Mark this Report - Uncertainty Lab
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Toronto Metropolitan University *
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Course
211
Subject
Industrial Engineering
Date
Dec 6, 2023
Type
Pages
9
Uploaded by CaptainWasp3881
Material Identification by Density
Course: PCS-211 Section 7
Instructor: Dr. Eric Da Silva
TA: Diana Ha
Date of Submission: September 17, 2023
Performed by: Ammar Siddiqui and Shayan Hajizad
Introduction
The purpose of this lab was to identify an unknown material which was given in the form
of a cuboid block. The volume of the material was measured using three distinct methods. Each
different method yielded three different volume values. These volumes were used to calculate
three different densities. By comparing the average density to a table of known values, the
unknown material was identified. This lab also accounted for the uncertainty of each measuring
method. The uncertainties were utilized to determine the consistency of the results.
Theory
There were two core physics concepts investigated in this lab, uncertainty in physics, and
density of materials.
Uncertainty
In physics, it is important to show that the tool used has some sort of uncertainty associated with
it. When expressing a measured value, x, then the measured value should be written in the
following way,
,
𝑥 ± ∆𝑥
where
is the associated uncertainty.
∆𝑥
Any measuring device is limited by the process it was manufactured. No process of
manufacturing is perfect. Due to this, when reporting measurements taken with instruments, the
associated uncertainty is the smallest increment of that tool divided by two.
Calculating the mean value of measurements makes sure that the results are consistent. In
this lab, the mean value
of the densities is calculated using the following formula,
(µ)
(1)
µ =
??? ?? ???????? ??????
?????? ?? ??????
When performing calculations with uncertain measurements, in this case
addition/subtraction followed by multiplication/division the following two rules are followed,
(2)
∆? =
∆?
2
+ ∆?
2
(3)
∆?
?
=
(
∆?
?
)
2
+ (
∆?
?
)
2
where
,
are the measurements and
is the result of the operation.
,
, and
are the
? ?
?
∆? ∆?
∆?
associated uncertainties with their respective measurements.
Density of Materials
Density is a measure of how much mass is present in a given amount of volume. The SI unit of
density is kg/m
3
(kilograms per cubic meter). The closer the particles in a substance are packed
together, the higher its density. Materials with uniform properties exhibit clearly defined
densities, making them distinguishable based on their density. The quality of uniform materials
having unique densities can be used to ascertain its identity. The density of a material (ρ) can be
determined by using the formula,
ρ =
,
(4)
?
?
where
stands for the measured volume of the material and
stands for the measured mass of
?
?
the material
(1)
.
Procedure
A glass beaker, a graduated cylinder, a standard metre stick, a vernier caliper, a triple beam
balance, and one unknown material were used in the following procedure.
The mass of the unknown material was measured using the triple beam balance. The counter
masses were adjusted so that the beam was balanced and a reading of the measurement was
recorded. The uncertainty of the triple beam balance was
0.05g.
±
The metre stick was then used to measure the dimensions of the unknown material, including the
length, width, and height. The uncertainty of the metre stick was
0.1 cm. These values were
±
later used to calculate the volume of the block.
Next, the vernier caliper was used to measure the dimensions of the unknown material, including
the length, width, and height. The uncertainty of the caliper was
0.005 cm. These values were
±
later used to calculate the volume of the block.
Lastly, the method of water displacement was used to measure the volume of the unknown
material. A beaker was used to measure out 300 mL of water. The unknown material was then
placed inside of the beaker. Next, the water was poured out of the beaker into a graduated
cylinder until the reading on the beaker was 300 mL. The graduated cylinder gives a more
accurate reading of the volume displaced with an uncertainty of
1 mL.
±
Results and Calculations
It is known that
when dealing with water, and Volume,
(2)
.
1? = 1??
3
? = 𝐿 * ? * 𝐻
Table 1:
Summary of dimensions and mass of block, including calculations for volumes, density,
and average density. [See Appendix A for data.]
Metre
Stick
Uncertainty
Vernier
Caliper
Uncertainty
Volume
Displacement
Uncertainty
Volume
(cm
3
)
12
0.6
10.8
0.05
12
0.5
Density
(g/cm
3
)
9.5
0.4
11.0
0.05
9.9
0.4
Average
Density
(g/cm
3
)
10.
1
Sample Calculations (Metre Stick):
●
Volume
? = 𝐿 * ? * 𝐻
? = 3. 1 * 3. 1 * 1. 3
?
=
12. 493 ??
3
●
Uncertainty
Using equation (3)
∆? =
(
0.05
3.1
)
2
+ (
0.05
3.1
)
2
+ (
0.05
1.3
)
2
* 12. 493
∆?
=
0. 55864546 ??
3
Therefore
? = 12 ± 0. 6 ??
3
●
Density
Using equation (4)
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