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Unformatted text preview: e that was poorly
calibrated. They analyzed their results and found the
mass of penny B to be 3.556 ± 0.004 g. This number is
more precise than their previous result since the
uncertainty is smaller, but the new measured value
of mass is very different from their previous value.
We might conclude that this new value for the mass Addition: Multiplication: (3.131 ± 0.008 g) + (3.121 ± 0.004 g) = ? (3.131 ± 0.013 g) x (6.1 ± 0.2 cm) = ? First, find the sum of the values:
3.131 g + 3.121 g = 6.252 g
Next, find the largest possible value:
3.139 g + 3.125 g = 6.264 g
The uncertainty is the difference between the
two:
6.264 g – 6.252 g = 0.012 g First, find the product of the values:
3.131 g x 6.1 cm = 19.1 gcm
Next, find the largest possible value:
3.144 g x 6.3 cm = 19.8 gcm
The uncertainty is the difference between
the two:
19.8 gcm  19.1 gcm = 0.7 gcm Answer: 6.252 ± 0.012 g. Answer: 19.1 ± 0.7gcm. Note: This uncertainty can be found by simply Note:
The percentage uncertainty in the
adding the individual uncertainties:
answer is the sum of the individual percentage
uncertainties:
0.004 g + 0.008 g = 0.012 g
0.013
0.2
0.7 100% 100% 100%
3.131
6.1
19.1 247 APPENDIX: ACCURACY, PRECISION AND UNCERTAINTY Subtraction: Division: (3.131 ± 0.008 g) – (3.121 ± 0.004 g) = ? (3.131 ± 0.008 g) ÷ (3.121 ± 0.004 g) = ? First, find the difference of the values:
3.131 g  3.121 g = 0.010 g
Next, find the largest possible difference:
3.139 g – 3.117 g = 0.022 g
The uncertainty is the difference between the
two:
0.022 g – 0.010 g = 0.012 g First, divide the values: Answer: 0.010±0.012 g. Answer: 1.003 ± 0.004 3.131 g 3.121 g = 1.0032
Next, find the largest possible value:
3.139 g ÷ 3.117 g = 1.0071
The uncertainty is the difference between
the two:
1.0071  1.0032 = 0.0039 Note: This uncertainty can be found by simply Note:
The percentage uncertainty in the
adding the individual uncertainties:
answer is the sum of the individual percentage
uncertainties:
0.004 g + 0.008 g = 0.012 g
0.008
0.004
0.0039 100% 100% 100%
3.131
3.121
1.0032 Notice also, that zero is included in this range, so
it is possible that there is no difference in the Notice also, the largest possible value for the
numerator and the smallest possible value for
masses of the pennies, as we saw before.
the denominator gives the largest result.
The same ideas can be carried out with more
complicated calculations. Remember this will always
give you an overestimate of your uncertainty. There
are other calculation techniques, which give better
estimates for uncertainties. If you wish to use them,
please discuss it with your instructor to see if they
are appropriate. These techniques help you estimate the random
uncertainty that always occurs in measurements.
They will not help account for mistakes or poor
measurement procedures. There is no substitute for
taking data with the utmost of care. A little
forethought about the possible sources of uncertainty
can go a long way in ensuring precise and accurate
data. 248 APPENDIX: ACCURACY, PRECISION AND UNCERTAINTY PRACTICE EXERCISES:
B1. Consider the following results for different experiments. Determine if they agree with the accepted
result listed to the right. Also calculate the precision for each result.
a) g = 10.4 ± 1.1 m/s2 g = 9.8 m/s2 b) T = 1.5 ± 0.1 sec T = 1.1 sec c) k = 1300 ± 50 N/m k = 1368 ± 45 N/m Answers: a) Yes, 11%; b) No, 7%; c) Yes, 3.3%
B2. The area of a rectangular metal plate was found by measuring its length and its width. The length was
found to be 5.37 ± 0.05 cm. The width was found to be 3.42 ± 0.02 cm. What is the area and the average
deviation?
Answer: 18.4 ± 0.3 cm2 B3. Each member of your lab group weighs the cart and two mass sets twice. The following table shows
this data. Calculate the total mass of the cart with each set of masses and for the two sets of masses
combined.
Cart
(grams) Mass set 1
(grams) Mass set 2
(grams) 201.3
201.5
202.3
202.1
199.8
200.0 98.7
98.8
96.9
97.1
98.4
98.6 95.6
95.3
96.4
96.2
95.8
95.6 Answers:
Cart and set 1:
Cart and set 2:
Cart and both sets: 249 299.3±1.6 g.
297.0±1.2 g.
395.1±1.9 g. APPENDIX: ACCURACY, PRECISION AND UNCERTAINTY 250 Appendix: Review of Graphs Graphs are visual tools used to represent relationships (or the lack thereof) among numerical
quantities in mathematics. In particular, we are interested in the graphs of functions. What is a graph?
In this course, we will be dealing almost exclusively with graphs of functions. When we
graph a quantity with respect to a quantity , we mean to put on the horizontal axis and
on the vertical axis of a twodimensional region and then to draw a set of points or curve
showing the relationship between them. We do not mean to graph any other quantity from
which
or can be determined. For example, a plot of acceleration versus time has
acceleration itself, ( ), on the vertical axis, not the corresponding velocity ( ); the time , of
course, goes on the horizontal axis. See Figure 1. (a)
(b)
Figure 1: Graphs of acceleration a and velocity v for an object in...
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 Spring '14

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