Dr. Peter R. Conwell
Physics 211, Physics for Scientists and Engineers
3/9/2016
2nd Problem Set: Measurement and Uncertainty
The Measurement Contest. This year's challenge is shown in the box below. Your activity guide outlines
the rules on page 15 and 16. Turn in your response to the contest on a separate piece of paper; not
attached to your answers to this assignment. Turn in one quiz response per team.
Converse Hall
On the east side of campus, lies
the flagship of the school,
Converse Hall. Your job is to
measure the vertical distance
from base of the bottom step on
the center-east side of the
building to the top of the center
triangle on the roof. Remember
that direct measurements are
forbidden by the contest rules. It
would be very dangerous to
attempt to climb to the top of
converse. Be careful of the
campus landscaping, and Watch
for cars in the traffic circle!!
Recommended Reading: The assigned problems are on types of error, standard deviation, the standard
deviation of the mean, and significant figures. I recommend that you read Appendix C of the Workshop
Physics Activity Guide before attempting the problems.
1) You are working in your spare time for a consumer organization that is checking on whether or not the
small house brand cereal boxes sold at the local supermarket each have the number of grams of cereal
listed on the box. The market manager claims that their food processors actually "over fills" the boxes
so that consumers are not cheated, but several members of your organization are skeptical. You
purchase six boxes of each of three types of cereal, Tasteeos (rated at 21g), Honey Nut Tasteeos (rated
at 25g), and Raisin Bran (rated at 35g). The packaging for all three types of cereal is identical and
consists of a cardboard box and an inner plastic wrapper. Using an electronic balance you determine
that the mass of a box and wrapper without the cereal is 17.7g. Next you find the total masses for each
of the boxes. Your results are shown in the following table.
Box 1
Box 2
Box 3
Box 4
Box 5
Box 6
a)
Tasteeos[g]
40.8
42.9
42.3
41.9
42.5
40.5
Honey Nut Tasteeos [g]
43.6
45.1
44.2
45.8
44.8
45.1
Raisin Bran [g]
65.6
63.1
67.7
66.0
66.4
67.7
Examine the three sets of data. Before doing any calculations, which set do you predict will have
the largest standard deviation? The smallest standard deviation? Explain your predictions. Note:
No points will be taken off for an incorrect predictions as long as the reasons you give for your
predictions are consistent with a correct understanding of the standard deviation. (Appendix C of
the Workshop Physics Activity Guide.)
Dr. Peter R. Conwell
Physics 211, Physics for Scientists and Engineers
3/9/2016
b) Using the method described in Appendix C, find the average and standard deviation by hand for
the mass of the cereal (excluding the packaging) in the boxes of Tasteeos. Notes: (1) N is not 12
in this case. (2) No credit will be given unless you show your calculations!
c)
Use a spreadsheet to find the average, Standard Deviation (sd or SD), and Standard Deviation of
the Mean (SDM) for the mass of the cereal (excluding the packaging) in the boxes of Tasteeos.
Do this also for the Honey Nut Tasteeos and the Raisin Bran. Just report the results. No need to
hand in the spreadsheet printout.
d) Do any of the boxes contain a mass of cereal that is under the rated mass? Are the boxes overfilled,
or underfilled?
e)
What is the Standard Deviation of the Mean (SDM) for each set of measurements? How did you
determine this?
f)
Suppose you are given just one additional box of Honey Nut Tasteeos. Do you have a 68.3%
chance that a determination of the mass of that single box will lie within one sd of the average.
Or, a 68.3% chance that your measurement of the mass will within one Standard Deviation of the
Mean (SDM)? Explain your answer.
2) Diane and Daniel calculated the averages and standard deviations for several types of measurements.
They got into a rote mood and copied all the digits that appeared on their scientific calculator. Please
boil these down to the “correct” number of significant figures and state how many significant figures is
being reported for each best estimate.
a)
b)
c)
d)
e)
.666666667 ± 0.028314389 Volts
–18.25723 ± 2.31684 degrees Celsius
0.0373749738 ± 0.005174523 meters
3546.78349 ± 188.549763 seconds (express this answer with and without scientific notation)
Explain why the result of rounding off in part (d) is ambiguous unless scientific notation is used.
3) Bridget and Chris are studying projectile motion and they have derived a theoretical equation that
allows them to predict how far a projectile will move as a function of its angle of launch, height above
the floor, and initial speed. Each of them agrees that for the initial angle of launch, height above the
floor and speed of the ball that they measured, the theoretical distance for a launched ball ought to be
4.58 meters. To verify the theoretical equation, Bridget tapes the center of a piece of paper at a
horizontal distance of 4.58m from the point of launch and shoots the projectile six times. She puts a
piece of carbon paper on top of her other paper so that each time the projectile hits the paper it leaves a
smudge. Chris thinks he can get less variation in his data by steadying the table and adjusting the angle
of launch more carefully. He does these things, puts down a clean paper, and launches the projectile
four more times.
Dr. Peter R. Conwell
Physics 211, Physics for Scientists and Engineers
3/9/2016
(a)
Assuming that the picture of each piece of paper that follows is its actual size, what is the average
distance of Bridget’s six launches? What is Chris's average distance? What is the standard
deviation (sd or SD) and standard deviation of the mean (SDM) of each set of measurements?
(b)
Discuss in essay form which of the two students has the least uncertainty associated with his or her
launches and is thus more precise. Which one of the students is the most accurate in the sense of
having best estimate that closely matches the theoretical calculation. Which one of the students
appears to have a systematic error associated with his or her launches? What factors in the
experimental technique used by each student might cause one to be more precise and another more
accurate?
Wanda
4.58 m
Chris
4.58 m
4)
Refer to the drunkard's walk activities you completed in Activity 2.9.2. Set up a spreadsheet
simulation that allows a drunk to take a hypothetical 30 second walk (instead of the hypothetical
20 second walks they took when you completed Activity 2.9.2). There are tips at the end of
Section 2.9 on constructing this type of spreadsheet simulation. Run your simulation and record
where the drunk is relative to the bar after a 30 second walk. (That is, how many "steps" is the
drunk to the right (+) or the left (-) of your starting point?). Submit a printout of your spreadsheet
with your name and the date submitted on it.
5)
Refer to the spreadsheet simulation you designed in the previous problem. Modify this spreadsheet
to take 10 drunks each on 30 second walks. Use the spreadsheet to record each drunk’s location
after each of these walks. In answering the following questions, there is no need to print out or
submit an additional spreadsheet. You can use a spreadsheet to do every thing in the
a) List the locations of the 10 drunks in "steps" right(+) or left(-) from the bar.
b) What is the Standard Deviation in steps for the simulated walks taken by the 10 drunks?
c) The average location of your 10 drunks should be close to zero. ( It may not be exactly zero
because you've only tested 10 drunks instead of an infinite number of them.) Assume that the
average distance in steps from the bar is actually zero. Within what distance, n, in steps from
the bar would you expect to have a 68% probability of finding the next drunk who staggers out
of the bar and walks for 16 seconds?
Dr. Peter R. Conwell
Physics 211, Physics for Scientists and Engineers
3/9/2016
If 68% of the drunks fell down in this region what is n?
0
-n
6)
7)
+n
Bar
Let's imagine taking a whole new data set with 24 more drunks taking walks. Not just one more as
in part (c) above. Within what distance from the bar in steps would you expect to have a 68%
probability of finding the average distance of 24 more drunkards? Hint: The standard deviation of
the mean (SDM) has something to do with this!
Take a second look at the normalized histograms in section 2.9 in your (modified) activity guide.
Esimate the probability that x  1 . (Hint: there is a simple way to do this based on the answer you
already computed in your activity guide.)