SCI107 Physics
Laboratory Exercise 1 - Measurement & Uncertainty
Location: Sippy Downs H.1.06/07 Science Labs
Moreton Bay A.2.08 Flexi-Lab
Required Safety Wear:
Closed footwear (for Moreton Bay and Sippy Downs campus students)
Lab coat, Safety glasses (for Sippy Downs students in PC1 Science Lab)
Date/time…………………
Group Members……………………………………………………………………………….
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(You should read the background and example before class)
1.
BACKGROUND
This exercise introduces some experimental techniques to perform measurements and
estimate uncertainty. In any measurement, there will be a degree of uncertainty. In general,
uncertainties may be caused by several factors. Let’s take look at some of these in the
following example.
Example: Measuring the length of a piece of wood
Suppose you’re measuring the length of a piece of wood using a metal measuring tape. One
of the first things you may notice is that the beginning end is free to move a bit to allow you
to make “inside” and “outside” measurements. Now, how do you know that the movement of
the end of the tape is just right? In other words, if you hook the tape over the end of your
piece of wood, will the zero on the tape correspond to the end of the wood? Any uncertainty
in this will be called a “zero error” and will result in a measurement that is either a bit small
or a bit big, no matter how good the rest of your measuring process may be.
Another source of uncertainty is the precision of the scale marked on the tape. There are
several factors that can affect this. Two important ones are how precisely was the scale
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marking when the tape was made and has the tape changed its length – either by stretching or
through thermal expansion? No matter how carefully you make measurements, these may
both lead to other types of uncertainty.
1. If the scale marking was not done with sufficient precision, then some of the intervals
on the tape will be too short and others will be too long. If there is no systematic
relationship between these variations, then we call them “random variations” and
any measurement will be different from the “true length” by an amount which we
call a “random uncertainty”.
2. Now, if the tape has stretched (or shrunk) uniformly, then any measurement will also
be too small (or too big) in a systematic way. By “systematic” we mean that there is a
relationship between the true length and the measured length. For example, if a 10.00
m long tape has stretched to 10.10 m, then any measurement that we make will be too
small by about 1% (ten millimetres in every metre!).
The last type of uncertainty that we will consider is due to the limit of accuracy of reading the
scale. There are two situations to consider:
1. If you are looking at the end of a piece of wood, can you identify a scale marking that
corresponds to the end? Is the end nice, smooth and square or is it rough and crooked?
In the case of a rough piece of wood, you may say it is 757 mm give or take 3 mm.
2. However, when the ends are smooth and square, you need only consider the limit to
which you can read the tape. We generally allow for an accuracy of half of the
smallest scale marking. So, if the tape was marked in 1 mm intervals, then you could
say that the piece of wood was 757 mm give or take 0.5 mm.
Lastly, one of the fundamental requirements of any experiment is reproducibility. So, you
should always make repeated measurements, until you have a good understanding of the size
of the uncertainties in your measurements.
2.
EXAMPLE: Measuring density of a block of unknown material
(You should review and attempt the questions in this example before class)
First, we have taken three measurements across three different orthogonal faces (Length 1,
Length 2, and Length 3) of a block which are recorded in Table 1. Uncertainty is half the
smallest measurement unit, which in this example was ± 0.01 mm. (This uncertainty depends
on the device that you use to measure it with!)
We can then calculate the average length and uncertainty for each face. The uncertainty in the
face length can be found from ± ½ Range, where the Range is the largest value minus the
smallest length. For example, for Length 1, the Range = 27.59 – 27.52 = 0.07 mm
Then, uncertainty in the average for Length 1 = ± (0.07 mm / 2) = ± 0.035 mm
(by convention we round this to one sig. fig. so the final value for uncertainty of Length 1
is ± 0.04 mm)
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Table 1 – Calculated average lengths and uncertainties for measurements on three
orthogonal faces of example block.
Average
Length 1 (mm)
27.55±0.01
27.59±0.01
27.52±0.01
27.55±0.04
Length 2 (mm)
26.34±0.01
26.32±0.01
26.36±0.01
26.34±0.02
Length 3 (mm)
57.25±0.01
57.25±0.01
57.25±0.01
57.25±0.01
Mass (g)
186.891±0.001
186.895±0.001
186.873±0.001
186.89±0.01
Now we will calculate the volume, density and their uncertainties!
1. The volume, V, is given by the formula: V
Hence volume is:
VP1
=
=
L1 × L2 × L3
L1 × L2 × L3
= 27.55 × 26.34 × 57.25
= 41544 mm 3
The uncertainty in the volume, ΔV, may be calculated using:
∆𝑉𝑉 = 𝑉𝑉 �
∆𝐿𝐿1 ∆𝐿𝐿2 ∆𝐿𝐿3
0.04
0.02
0.01
� = 41544 �
� = 99.12 𝑚𝑚𝑚𝑚3
+
+
+
+
𝐿𝐿1
𝐿𝐿2
𝐿𝐿3
27.55 26.34 57.25
There is a convention that uncertainties should be given with one significant figure
(and the final result rounded to the same place value as uncertainty) unless you
have convincing statistics that can be used to justify more! Therefore, if we were
determining volume, we should round our value of ΔV to 100 mm3 and round V to the
same place value as the uncertainty so that the final result for the volume is:
𝑉𝑉𝑃𝑃1 = 41500 ± 100 𝑚𝑚𝑚𝑚3
(But for the following calculation of density we should use the unrounded volume
result in our calculation or else it introduces error)
m
,
V
Let’s work in SI units so, m = 0.18689 kg
And V = 4.1544 × 10-5 m3
2. To calculate density, ρ
𝜌𝜌 =
=
0.18689 𝑘𝑘𝑘𝑘
= 4499 𝑘𝑘𝑘𝑘/𝑚𝑚3
4.1544 × 10−5 𝑚𝑚3
The uncertainty in the density is:
∆𝑚𝑚 ∆𝑉𝑉
0.01
99.12
� = 10.97 𝑘𝑘𝑘𝑘/𝑚𝑚3
∆𝜌𝜌 = 𝜌𝜌 �
+ � = 4499 �
+
𝑚𝑚
𝑉𝑉
186.89 41544
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SCI107 Physics
Again, we round uncertainty to one significant figure and the result for density is rounded to
the same place so after rounding we get:
𝜌𝜌𝑃𝑃𝑃𝑃 = (4500 ± 10) 𝑘𝑘𝑘𝑘/𝑚𝑚3
Do you agree with the above working? …………………………………………
What do you think this block is made from? ……………………………………………
Why do you make this claim? ………………………………………………………………..
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(2 marks)
3. EXPERIMENTAL AIM
The aim is to determine the identity of three blocks of material based on measurements of
their density including uncertainties.
4. EXPERIMENTAL PROCEDURE
You will be given three blocks made of different materials. Measure the blocks as
accurately as possible and record results in Table 2 so that you can calculate their volumes.
Weigh the blocks and calculate the density including uncertainty for each block based on
your measurements and record them in Table 3. You should compare your calculated
densities with values provided in Table 4 to identify the blocks.
Complete the tables with your correctly rounded values and show full working for one of
your blocks in the space provided or on separate pages.
PLEASE KEEP IT NEAT!
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5.
RESULTS
Table 2 – Measurements of block side lengths and mass
Block ID
Length 1
(mm)
Length 2
(mm)
(2 marks)
Length 3
(mm)
Mass
(g)
Average
Average
Average
Working / Calculations:
(2 marks)
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Fill in this table with your final correctly rounded results for each block:
Table 3 – Calculated volumes and densities including uncertainties and identification of
materials.
(2 marks)
Block ID /
Material
Example
Volume, V
(mm3)
41500
ΔV
(mm3)
100
Density, ρ
(kg/m3)
4500
Δρ
(kg/m3)
10
Material:
Material:
Material:
Material:
Table 4 - Approximate densities of common materials.
Material
Aluminium
Iron
Cork
Tin
Austenitic Steel
Silver
Perspex
Density (kg/m3)
2710
7900
160-250
7300
7600
10300
1180
Material
Brass
Copper
Wood
Titanium
Gold (22 carat)
Zinc
Glass
Density (kg/m3)
8450
8960
475
4506
17500
7100
2500
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6.
CONCLUSIONS
(2 marks)
Summarise your findings commenting on your measurements and the uncertainties and
identify the block materials including justifications for your selections.
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Evaluating Individual Participation
Note: The laboratory experimentation and preparation of reports during the lab classes are a
team effort. Normally, we expect that all group members will contribute equally so that each
receives the same result which is based on the quality of experimental results and responses.
In circumstances where team members are unable to contribute equally, individual marks can
be varied based on an agreed participation rate provided in the table below. Where all team
members contribute equally then the participation rate is 100% for each team member and
there is no need to fill in this table.
Team Member’s Name
Signature
Overall Participation (%)
If there is disagreement on the participation rate, class tutors will adjudicate based on their
observations and from discussions with the individual group members. Marks will be scaled
according to the agreed participation rates.
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