Foundation Lesson II
Numbers in Science
Exploring Measurements, Significant Digits, and
Dimensional Analysis
OBJECTIVE
Students will be introduced to correct measurement techniques, correct use of significant digits, and
dimensional analysis.
T E A C H E R
P A G E S
LEVEL
All
NATIONAL STANDARDS
UCP.1, UCP.3, A.1, G.2
TEKS
6.1(A), 6.2(A), 6.2(B), 6.2(C), 6.2(D), 6.4(A)
7.1(A), 7.2(A), 7.2(B), 7.2(C), 7.2(D), 7.4(A)
8.1(A), 8.2 (A), 8.2(B), 8.2(C), 8.2(D), 8.4(A)
IPC: 1(A), 2(A), 2(B), 2(C), 2(D)
Biology: 1(A), 2(A), 2(B), 2(C), 2(D)
Chemistry: 1(A), 2(A), 2(B), 2(C), 2(E)
Physics: 1(A), 2(A), 2(B), 2(C), 2(D), 2(F)
CONNECTIONS TO AP
Students are expected to report measurements and perform calculations with the correct number of
significant digits.
TIME FRAME
180 minutes, depending on level
MATERIALS
(For a class of 28 working in pairs)
14 small cubes
14 metric rulers
14 200 mL beakers
14 large graduated cylinders
14 spherical objects
14 tweezers
14 flexible tape measures
balances
TEACHER NOTES
Small wooden alphabet blocks or dice should be inexpensive and easy to obtain. Cube shaped ice could
also be used. Be sure to find a cube/graduated cylinder combination that ensures total submersion of the
cube since its volume will be determined by water displacement.
44
Laying the Foundation in Physics
Foundation Lesson II
Spherical objects could be a large marble or small rubber ball. Again, be sure to check the
sphere/cylinder size to ensure that total submersion of the sphere is possible.
For the flexible tape measure, photocopy the metric side of a tape measure and have students cut it out
on paper. Or, provide students with a length of string and metric ruler. The string can be wrapped around
the sphere, marked, and then removed and measured.
This lesson is designed to introduce or reinforce accurate measurement techniques, correct usage of
significant digits, and dimensional analysis. Dimensional analysis is also called the Factor-Label method
or Unit-Label method and is a technique for setting up problems based on unit cancellations. Lecture as
well as guided and independent practice of these topics should precede this activity. Students should be
provided with reference tables containing metric and English conversion factors.
POSSIBLE ANSWERS TO CONCLUSION QUESTIONS AND SAMPLE DATA
5.75 mL
Middle:
3.0 mL
Right:
0.33 mL
Laying the Foundation in Physics
P A G E S
Introduction
Left:
T E A C H E R
The purpose of significant digits is to communicate the accuracy of a measurement as well as the
measuring capacity of the instrument used. Remind students repeatedly to take measurements including
an estimated digit and to perform their calculations with the correct number of significant digits.
Emphasize that points will be deducted for answers containing too many or too few significant digits.
The correct number of significant digits to be reported by your students will depend entirely upon your
equipment.
45
Foundation Lesson II
DATA AND OBSERVATIONS
Data Table
Cube Data
Mass: 15.05 g (4 sd)
Dimensions
length: 3.68 cm (3 sd) width: 3.65 cm (3 sd) height: 3.67 cm (3 sd)
Volume
Beaker
Beaker
initial volume:
100 mL (1 sd) Final volume: 150 mL (2 sd)
Graduated cylinder
Graduated cylinder
initial volume:
175.0 mL (4 sd) final volume:
225.1 mL (4 sd)
Sphere Data
T E A C H E R
P A G E S
Mass:
19.38 g (4 sd)
Dimensions Circumference: 7.62 cm (3sd)
Volume
Beaker initial volume: 100 mL (1 sd) Beaker final volume: 110 mL (2 sd)
Graduated cylinder
Graduated cylinder
initial volume:
175.0 mL (4 sd) final volume:
182.3 mL (4 sd)
Formula for calculating the volume of a cube:
Formula for calculating the circumference of a circle:
Formula for calculating the diameter of a circle:
Formula for calculating the volume of a sphere:
V = length × width × height
C = πd
d = 2r
4
V = π r3
3
ANALYSIS
•
Show your organized work on a piece of notebook paper. Transfer your final answers to the
blanks beside each question. Staple your work to your answer sheet before turning it in.
• Remember to follow the rules for reporting all data and calculated answers with the correct
number of significant digits.
• You may need tables of metric and English conversion factors to work some of these problems.
1. For each of the measurements you recorded above, go back and indicate the number of significant
digits in parentheses after the measurement. Ex: 15.7 cm (3 sd)
• The number of significant digits will be determined by the equipment you are using.
2. Use dimensional analysis to convert the mass of the cube to (a) mg and (b) ounces.
(a)
15.05 g ×
1000 mg
= 15, 050 mg
1g
(b) 15.05 g × 1 lb × 16 oz = 0.5304 oz
454 g 1 lb
3. Calculate the volume of the cube in cm3.
• V = l ×w ×h
V = 3.68 cm × 3.65 cm × 3.67 cm = 49.3 cm3
46
Laying the Foundation in Physics
Foundation Lesson II
4. Use dimensional analysis to convert the volume of the cube from cm3 to m3.
1m
1m
1m
• 49.3 cm 3 ×
×
×
= 4.93 ×10 –5 m3
100 cm 100 cm 100 cm
5. Calculate the volume of the cube in mL as measured in the beaker. Convert to cm3 knowing that
1 cm3 = 1 mL.
• V = Vfinal – Vinitial
V = 150 mL – 100 mL
V = 50 mL = 50 cm 3
6. Calculate the volume of the cube in mL as measured in the graduated cylinder. Convert to cm3
knowing that 1 cm3 = 1 mL.
T E A C H E R
• V = Vfinal – Vinitial
V = 225.1 mL – 175.0 mL
V = 50.1 mL = 50.1 cm3
mass
, calculate the density of the cube as determined by the
volume
(a) ruler (b) beaker (c) graduated cylinder.
7. Using the density formula D =
D=
15.05 g
49.3 cm 3
D = 0.305
(b)
D=
g
cm 3
(c)
D=
15.05 g
50.1 cm 3
D = 0.300
P A G E S
(a)
g
cm 3
15.05 g
50 cm 3
D = 0.3
g
cm 3
8. Use dimensional analysis to convert these three densities into kg/m3.
(a)
0.305
(b)
0.3
g
1 kg
100 cm 100 cm 100 cm
kg
×
×
×
×
= 305 3
3
cm
1000 g
1m
1m
1m
m
g
cm
(c) 0.300
3
×
1 kg
100 cm 100 cm 100 cm
kg
×
×
×
= 300 3
1000 g
1m
1m
1m
m
g
1 kg
100 cm 100 cm 100 cm
kg
×
×
×
×
= 300 3
3
cm
1000 g
1m
1m
1m
m
Laying the Foundation in Physics
47
Foundation Lesson II
•
The bar above the last zero of the number 300 communicates it is a significant zero transforming
the recorded answer from one significant digit to three. It is equally appropriate to teach your
students to use scientific notation to effectively communicate three significant digits. The
number would be correctly written as 3.00 × 102. Another way to communicate a number
accurate to the ones position is to use a decimal at the end of the number. The number could be
written as 300. representing that this measurement is accurate to the last digit.
T E A C H E R
P A G E S
9. Convert the mass of the sphere to (a) kg and (b) lbs.
(a)
19.38 g ×
1 kg
= 0.01938 kg
1000 g
(b)
19.38 g ×
1 lb
= 0.04269 lbs
454 g
10. Using the measured circumference, calculate the diameter of the sphere.
• C = πd
C
π
7.62 cm
= 2.43 cm
d=
3.14
d=
11. Calculate the radius of the sphere.
• d = 2r
d
2
2.43 cm
= 1.22 cm
r=
2
r=
12. Calculate the volume of the sphere from its radius.
• V = 4 πr 3
3
4
V = (π )(1.22)3
3
V = 7.61 cm3
13. Calculate the volume of the sphere in mL as measured in the beaker. Convert to cm3 knowing that
1 cm3 = 1 mL.
• V = Vfinal − Vinitial
V = 110 mL − 100 mL
V = 10 mL = 10 cm3
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Laying the Foundation in Physics
Foundation Lesson II
14. Calculate the volume of the sphere in mL as measured in the graduated cylinder. Convert to cm3
knowing that 1 cm3 = 1 mL.
• V = Vfinal − Vinitial
V = 182.3 mL − 175.0 mL
V = 7.3 mL = 7.3 cm3
mass
, calculate the density of the sphere as determined by the
volume
(a) tape measure (b) beaker (c) graduated cylinder.
15. Using the density formula D =
D=
(b)
D=
19.38 g
7.3 cm3
g
D = 2.7
cm3
(c)
D=
T E A C H E R
19.38 g
7.60 cm3
g
D = 2.55
cm3
(a)
19.38 g
10 cm3
g
D=2
cm3
P A G E S
16. Use dimensional analysis to convert the densities into lbs/ft3.
3
(a)
(b)
3
3
⎛ 2.54 cm ⎞ ⎛ 12 in ⎞
1 lb
lbs
×
×⎜
⎟ ×⎜
⎟ = 100 3
3
cm
454 g ⎝ 1 in ⎠ ⎝ 1 ft ⎠
ft
2g
3
(c)
3
2.55 g
⎛ 2.54 cm ⎞ ⎛ 12 in ⎞
1 lb
lbs
×
×⎜
⎟ ×⎜
⎟ = 159 3
3
cm
454 g ⎝ 1 in ⎠ ⎝ 1 ft ⎠
ft
3
2.7 g
⎛ 2.54 cm ⎞ ⎛ 12 in ⎞
1 lb
lbs
×
×⎜
⎟ ×⎜
⎟ = 170 3
3
cm
454 g ⎝ 1 in ⎠ ⎝ 1 ft ⎠
ft
Laying the Foundation in Physics
49
Foundation Lesson II
CONCLUSION QUESTIONS
T E A C H E R
P A G E S
1. Compare the densities of the cube when the volume is measured by a ruler, beaker and graduated
cylinder. Which of the instruments gave the most accurate density value? Use the concept of
significant digits to explain your answer.
• The density of the cube had 3 significant digits when measured with the ruler. After subtracting
to find the difference between the initial and final water levels in the graduated cylinder and
beaker there are 2 significant digits when measured with the graduated cylinder but only 1
significant digit when measured with the beaker.
• The ruler is the more accurate measure of the volume when compared to the volume obtained by
water displacement using the graduated cylinder. The tweezers used to submerge the cube will
contribute a small amount to the volume recorded since they contribute to the TOTAL amount of
water displaced. See if you can get your students to come up with this concept!
• Student answers may vary in significant digits depending on equipment used.
2. A student first measures the volume of the cube by water displacement using the graduated cylinder.
Next, the student measures the mass of the cube before drying it. How will this error affect the
calculated density of the cube? Your answer should state clearly whether the calculated density will
increase, decrease or remain the same and must be justified.
• The calculated density of the cube would increase.
• Measuring a wet block will make the mass appear greater. Since mass is in the numerator of the
mass
equation D =
, the density value reported will be too large.
volume
3. A student measures the circumference of a sphere at a point slightly higher than the middle of the
sphere. How will this error affect the calculated density of the sphere? Your answer should state
clearly whether the calculated density will increase, decrease, or remain the same and must be
justified.
• The density of the sphere would increase.
• If the student measured the circumference at any point other than the center, the circumference
would be reported as too low.
• If the circumference is too small then the diameter will be too small.
C
• If ↓ = d ∴ d↓ the diameter is reported as too small, the radius will also be reported as too
π
small.
d
• If ↓ = r ∴ r↓ the radius is reported as too small then the volume will be reported as too small.
2
4
• If V = πr↓3 ∴ V↓ the volume is reported as too small the density will be reported as too large.
3
Density =
50
m
∴ Density↑
V↓
Laying the Foundation in Physics
Foundation Lesson II
Numbers in Science
Exploring Measurements, Significant Digits, and
Dimensional Analysis
TAKING MEASUREMENTS
The accuracy of a measurement depends on two factors: the skill of the individual taking the
measurement and the capacity of the measuring instrument. When making measurements, you should
always read to the smallest mark on the instrument and then estimate another digit beyond that.
For example, if you are reading the length of the steel pellet pictured above using only the ruler shown
to the left of the pellet, you can confidently say that the measurement is between 1 and 2 centimeters.
However, you MUST also include one additional digit estimating the distance between the 1 and 2
centimeter marks. The correct measurement for this ruler should be reported as 1.5 centimeters. It would
be incorrect to report this measurement as 1 centimeter or even 1.50 centimeters given the scale of this
ruler.
What if you are using the ruler shown on the right of the pellet? What is the correct measurement of the
steel pellet using this ruler? 1.4 centimeters? 1.5 centimeters? 1.40 centimeters? 1.45 centimeters? The
correct answer would be 1.45 centimeters. Since the smallest markings on this ruler are in the tenths
place we must carry our measurement out to the hundredths place.
If the measured value falls exactly on a scale marking, the estimated digit should be zero.
The temperature on this thermometer should read 30.0°C. A value of 30°C would imply this
measurement had been taken on a thermometer with markings that were 10° apart, not 1°
apart.
Laying the Foundation in Physics
51
Foundation Lesson II
When using instruments with digital readouts you should record all the digits shown. The instrument has
done the estimating for you.
When measuring liquids in narrow glass graduated cylinders, most liquids form a slight dip in the
middle. This dip is called a meniscus. Your measurement should be read from the bottom of the
meniscus. Plastic graduated cylinders do not usually have a meniscus. In this case you should read the
cylinder from the top of the liquid surface. Practice reading the volume contained in the 3 cylinders
below. Record your values in the space provided.
Left:________________
Middle: _____________
Right:_______________
SIGNIFICANT DIGITS
There are two kinds of numbers you will encounter in science, exact numbers and measured numbers.
Exact numbers are known to be absolutely correct and are obtained by counting or by definition.
Counting a stack of 12 pennies is an exact number. Defining 1 day as 24 hours are exact numbers. Exact
numbers have an infinite number of significant digits.
Measured numbers, as we’ve seen above, involve some estimation. Significant digits are digits believed
to be correct by the person making and recording a measurement. We assume that the person is
competent in his or her use of the measuring device. To count the number of significant digits
represented in a measurement we follow 2 basic rules:
1. If the digit is NOT a zero, it is significant.
2. If the digit IS a zero, it is significant if
a. It is a sandwiched zero.
OR
b. It terminates a number containing a decimal place.
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Laying the Foundation in Physics
Foundation Lesson II
Examples:
3.57 mL has 3 significant digits (Rule 1)
288 mL has 3 significant digits (Rule 1)
20.8 mL has 3 significant digits (Rule 1 and 2a)
20.80 mL has 4 significant digits (Rules 1, 2a and 2b)
0.01 mL has only 1 significant digit (Rule 1)
0.010 mL has 2 significant digits (Rule 1 and 2b)
0.0100 mL has 3 significant digits (Rule 1 and 2b)
3.20 × 104 kg has 3 significant digits (Rule 1 and 2b)
SIGNIFICANT DIGITS IN CALCULATIONS
A calculated number can never contain more significant digits than the measurements used to
calculate it.
Calculation rules fall into two categories:
1. Addition and Subtraction: answers must be rounded to match the measurement with the least number
of decimal places.
37.24 mL + 10.3 mL = 47.54 (calculator value), report as 47.5 mL
2. Multiplication and Division: answers must be rounded to match the measurement with the least
number of significant digits.
1.23 cm × 12.34 cm = 15.1782 (calculator value), report as 15.2 cm2
DIMENSIONAL ANALYSIS
Throughout your study of science it is important that a unit accompanies all measurements. Keeping
track of the units in a problem can help you convert one measured quantity into its equivalent quantity of
a different unit or set up a calculation without the need for a formula.
In conversion problems, equality statements such as 1 ft = 12 inches, are made into fractions and then
strung together in such a way that all units except the desired one are canceled out of the problem.
Remember that defined numbers, such as the 1 and 12 above, are exact numbers and thus will not affect
the number of significant digits in your answer. This method is also known as the Factor-Label method
or the Unit-Label method.
To set up a conversion problem follow these steps.
1. Think about and write down all the “=” statements you know that will help you get from your
current unit to the new unit.
2. Make fractions out of your “=” statements (there could be 2 fractions for each “=”). They will be
reciprocals of each other.
3. Begin solving the problem by writing the given amount with units on the left side of your paper and
then choose the fractions that will let a numerator unit be canceled with a denominator unit and vice
versa.
Laying the Foundation in Physics
53
Foundation Lesson II
4. Using your calculator, read from left to right and enter the numerator and denominator numbers in
order. Precede each numerator number with a multiplication sign and each denominator number with
a division sign. Alternatively, you could enter all of the numerators, separated by multiplication
signs, and then all of the denominators, each separated by a division sign.
5. Round your calculator’s answer to the same number of significant digits that your original number
had.
Example: How many inches are in 1.25 miles?
Solution:
1 ft
12 in.
OR
12 in.
1 ft
1 ft = 12 in.
5280 ft = 1 mile
1.25 miles ×
5280 ft
1 mile
OR
1 mile
5280 ft
5280 ft 12 in.
×
= 79,200 in.
1 mile
1 ft
As problems get more complex the measurements may contain fractional units or exponential units. To
handle these problems treat each unit independently. Structure your conversion factors to ensure that all
the given units cancel out with a numerator or denominator as appropriate and that your answer ends
with the appropriate unit. Sometimes information given in the problem is an equality that will be used as
a conversion factor.
Example: Suppose your automobile tank holds 23 gal. and the price of gasoline is 33.5¢ per L. How
many dollars will it cost you to fill your tank?
Solution: From a reference table we will find,
1 L = 1.06 qt
4 qt = 1 gal.
We should recognize from the problem that the price is also an equality, 33.5¢ = 1 L
and we should know that 100¢ = 1 dollar
Setting up the factors we find,
23 gal.×
54
33.5¢
4 qt
1L
$1
×
×
×
= $29
1 gal. 1.06 qt
1L
100¢
Laying the Foundation in Physics
Foundation Lesson II
In your calculator you should enter 23 × 4 ÷ 1.06 × 33.5 ÷ 100 and get 29.0754717. However, since
the given value of 23 gal. has only 2 significant digits, your answer must be rounded to $29.
Squared and cubed units are potentially tricky. Remember that a cm2 is really cm × cm. So, if we need to
convert cm2 to mm2 we need to use the conversion factor 1 cm = 10 mm twice so that both centimeter
units cancel out.
Example: One liter is exactly 1000 cm3. How many cubic inches are there in 1.0 L?
Solution: We should know that
1000 cm3 = 1 L
From a reference table we find,
1 in. = 2.54 cm
Setting up the factors we find,
1000 cm × cm × cm
1 in.
1 in.
1 in.
×
×
×
= 61 in.3
1L
2.54 cm
2.54 cm
2.54 cm
(The answer has 2 significant digits since our given 1.0 L contained two significant digits.)
1.0 L ×
As you become more comfortable with the concept of unit cancellation you will find that it is a very
handy tool for solving problems. By knowing the units of your given measurements, and by focusing on
the units of the desired answer you can derive a formula and correctly calculate an answer. This is
especially useful when you’ve forgotten, or never knew, the formula!
Example: Even though you may not know the exact formula for solving this problem, you should be
able to match the units up in such a way that only your desired unit does not cancel out.
What is the volume in liters of 1.5 moles of gas at 293 K and 1.10 atm of pressure?
0.0821 L • atm
The ideal gas constant is
mol • K
Solution: It is not necessary to know the formula for the ideal gas law to solve this problem correctly.
Working from the constant, since it sets the units, we need to cancel out every unit except L. Doing this
shows us that moles and Kelvins need to be in the numerator and atmospheres in the denominator.
0.821 L • atm 1.5 mol 293 K
×
×
×
= 33 L
mol • K
1.10 atm
(2 significant digits since our least accurate measurement has only 2 sig. digs.)
Laying the Foundation in Physics
55
Foundation Lesson II
**NOTE: NEVER rely on the number of significant digits in a constant to determine the number of
significant digits for reporting your answer. Consider ONLY the number of significant digits in given or
measured quantities.
PURPOSE
In this activity you will review some important aspects of numbers in science and then apply those
number handling skills to your own measurements and calculations.
MATERIALS
small cube
metric ruler
200 mL beaker
large graduated cylinder
spherical object
tweezers
flexible tape measure
balance
PROCEDURE
*Remember when taking measurements it is your responsibility to estimate a digit between the two
smallest marks on the instrument.
1. Mass the small cube on a balance and record your measurement in the data table on your student
page.
2. Measure dimensions (the length, width and height) of the small cube in centimeters, being careful to
use the full measuring capacity of your ruler. Record the lengths in your data table.
3. Fill the 200 mL beaker with water to the 100 mL line. Carefully place the cube in the beaker and use
the tweezers to gently submerge the cube. The cube should be just barely covered with water.
Record the new, final volume of water. Dry the cube.
4. Fill the large graduated cylinder ¾ of the way full with water. Record this initial water volume.
Again, use the tweezers to gently submerge the cube and record the final water volume.
5. Mass the spherical object on a balance and record your measurement in the data table.
6. Use the flexible tape measure to measure the widest circumference of the sphere in centimeters. Be
careful to use the full measuring capacity of the tape measure.
7. Fill the 200 mL beaker with water to the 100 mL line. Carefully place the spherical object in the
beaker and, if needed, use the tweezers to gently submerge the sphere. Record the final volume of
water from the beaker. Dry the spherical object.
8. Fill the large graduated cylinder ¾ of the way full with water. Record this initial water volume.
If needed, use the tweezers to gently submerge the sphere and record the new water volume.
9. Clean up your lab area as instructed by your teacher.
56
Laying the Foundation in Physics
Foundation Lesson II
Name _____________________________________
Period ____________________________________
Numbers in Science
Exploring Measurements, Significant Digits, and
Dimensional Analysis
DATA AND OBSERVATIONS
Data Table
Cube Data
Mass:
Dimensions
length:
width:
Volume
Beaker
initial volume:
100 mL
Graduated cylinder
initial volume:
height:
Beaker
final volume:
Graduated cylinder
final volume:
Sphere Data
Mass:
Dimensions
Circumference:
Volume
Beaker initial volume: 100 mL
Beaker final volume:
Graduated cylinder
initial volume:
Graduated cylinder
final volume:
Formula for calculating the volume of a cube:
Formula for calculating the circumference of a circle:
Formula for calculating the diameter of a circle:
Formula for calculating the volume of a sphere:
Laying the Foundation in Physics
57
Foundation Lesson II
ANALYSIS
•
•
•
Show your organized work on a piece of notebook paper. Transfer your final answers to the
blanks beside each question. Staple your work to your answer sheet before turning it in.
Remember to follow the rules for reporting all data and calculated answers with the correct
number of significant digits.
You may need tables of metric and English conversion factors to work some of these problems.
1. For each of the measurements you recorded above, go back and indicate the number of significant
digits in parentheses after the measurement. Ex: 15.7 cm (3sd)
2. Use dimensional analysis to convert the mass of the cube to
a.__________ mg
b. _________ ounces
3. ____________ Calculate the volume of the cube in cm3.
4. ____________ Use dimensional analysis to convert the volume of the cube from cm3 to m3.
5. ____________ Calculate the volume of the cube in mL as measured in the beaker. Convert the
volume to cm3 knowing that 1 cm3 = 1 mL.
6. ____________ Calculate the volume of the cube in mL as measured in the graduated cylinder.
Convert to cm3 knowing that 1 cm3 = 1 mL.
7. Using the density formula D =
mass
, calculate the density of the cube as determined by the
volume
a.__________ ruler
b. _________ beaker
c.__________ graduated cylinder
8. Use dimensional analysis to convert these three densities into kg/m3.
9. Convert the mass of the sphere to
a.__________ kg
b. _________ lbs
10. ____________ Using the measured circumference, calculate the diameter of the sphere.
11. ____________ Calculate the radius of the sphere.
12. ____________ Calculate the volume of the sphere from its radius.
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Laying the Foundation in Physics
Foundation Lesson II
13. ____________ Calculate the volume of the sphere in mL as measured in the beaker. Convert to cm3
knowing that 1 cm3 = 1 mL.
14. ____________ Calculate the volume of the sphere in mL as measured in the graduated cylinder.
Convert to cm3 knowing that 1 cm3 = 1 mL.
15. Using the density formula D =
mass
, calculate the density of the sphere as determined by the
volume
a. __________ tape measure
b.__________ beaker
c. __________ graduated cylinder
16. ____________ Use dimensional analysis to convert these three densities into lbs/ft3.
CONCLUSION QUESTIONS
1. Compare the densities of the cube when the volume is measured by a ruler, beaker and graduated
cylinder. Which of the instruments gave the most accurate density value? Use the concept of
significant digits to explain your answer.
2. A student first measures the volume of the cube by water displacement using the graduated cylinder.
Next, the student measures the mass of the cube before drying it. How will this error affect the
calculated density of the cube? Your answer should state clearly whether the calculated density will
increase, decrease or remain the same and must be justified.
3. A student measures the circumference of a sphere at a point slightly higher than the middle of the
sphere. How will this error affect the calculated density of the cube? Your answer should state
clearly whether the calculated density will increase, decrease, or remain the same and must be
justified.
Laying the Foundation in Physics
59