MATH_2250_001_FALL_2023_LAB10
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School
University of Utah *
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Course
2250
Subject
Mathematics
Date
Jan 9, 2024
Type
Pages
4
Uploaded by tinalove801
University of Utah
Fall 2023
MATH 2250-001 Lab 10 Specification
TA: Devanshi Merchant
Subject to Change; Last Updated: 2023-11-15
The printed version of this document may contain errors or
outdated information. Always refer to the digital version on
Canvas for the most accurate and up-to-date content.
1
Background
We see some applications of the oscilator, in Physics.
2
What to Do
Solve the following problems.
1. Consider the following harmonic oscillator with two masses, three springs and
no dashpots (nor any other friction forces):
Using
Newton’s Second Law of Motion
together with
Hooke’s law
that states
that the force exerted by a spring is proportional to how much it’s expanded or
contracted, with the proportionality constant being the spring constant, derive
the system of ODE’s corresponding to the harmonic oscillator. The unknowns
of the system of ODE ought to be both the displacement
x
1
of the first mass
from its equilibrium position and the displacement
x
2
of the second mass from
its equilibrium position.
1
University of Utah
Fall 2023
2. Consider three
100
-gal fermentation vats
T
1
,
T
2
and
T
3
connected as follows:
Suppose the mixtures in each tank are kept uniform by stirring and that the
mixture circulates in the direction indicated above between the tanks at the
rate of
10
gallons per minute. Derive the system of ODEs describing the change
of amounts of alcohol (in pounds) in each tank. The system ought to have three
scalar functions as unknowns, one for the amount of alcohol in each tank.
3. Consider two particles each of mass
m
attached to a string under constant
tension
T
in the following configuration:
Assume that both of the particles oscillate vertically along the
y
axis with am-
plitudes small enough that the sines of the angles shown are well-approximated
by their tangents. Using
Newton’s Second Law of Motion
, derive equations of
motion for these two particles.
The unknowns of the ODE ought to be two
scalar functions, each representing the vertical displacement of one of the par-
ticles. Note that since there is no horizontal motion the total forces acting on
either one of the masses ought to cancel out.
Devanshi Merchant
2
u1474124@utah.edu
University of Utah
Fall 2023
3
ChatGPT Regulations
This section is in case you decide to use ChatGPT in this lab worksheet. If you will
not be using ChatGPT you may skip this section.
3.1
ChatGPT Versions
•
You may use either the GPT3.5 (freely available with a ChatGPT account) or
GPT4 (available with a ChatGPT Plus account).
•
Turn off all Custom Instructions before you start a chat. If you don’t, it will
be apparent in the archived version that you didn’t.
3.2
Chat Guidelines
•
Your first message in any given chat must be the following guardrails paragraph;
you may copy and paste it:
Hello. I am working on a differential equations and linear algebra
problem as part of a university class. My instructor has permitted
the use of ChatGPT, but only under specific guidelines to
encourage independent critical thinking. Please assist me by asking
probing questions, encouraging reflection, and providing general
insights about the concepts involved. Do not offer direct hints,
strategies, solutions, or step-by-step guidance. I seek to understand
the underlying principles and want to develop my own approach to
the problem. Your role is to facilitate my learning process without
directly leading me to the answer. Thank you!
•
You may copy and paste parts of this specification document, as well as parts
of the textbook or other sources.
•
You may not ask ChatGPT to write for you the solution for any one of the
problems in complete detail.
3.3
Archiving Chats
•
Once you are done with a chat with ChatGPT, click the "Share chat" icon on
the top righthand corner; see the
documentation
for details. In your chat don’t
include any personal information, and keep your user name hidden when you
are creating a link for anonymity.
•
Next you will use
Wayback Machine
to take a "snapshot" of your chat, see the
documentation
for the "Save Page Now" feature. You do not need an
Internet
Archive
account to do this, but having such an account (which is free) would
provide you with further options.
Devanshi Merchant
3
u1474124@utah.edu
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University of Utah
Fall 2023
•
You have to take a snapshot of each one of your relevant chats separately, and
share the links to their archived versions in the form for this problem set.
•
To see an example of the outcome, see the Acknowledgements section in the
course syllabus. Note that the staff did use Custom Instructions in this case.
4
How to Submit
•
Step 1 of 2:
Submit the form at the following URL:
https://forms.gle/mVouPWxivgJ8kf6w5
.
Your submission on Gradescope will receive a zero if you don’t submit this form
before the deadline.
•
Step 2 of 2:
Submit your work on Gradescope at the following URL:
https://www.gradescope.com/courses/565427/assignments/3587550
,
see the Gradescope
documentation
for instructions. As you upload your work,
please separate it into individual problems.
This will expedite the response
and feedback process. Make sure that you are submitting your work under the
correct assignment on Gradescope.
5
When to Submit
This lab worksheet is due on November 23, 2023 at 11:59 PM.
Devanshi Merchant
4
u1474124@utah.edu