Physics 1020
Laboratory #1
Measurement and Uncertainty
Measurement and Uncertainty
Any experimental measurement or result has an uncertainty associated with it. In todays lab
you will perform a set of very simple measurements. You will have to estimate the random
uncertainty associated with each of them. As a rule of thumb the precision of your measuring
device (for example a ruler) is always a very good starting value for your uncertainty.
Furthermore you will be asked to perform some calculations using the values you just
measured. The results of those calculations will also have an uncertainty associated with
them. To obtain those values you will have to follow a set of rules. They are explained on
page X of this book in “Making Measurements in Physics” section.
Finally you will collect a set of 10 measurements of the same quantity from your classmates
and asked to calculate their average. That result also has an uncertainty associated with it.
Your instruction will tell you how to calculate it.
Also recall that the perimeter 𝑃 of a rectangular object is the total length of all the sides, i.e
𝑃 = 𝐿 + 𝐿 + 𝑊 + 𝑊 = 2𝐿 + 2𝑊
where 𝐿 is the length of the object and 𝑊 is the width.
Similarly the area 𝐴 of a solid rectangular object is
𝐴=𝐿×𝑊
Below is a summary of some of the rules you will need for your uncertainty calculations:
Addition/Subtraction Rule
When finding a result by adding or subtracting uncertain quantities, the absolute
experimental uncertainty in the result is the sum of the absolute experimental uncertainties
of the uncertain quantities.
𝐼𝐹 𝒛 = 𝒙 + 𝒚 𝑂𝑅 𝒛 = 𝒙 − 𝒚 𝑇𝐻𝐸𝑁
𝜹𝒛 = 𝜹𝒙 + 𝜹𝒚
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Physics 1020
Laboratory #1
Measurement and Uncertainty
Example:
𝐴 = (122 ± 5) 𝑚,
𝐵 = (207 ± 2) 𝑚
𝐶 = 𝐴 + 𝐵 = 122 𝑚 + 207 𝑚 = 329 𝑚
𝛿𝐶 = 𝛿𝐴 + 𝛿𝐵 = 5 𝑚 + 2 𝑚 = 7 𝑚
𝐶 = (329 ± 7) 𝑚
Multiplication/Division Rule
When finding a result by multiplying or dividing uncertain quantities, the relative
experimental uncertainty in the result is the sum of the relative experimental uncertainties
of the uncertain quantities.
𝐼𝐹 𝒛 = 𝒙 × 𝒚 𝑂𝑅 𝒛 =
𝒙
𝑇𝐻𝐸𝑁
𝒚
𝜹𝒛 𝜹𝒙 𝜹𝒚
=
+
𝒛
𝒙
𝒚
Example:
𝐴 = (120 ± 5)𝑚, 𝐵 = (200 ± 10)𝑚
𝐶 = 𝐴 × 𝐵 = 120 𝑚 × 200 𝑚 = 24000 𝑚2
𝛿𝐶 𝛿𝐴 𝛿𝐵
5𝑚
10 𝑚
=
+
=
+
= 0.0917
𝐶
𝐴
𝐵
120 𝑚 200 𝑚
𝛿𝐶
𝛿𝐶 =
× 𝐶 = 0.0917 × 24000 𝑚2 = 2200 𝑚2
𝐶
𝐶 = (24000 ± 2000)𝑚2 𝑜𝑟 (2.4 ± 0.2) × 104 𝑚2
Note: The uncertainty 𝛿𝐶 = 2200 𝑚2 has been rounded to one significant figure, so it
becomes 2000 𝑚2.
Important: the only difference in the two rules is one uses absolute experimental uncertainty
and one uses relative experimental uncertainty. If you use the multiplication rule, you can
always get the absolute experimental uncertainty from the relative experimental uncertainty.
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Physics 1020
Laboratory #1
Measurement and Uncertainty
Prelab Questions
These questions need to be completed before
entering the lab. Please show all workings.
Marker’s
Initials
Prelab 1
A rectangular object has length L = 25.3  0.2 cm and width W = 18.6  0.5 cm.
•
•
What are the absolute uncertainties in length and width?
Find the relative uncertainties in length and width.
Prelab 2
Given a set of length measurements: 60.4, 60.0, 61.1, 60.8, 60.6 cm.
•
•
Find the average (mean) length
Find the standard error (refer to the introductory pages of your Lab
Workbook).
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Physics 1020
Laboratory #1
Measurement and Uncertainty
Laboratory Worksheet
Name and Student Number:
___________________________
Partner:
___________________________
Date:
___________________________
Table 1:
Value
Length
Width
QUESTION 1:
QUESTION 2:
𝛿𝐿 =
𝛿𝑊 =
QUESTION 3:
𝐿=
𝑊=
QUESTION 4:
𝛿𝐿
𝐿
=
𝛿𝑊
𝑊
=
Units
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Physics 1020
Laboratory #1
QUESTION 5:
CHECKPOINT: Instructor Initial
QUESTION 6:
𝑷=
QUESTION 7:
𝜹𝑷 =
QUESTION 8:
𝑷=
QUESTION 9:
Measurement and Uncertainty
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Physics 1020
Laboratory #1
Measurement and Uncertainty
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Table 2:
Value
Length
QUESTION 10:
QUESTION 11:
𝑨= 𝑳×𝑾 =
QUESTION 12:
𝜹𝑨
𝑨
=
QUESTION 13:
Uncertainty
Units
Physics 1020
QUESTION 14:
𝜹𝑨 =
𝑨=
Laboratory #1
Measurement and Uncertainty
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Physics 1020
Laboratory #1
graph paper is 1 cm  1cm squares
Measurement and Uncertainty
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Physics 1020
Laboratory #1
Measurement and Uncertainty
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Table 3:
Value
Area
QUESTION 15:
QUESTION 16:
Uncertainty
Units

Physics 1020
Laboratory #1
Measurement and Uncertainty
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Table 4:
Li (cm)
Wi (cm)
W
L
L 
(cm)
N=


L 
(no units)
L
N


QUESTION 17:
𝑳̅ =
𝜹𝑳̅
𝑳̅
=

(cm)

W 
(cm)
N=
W 
(no units)
W
N

(cm)
Physics 1020
Laboratory #1
QUESTION 18:
̅𝑾
̅̅ =
̅̅̅
𝜹𝑾
̅𝑾
̅̅
=
QUESTION 19:
QUESTION 20:
QUESTION 21:
Measurement and Uncertainty
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