+1 862 207 3288 Error Propagation: Volumes
Linearized plot: Density
Read the Lab Manual and complete the PreLab at the end of this document. PreLab is
due at the beginning of the lab.
Introduction
In this lab you will practice error analysis by measuring volumes of a few regular shapes
objects and graph linearization to define the density of the clay. You need to understand the
“ Data Analysis” and “Data fits” document s to successfully complete this lab activity . The
aim of the first part of the experiment is to understand how errors propag ate through
formulas . For example, the volume of a box is V = LWH. If we measure the width W, height
H, and l ength L many times to establish errors for each quantity ∆ W,∆ H,∆ L, what will the
error in the volume ∆ V be when we multiply the average values of the three quantities
together? Or in the other words how does the error propagate? The need to answer this type
of question arises in all areas of the physical sciences. (F or more details read: Philip
Bevington and D. Keith Robinson – “Data Reduction and Error Analysis for the Physical
Science”).
In the second part of the lab the density of the clay will be determined. The aim of the
second part of the lab activity is also to practice the error propagation and to practice build a
linearized graphs to determine the physics quantity.
The density is the mass per unit of volume:
V
m
= ρ (1)
The SI unit is kg/m
3
or g/cm
3
.
In this lab course, the partial derivatives approach will be used to propagate the error:
(2)
In the above equations Δx and Δy represents the standard errors (uncertainty/standard
deviation of the mean) of the quantity x and y accordingly. The (standard) error is also
called the standard deviation of the mean. It is obtained by calculating the standard
deviation σx
from a series of measurements of the quantity x first. The program Graphical
Analysis will calculate standard deviation σx
for you. Then use the equation for st. dev. of
the mean :
N
x
x
σ
σ = , (3)
where N is the total number of measurements
made of quantity x to calculate the standard
error .
The values of x and y are their mean
values.
All uncertainties in the final results should be
reported with one significant figure (unless it
equals to one, then two significant figures need
to be reported in the uncertainty ). The number
of significant figures to report in the mean value
is determine d by the uncertainty, which means
that the number of the decimal places in the
final mean value should be equal to the number
of the decimal places in the uncertainty.
2
2
2
⎟
⎠
⎞
⎜
⎝
⎛
Δ
∂
∂
+ ⎟
⎠
⎞
⎜
⎝
⎛
Δ
∂
∂
= Δ y
y
f
x
x
f
f
Equipment: plastic cup, a hollow cylinder, a bullet shaped object , plastic bag with a set of
clay spheres, balance, set of masses, verniers’ calipers, set of cubes with known densities,
measuring cylinder filled with water.
Procedur e
Part 1: Volumes
In the first part of the lab the volumes and their error s for a small plastic cup, a hollo w
cylinder (fin d the volume of solid material), triangle prism and a bullet shaped object will be
found. First, make sure everyone in the group learns to use the vernier calipers and
understands how to read the measurements with its scales. Use Vernier calipers to measure
relevant dimensions of the provided objects. The Vernier is precise to 0.05 mm, so many of
your measurements may give the same results. Take five measurements of each dimension.
Compute the mean and standard deviation of these readings for each dimension using
Graphical Analysis. Then f ind the volume of each of the provided objects and propagate the
erro r in each volume. Report final results in cm
3
. To calculate volume the mean values of
each dimension are used. The error in each dimension is its standa rd deviation of the mean
(3). The uncertainty in the volume propagates using equation (2). Note that the error in the
measurements for the bullet shaped object given to be 0.05 mm. Be sure you present all of
the volume calculations as well as error propaga tion equations in the Data Anal ysis section
in the lab report. The lab report expected to include all calculated values of volumes and
their errors. . Remember results must be tabulated and reported in the correct format.
Part 2: Density
Scientists work with “models” of physical phenomena. “Models” establish the relationship
between physical quantities. After the relationship is found it is expressed as a mathematical
equation. The relationship between mass and diameter of a sphere is one example of
“mode l”. This well -known cubic relationship is studied in the lab experiment.
Make one measurement of the mass of each of the provided clay spheres with nominal
diameters 1 -7 cm. Remember the error in mass measurements is written on the electronic
balance. Use vernier caliper to take 5 independent readings of a diameter of each of the
spheres. Roll the sphere after each measurement is taken. Record measuremetns and
calculate the mean value of each of the diameters.
Fill in table prepared in Logg er Pro/GA. First column is mass and the second one is mean
diameters . Remember to use the appropriate units. Third column is for 5 measurements of
the diameters done for one of the spheres of your choice.
Make mass vs diameter graph, apply the curve fit to identify the parameters. Analyze the
parameters to check if the cubic relationship between the mass and diameter of a sphere
confirmed. Then present this cubic relationship as a linear graph diameter vs mass. In this
experiment diameter is less accurate measurement that’s why it needs to be plotted on the
vertical access and mass is on horizontal one. To present diameter vs mass as a linear
relationship the diameter cube needs to be plotted on y -axis. Create calculated column to
find diameter cube. Using the slope of the graph: diameter cube vs mass calculate the
density of the clay and its uncertainty.
The other approach to linearize graph is to make natural log graph ln (D) vs ln(m) . Two
new calculated columns ln (d) and ln (m) need to be created. Using the slope of the natural
log graph find the power of the diameter and its uncertainty to confirm the model of the
mass and sphere’s diameter relationship. Using y-intercept find the density of the clay.
In the last part of the experiment for the selected sphere of your choice calculate the density
of the clay using the density definition as mass per unit of volume (equation 1) . Use
statistics to obtain the mean value of the diameter for the selected sphere and its error. When
the density found propagat e its err or to be reported in the final results.
At the end of the lab section students are expected to have 4 graphs for further analysis.
TA will sign graphs and all the experimental data collected during the experiment: either
handwritten in the notebook or all typed in the Logger Pro. These are not the final result of
the lab yet. It is printed for further analysis and calculations to be done outside the class
time.
The lab report expected to include the found values of the power for the mass diameter
rela tionship and the values of clay density with its error found with three different ways.
Remember results must be tabulated and reported in the correct format.
Please review the Course Syllabus and recorded videos for the details about what
should be included in the Lab report in general and in the discussion section
specifically.
Prelab_Volumes and Density
PHY 122
Name: Section number:
Volumes:
1) This question will ask you to determine the area of a flat piece of metal according
to the data in the table below. The answer will need to be correctly state as (mean
value± σm)
units, for example: Area = 3.2 ± 0.2 cm
2
.
Note: 1) uncertainty has to be reported with only one significant figures; 2) the mean value should be
reported with the number of the decimal places that matches the number of the decimal places in the
uncertainty. (Hint: watch Introduction video: Error propagation, results. discussion)
Length (cm) Width (cm)
8.4 6.3
8.6 5.9
8.3 6.2
8.7 6.5
8.5 6.3
8.8 6.1
(2 points)
a) Determine the mean, standard deviation, and the standard deviation of the mean
for the length measurements. Show all your work (equations and calculations).
Note: Equations are provided in the ppt presentation “Results and Uncertainties_PHY 122” that is posted on
the Bb.
(1 points)
b) Determine the mean, standard deviation, and the standard deviation of the mean
for the width measurements.
Note: for this part of the question only, you can use scientific calculator or Excel to find the standard
deviation and the standard deviation of the mean.
(2 points)
c) Using the mean values of the length and width determine the area of the plate and
its uncertainty using the error propagation rules. Show all your work (equations and
calculations). Properly report the area and its uncertainty.
Note: The rules of error propagation were presented in the first class. Read also the reference materials
posted on the BB. (Hint: watch introduction video: Error Propagation)
5
Density
In this part of the experiment you will measure the mass and diameter of several clay
spheres to find the density of the clay.
1) (5 points) For a spherical object, what is the equation to find mass in terms of
density of the material, its diameter and any necessary constants? How should
the data of mass and diameter be plotted to obtain a linear graph, other than
log- log graph? Make a sketch of this linear graph with both axes labeled. How
is the density of the sphere related to the slope of the graph? Write an equation
for the density with the slope as a variable.
(Hint: watch introduction video: “Density”)
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