Mass & Volume: Penny Lab
 The purpose of this lab is to graphically
determine the mathematical relationship
between mass and volume for the pre-1982
pennies that we used.
 Your graph needs to have the mass (in grams)
of your groups of pennies on the y-axis.
 It should have the volume (in milliliters) of
your groups of pennies on the x-axis.
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Mass & Volume: Penny Lab
 The volume of water used does not
appear on the graph.
 The number of pennies does not
appear on the graph.
 The graph can be computer-generated
or hand-written on graph paper.
 Graphs written on notebook paper will
not receive credit.
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Mass & Volume: Penny Lab
 You were encouraged to use groups
of pennies (not one at a time) for two
reasons.
 It is incredibly difficult to measure the volume
of one penny accurately by water displacement
in the graduated cylinder.
 Using a larger number of pennies averages out
small imperfections and differences (dirt,
distortions, etc) and allows more precise data.
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Mass & Volume: Penny Lab
 There are several different completely
valid ways in which to gather data for
this lab.
For the sake of simplicity, I will just
focus on two. (You may have used
some other procedure, and that is
fine.)
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Mass & Volume: Penny Lab
The Start-Again-From-Scratch method goes like
this:
1) I weighed 5 pennies and recorded their mass.
 2) I filled a graduated cylinder partway with water,
then recorded the volume as Vi.
 3) I added the pennies I’d already weighed to the
cylinder, then recorded the new volume as Vf.
 4) I dumped everything out, dried the pennies, then
repeated these steps with 10, 15, 20, 25, and 30
pennies.
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Mass & Volume: Penny Lab
 My data table might look like this:
# pennies
mass
Vi
Vf
(g)
(mL)
(mL)
5
12.5
51.2
52.7
10
24.9
50.3
53.4
15
37.4
49.9
54.4
20
49.9
52.0
58.0
25
62.5
51.1
59.2
30
74.9
51.7
61.9
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Mass & Volume: Penny Lab
 In order to make a graph, I’ll need to find the
volume for each number of pennies.
 (I need to subtract out the initial volume of
water ; Vf-Vi = Vpennies)
 Since finding the volume of the pennies
requires math, my Vpennies values will go in the
Evaluation of Data section.
Vpennies
(Vf - Vi)
(mL)
1.5
3.1
4.5
6.0
8.1
10.2
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Mass & Volume: Penny Lab
The Addition method works this way:






1) I weighed 5 pennies and recorded their mass.
2) I filled a graduated cylinder partway with water, then recorded
the volume.
3) I added the pennies I’d already weighed to the cylinder, then
recorded the new volume as Vf.
4) I weighed 5 additional pennies (as in step one) and added
them to the graduated cylinder that already contained pennies
and water. I recorded the new volume as Vf for 10 pennies.
5) I repeated step 4 until I had six sets of data.
6) I dumped everything out and dried the pennies.
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Mass & Volume: Penny Lab
 My data table will need to label the data some
other way than in the first method. I can’t list
data by “number of pennies” anymore
because the masses are each for only five
pennies, but the volumes are for all the
pennies together.
 I decided to just label things as “trials” and
leave the number of pennies out. Trial 0 is for
zero pennies. (This is where I recorded the
volume of water in the graduated cylinder
before I started dropping pennies in.)
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Mass & Volume: Penny Lab
 My data table might look like this:
Trial #
massadded
Vf
(g)
(mL)
0
0
51.2
1
12.5
52.7
2
12.4
54.3
3
12.5
55.7
4
12.5
57.2
5
12.6
59.3
6
12.4
61.4
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Mass & Volume: Penny Lab
 Using this method is definitely more efficient
during the lab—I don’t need to stop and dry
pennies several times—but it requires more
Evaluation of Data.
 In order to make a graph, I still need Vpennies
and mpennies for six data points.
 To calculate the volumes, I’ll subtract out the
same initial volume every time (Vf from trial 0,
when only water was in the graduated
cylinder).
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Mass & Volume: Penny Lab
To calculate the volumes, I’ll subtract out the same
initial volume every time (Vf from trial 0, when
only water was in the graduated cylinder).
(Vf - 51.2)
Vpennies
(mL)
(mL)
52.7-51.2
1.5
54.3-51.2
3.1
55.7-51.2
4.5
57.2-51.2
6.0
59.3-51.2
8.1
61.4-51.2
10.2
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Mass & Volume: Penny Lab
 I also need to make a column in “Evaluation
of Data” where I calculate the total mass of
pennies involved in each trial.
massTOT
(g)
12.5
24.9
37.4
49.9
62.5
74.9
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Mass & Volume: Penny Lab
 Now I have the data I need to generate a
graph.
Note: I titled the graph, and put both a variable name
and a unit on each axis!
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Mass & Volume: Penny Lab
 Now I need to draw a “best-fit” line through
my data points so that I can find the equation
of the line.
 If I’m using graph paper, I can draw the line
with a ruler ,then calculate the slope of the line
myself.
 If I’m using a computer program, I can have
the computer do the best fit line and find the
equation for me.
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Mass & Volume: Penny Lab
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Mass & Volume: Penny Lab
 When I found an equation for my line, I used mass as
the “y variable” because I’m graphing mass on the yaxis. I used Volume as the “x variable” because I’m
graphing volume on the x-axis. I included units on
both of those variables.
 The slope of my graph had a value of 7.7 g/mL. I can
only report my slope with two significant figures
because most of my Vpennies measurements only had
two.
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Mass & Volume: Penny Lab
 The units of my slope are g/mL (grams per milliliter)
because of the units of the axes of my graph, and
because mass and volume are two different things
and cannot cancel out.
 My equation shows that for these pennies (pre-1982
pennies, which are made of copper), there is a direct
proportion relating mass and volume.
 A slope of 7.7 g/mL means that one milliliter of copper
has a mass of 7.7 grams.
 My intercept is zero because zero milliliters of copper
should have a mass of zero. (If I don’t have any
copper it doesn’t have any mass.)
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Mass & Volume: Penny Lab
 I can use this graph (a density graph) to answer
questions about other masses and volumes of
copper.
 Even though I didn’t have a data point at 4.0 mL, I can
determine the mass of 4.0 mL of copper from the
graph as follows:
Pennies: Mass vs.
1
Volume
0.9
0.8
0.7
m (g) = 7.7 (g/mL) * V (mL)
mass
0.6
0.5
0.4
0.3
0.2
0.1
0
0.0
20.0
40.0
Volume
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60.0
80.0
Mass & Volume: Penny Lab
 I drew a straight line from the 4.0 mL mark up to my
best-fit line, then found the mass that corresponded
to it. The mass is above 30 g and below 40 g. I’m
going to estimate 32 g.
 If you are asked to solve GRAPHICALLY, this is what
you do. Track the given data to the best-fit line, then
see what value it corresponds to on the other axis.
Pennies: Mass vs.
Volume
1
0.8
m (g) = 7.7 (g/mL) * V (mL)
mass
0.6
0.4
0.2
0
0.0
20.0
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40.0
60.0
Volum
80.0
Mass & Volume: Penny Lab
 What if I were asked for the mass of 20.0 mL of
copper?
 That value doesn’t show on my graph. I can
extrapolate (extend the graph along its current
trajectory) to find the answer.
Pennies: Mass vs. Volume
1
0.9
0.8
m (g) = 7.7 (g/mL) * V (mL)
0.7
mass
0.6
0.5
0.4
0.3
0.2
0.1
0
0.0
20.0
40.0
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Volume
60.0
80.0
Mass & Volume: Penny Lab
 OR I can solve using the equation of the line!
 mass(g) = 7.7 (g/mL) * Volume (mL)
 If the volume is 20.0 mL, the equation looks like this:
 mass (g) = 7.7 g/mL * 20.0 mL
 7.7*20.0 = 154
 (g/mL) * mL = g
 So 154 g. (150 g, since I only have 2 sig figs.)
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Mass & Volume: Penny Lab
 Since my y-intercept was zero, I can also use the
density from the equation (7.7g/mL) as a conversion
factor.
 A conversion factor is a fraction that has a value of
one, but different numbers in the numerator and
denominator (due to different units)
 12in/ft means 12inches per 1 foot
 It has a VALUE of 1 (which means multiplying by it
doesn’t change the value of a measurement)
 But 12 and 1 are different numbers… because
inches and feet are different units.
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Mass & Volume: Penny Lab
 Any number with divided units is actually a
conversion factor between the units.
 7.7 g/mL means
 7.7 g (of copper) = 1.0 mL (of copper)
• 7.7 g/1.0 mL has a value of 1 (for copper)
• 1.0 mL/7.7g also has a value of 1
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