Name:
Measurements and Calculations
Introduction
Much of what we do in this laboratory exercise may seem rather distantly related to
astronomy. After all, you won’t be looking at a single star all night. So why do it?
Because you’ll be making many measurements and doing many calculations over the
course of this semester, and we need to make sure that you’re comfortable with how to
make measurements, how to use your calculators, how to do unit conversions, and how to
report your answers.
What you’ll need
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Meter stick
Scientific calculator (the same one that you will be using all semester)
This lab exercise.
A ball from the front table.
Note paper.
Procedure
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READ the lab.
Make sure that you understand what it says.
Answer the questions in the spaces provided.
Hand in these sheets. This is an informal lab, and so you do not need to follow
the procedures for a formal lab write-up.
A) Using scientific notation
Scientific notation is a form of shorthand. For example, it would rapidly cause writer’s
cramp if I had to constantly write the mass of the Sun as:
2,000,000,000,000,000,000,000,000,000,000 kg
That’s a 2 followed by 30 zeroes. It’s much easier to write it as:
2x1030 kg
What the shorthand (scientific notation) form says is that I’m multiplying 2 by 10, thirty
times. Let’s look at this using a simpler example, say the number:
2000
I can re-write that as:
2x10x10x10,
because 1000 is 10 multiplied by itself three times. Well, that’s actually more painful
than writing 2000, and so we replace the three multiplications by ten as
2x103
To convert 2000 into scientific notation is pretty straightforward. I start by actually
putting in the decimal point where it belongs, as
2000.
I then count the spaces that I would shift the decimal to put it just right of the leftmost
number, or:
2000.
The number of spaces becomes the exponent of ten in my scientific notation (three, in
this case). Most numbers, however, will look more like this:
15348.226
To convert that to scientific notation, follow the same rule: count the number of spaces
between the decimal point and the leftmost number, or:
15348.226
It took four shifts to get the decimal to the right of the “1,” and so the scientific notation
version would be:
1.5348226x104
For numbers less than one, the exponent of the ten is negative. For example, the number
one one-hundredth:
0.01 =
1
1
=
= 1×10−2
100 10 ×10
The negative exponent is just a shorthand way of indicating that we’re actually dividing
by ten twice. To convert a number less than one into scientific notation, I again count the
€ number of spaces that I have to move the decimal in order to put it to the right of the
leftmost non-zero number. For example:
0.000254
Because I had to shift the decimal four places to put it to the right of the “2,” the
scientific notation form of the number above is:
2.54x10-4
Exercise: Convert the following into scientific notation. Show your work in the
space provided.
1. 15.448
2. 0.0314156
3. 1666684.2
4. 0.00000449
B) Computing with scientific notation
Multiplying numbers written in scientific notation by hand takes a bit more thought than
normal multiplication. When multiplying two powers of ten, you add the exponents. For
example:
103 x 104 = 103+4 = 107
Or just one thousand times ten thousand equals ten million.
When the number has both parts of the scientific notation, you multiply the non-exponent
parts as you would normally. For example:
1.5 ×10 4 ∗1.6 ×10 5 = 1.5 ∗1.6 ∗10 4 ∗10 5 = 2.4 ∗10 4 +5 = 2.4 ×10 9
To help myself out, I re-arranged the numbers to put the powers of ten next to each other.
€ In many cases, multiplying the non-exponents together gives me another power of ten.
For example:
3.5 ×10 3 ∗ 4 ×10 2 = 3.5 ∗ 4 ∗10 3 ∗10 2 = 14 ∗10 3+2 = 14 ×10 5 = 1.4 ×101 ×10 5 = 1.4 ×10 6
Exercise: Do the following computations by hand (you can check your answer using
a calculator). Write your answers in scientific notation. Show your work in the
space provided.
€
5. 1.2x106 * 2.8x107
6. 3.4x104 * 2.2x10-5
7. 4.5x103 * 6.0x108
8. 8.8x104 * 5.5x10-5
C) Make sure that you’re familiar with your calculator
Your scientific calculator has its own shorthand way of handling scientific notation,
which you should use. For example, in doing the following:
1.33 ×10 3 ∗1.4 ×10 4
many people will press the literal sequence of keys on their calculator:
€
“1.33” “x” “10” “^” “3” “x” “1.4” “x” “10” “^” “4”
Sometimes this works, and sometimes your calculator will give you odd results. Avoid
the potential of odd results by using the “E” or “EE” key on your calculator. The “E” or
“EE” stands for “enter exponent,” and tells the calculator that you’re entering a number
in scientific notation. So, to enter the number 1.33x103, you would press the following
keys:
“1.33” “EE” “3”
What your calculator displays depends on the model. On some (expensive models), you
will see the actual scientific notation form of the number. On others, it may look like:
1.33E3
That’s your calculator’s shorthand version of the shorthand. Instead of writing “x10” it
just writes “E” (for “exponent”) in order to save display space, and you have to realize
that the 3 after the E is the exponent of 10 in the scientific notation.
Do the following calculations on your calculator, and make sure that you agree with
the answer provided in parentheses. See your instructor if you do not see how to
make your number agree. In the space provided, indicate whether or not your
answer agrees, and write in what key your calculator uses to enter powers of ten.
9. 4.3312x1021 * 6.6602x10-23 (2.885x10-1)
10. 2x1030/1.67x10-27 (1.198x1057)
D) Accuracy and precision
Accuracy and precision often get confused with each other, so let’s start with two rough
definitions:
Accuracy: This is describes of how well your measurements compare with the
actual value. The closer your measurements are to reality, the more accurate they
are.
Precision: This describes how well you are able to repeat your measurements. The
closer your repeated measurements are to each other, the more precise they are.
It is possible to be accurate, precise, neither, or both. To picture it, think of a shooting
target, where you want to hit the bullseye.
Let’s say that you take twenty shots. The following is neither accurate nor precise:
The shots are all over the place, and the scatter isn’t even centered on the bullseye
(though one hit, by dumb luck). The following would represent high precision, but low
accuracy:
The shots are tightly grouped, but nowhere near the bullseye. The following, however,
would represent both high precision, and high accuracy:
We can usually get a sense of the precision of our measurements. Let’s use the meter
stick for that:
Look at a meter stick and answer the following questions in the space provided.
11. What is the smallest unit of measurement on the meter stick?
12. What is the smallest length that you could measure accurately using the
meter stick?
13. What is the precision of measurements that you make using the meter stick
(huge hint: it’s half of the smallest division of the meter stick)?
Accuracy is tougher to get a handle on. Lots of things can mess up your accuracy,
without you even knowing it. For example, does your meter stick go all the way to zero,
or is the end a bit banged up? When you weigh yourself on some scales at the gym, are
you sure that they are properly calibrated (they might not measure pounds properly, or
their zero point might be off)? Human factors can enter in as well. For example,
somebody might not be very careful in using the meter stick, or they might view it at an
angle and so get a bad reading.
14. Consult with your group members, and/or members of other groups.
Imagine that you are timing the swing of a pendulum with a stopwatch
(you’ll do this in a lab later this semester). Think of at least three factors
that might affect the accuracy of your measurements. Write them here.
E) Significant figures
Significant figures are just that--numbers that have significance, as determined by your
accuracy and precision. Let’s say that I measure the width of three identical tables that
are side-by-side. I use a meter stick with a precision of 0.5 mm, and I measure the width
as 3423.4 mm. To get the width of one table, I divide my answer by three. If I blindly do
this on my calculator, it gives me the following answer:
1141.133333 mm
If I record that number, then I’m claiming that I know the width of one table to an
accuracy of 10-6 mm. This is more than 100 times smaller than the size of the average
bacterial cell! If I write down that kind of precision, I’ve automatically lost a point on
my lab report (or if I do it at work, my boss laughs at me, and my next Annual
Performance Evaluation suffers).
I cannot claim that my measurement for one table is any better than one-third of my
original precision (because there are three tables, the possible measurement error is
“spread out” across the three). So, the possible error of my measured size of one table is
0.5/3 mm, which rounds off to 0.2 mm. Because my precision is at the level of tenths of
a millimeter, I cannot quote my measurement out further than that, and I have to round
off my answer to:
1141.1 mm
This indicates the precision of my measurement. The number of significant figures here
is 5, the number of digits that I can report as real, measured numbers. Note, though, that
if I thought that my meter stick measurements were off by as much as 1 mm, then I
would have to round my answer off to 1141 mm.
Scientific notation helps with reporting significant figures. For example, how many
significant figures are in the following number?
2000
The answer is that we don’t know. The zeroes might be real measured numbers, or just
placeholders for the two. If, on the other hand, I had written it as
2000.0
then the added zero after the decimal tells everybody that there are actually five
significant figures here (the zero after the decimal is one of them). On the other hand,
let’s say that there are only two significant figures there. Then it would be better for me
to use scientific notation to include only the significant figures:
2.0x103
The 10 and its exponent don’t count as significant figures; all they do is tell everybody
how large the number is. It’s the numbers ahead of the 10 that are significant figures
(two figures in this case).
For numbers less than one, leading zeros again are only placeholders, and not significant
figures. For example, the following has only three significant figures:
0.000936
Round off the following numbers to two significant figures. Write your answers in
scientific notation.
15. 1.372x105
16. 3042.5
17. 0.049887
18. 1.989x1030
F) Uncertainty
ALL measurements have uncertainty. The uncertainty is an estimate of accuracy. Go
back to the table measurement example above. If I’m good at reading it, and the meter
stick is in good condition, then the smallest possible uncertainty in my ability to make a
single measurement is 0.5 mm (the same as my precision). That was the uncertainty for
the width of three tables. For one table, I divide that by three, which rounds off to an
uncertainty of 0.2 mm. I would therefore quote my answer as:
1141.1 +/- 0.2 mm
The last number, after the +/- sign, is the amount by which my measurement is uncertain.
This might not be an honest representation of my true accuracy, though. What if the
meter stick is 20 years old, and very battered? What if I have poor eyesight? What if I
tend to look at an angle? I would have to think of these and probably increase my quoted
uncertainty to make a more honest assessment of my measurement. If, for example, I
didn’t have confidence that I could actually make a measurement to better than 1 mm,
then I would need to quote my answer as:
1141 +/- 1 mm
Note that I never quote a number to a higher level of precision than my accuracy. So, in
the above example, if my uncertainty is 1 mm, I don’t quote my measurement to a
smaller decimal place than that.
Do the following (answer questions in the space provided):
19. Take a ball from the front table. Using your meter stick, measure its diameter.
Repeat for a total of three measurements, and record each in the space below.
Also, compute the average of your measurements, and record it in the space
below. What is the uncertainty of each one of your measurements? We take more
than one measurement to decrease our uncertainties. With three measurements,
the uncertainty of your average is the uncertainty of each measurement, divided
by three. Record that number below (round off to one significant figure). What is
your final answer (written as I did for the table width above)?
F) Unit conversions
Another source of problems for many of us is unit conversions. The best way to avoid
problems here is to treat units as if they were numbers, and can be cancelled in
numerators and denominators. Let’s say that I want to convert a measurement in inches
to cm. There are 2.54 cm in 1 in. If I measured something to be 11 in. in length, then the
conversion is:
11 in ×
2.54 cm
= 28 cm
1 in
I treat the inch units as if they were numbers, and cancel the one in the numerator with
the one in the denominator, leaving cm. You’ll also notice that I write 28 cm, and not
€ 27.94 cm, which my calculator told me. That’s because I can’t quote the converted
number to more significant figures than the original number.
You can run large strings of these. Let’s say that I want to convert a measurement of 2.0
miles to centimeters. Well, there are 5280 feet in a mile, 12 inches in a foot, and 2.54 cm
in an inch, so the answer is:
2.0 miles ×
€
5280 ft
12 in
2.54 cm
×
×
= 3.2 ×10 5 cm
1 mile
1 ft
1 in
Do the following conversions. Show your work and write your answers in the space
provided.
20. Convert a measurement of 5.2 pounds to kg (there are 2.2 lb in one kg).
21. Convert a time of 2.1 years to fortnights (there are 365.24 days per year, and
14 days in one fortnight).
22. Convert a weight of 19 stone into ounces (1 stone = 14 lb, and 16 ounces = 1
lb.).