph (show your

work).

C1 = ________ C2 = ________ DC = ________

D1 = ________ D2 = ________ DD = ________

slope (m) = _________

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Q1: What is the equation you generally use to find the circumference of a circle from the

diameter?

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Q2: Find the percentage error between the slope you calculated for the circumference-versus-

diameter graph and the value of the constant in the above equation. Ask your instructor what

percentage error is within experimental error to decide if the values of the slope and the constant

you determined are the same. If you measured carefully, this value of the slope should be very

familiar to you. What is this constant called?

Q3. Considering the things you did to measure circumference and diameter, what might cause

uncertainty in your result? Do not forget to consider the objects themselves.

Investigation 1-2

Part D:

# of nuts Mass of bolt with nut (g)

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Q4: Briefly describe the relationship between mass and number of nuts shown on your graph.

Q5: Determine the slope of the graph. Include units

m1 = ________ m2 = ________ Dm = ________

# of nuts 1 = ________ # of nuts 2 = ________ D# of nuts = ________

slope = _________

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Q6. Write an equation (of the form of the straight line equation, y=mx+b), that describes the above

graph.

Q7. Considering the things you did to obtain your measurements, what might cause uncertainty in

your result? Do not forget to consider the objects themselves.

Investigation 1-3

Part E: Measurements of Radius and Area.

Circle

Area

( )

Radius

( )

Radius Squared

( )

1 (smallest)

2

3

4 (largest)

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Part F: Graph radius vs area

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Part G: Graph radius2 vs. area

Part H: Determination of the slope of the area-versus-radius squared graph.

A1 = A2 = DA =

= = D(r2) =

slope =

r1

2 r2

2

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Q9. What is the equation you generally use to find the area of a circle?

Q10. Find the percentage error between the slope you calculated for the area-versus-radius

squared graph and the value of the constant in the above equation. Ask your instructor what

percentage error is within experimental error to decide if the values of the slope and the constant

you determined are the same.