1. Coffee cup and mass (Ex. 6.27), 20 marks. (a) (10 marks) A coffee cup of mass M is connected to a mass m by a string. The coffee cup hangs over a frictionless pulley of negligible size, and the mass m is initially held with the string horizontal, as shown below. The mass m is then released. Find the equations of motion for r (the length of string between m and the pulley) and (the angle that the string to m makes with the horizontal). Assume that m somehow does not run into the string holding the cup up. (b) (10 marks) The coffee cup will initially fall, but it turns out that it will reach a lowest point and then rise back up. Write a program (see, for example, Morin, Section 1.4) that numerically determines the ratio of the r at this lowest point to the r at the start, for a given value of m/M. To check your program, a value of m/M=1/10 yields a ratio 0.208. At what time t is this minimum achieved? What is the corresponding value of (t*)? You can write your program in Maple, as suggested in the text (widely available at SFU) or in any other language that you may know (Python, Mathematica, Matlab, R, C, Java, Igor, etc.). Please write the program to use an explicit loop, in the manner suggested by Morin 1.4. If you have not yet programmed, please allow enough time for this part. Being able to do simple programming tasks is an essential "life skill" for a physicist. If you follow the model in Morin 1.4, this should not be a hard assignment. Ask your TA or instructor for assistance, if you are stuck on the programming part. Ask your friends for help in getting one of these languages set up on your own computer (or on accessing an installation on a University machine). m M Finally, note that most technical languages have "professional" routines to inte- grate differential equations. These can be very sophisticated. For example, they can adapt their time step, taking large steps in regions where the solution does not change much and slowing down (small time steps) in regions of rapid varia- tion. For simple problems such as these, it would probably take you more time to learn to use those routines properly (they have a lot of set up). Here, please use the simple way - it is easier to get to work and much more educational, even if other methods are ultimately more "efficient."
1. Coffee cup and mass (Ex. 6.27), 20 marks. (a) (10 marks) A coffee cup of mass M is connected to a mass m by a string. The coffee cup hangs over a frictionless pulley of negligible size, and the mass m is initially held with the string horizontal, as shown below. The mass m is then released. Find the equations of motion for r (the length of string between m and the pulley) and (the angle that the string to m makes with the horizontal). Assume that m somehow does not run into the string holding the cup up. (b) (10 marks) The coffee cup will initially fall, but it turns out that it will reach a lowest point and then rise back up. Write a program (see, for example, Morin, Section 1.4) that numerically determines the ratio of the r at this lowest point to the r at the start, for a given value of m/M. To check your program, a value of m/M=1/10 yields a ratio 0.208. At what time t is this minimum achieved? What is the corresponding value of (t*)? You can write your program in Maple, as suggested in the text (widely available at SFU) or in any other language that you may know (Python, Mathematica, Matlab, R, C, Java, Igor, etc.). Please write the program to use an explicit loop, in the manner suggested by Morin 1.4. If you have not yet programmed, please allow enough time for this part. Being able to do simple programming tasks is an essential "life skill" for a physicist. If you follow the model in Morin 1.4, this should not be a hard assignment. Ask your TA or instructor for assistance, if you are stuck on the programming part. Ask your friends for help in getting one of these languages set up on your own computer (or on accessing an installation on a University machine). m M Finally, note that most technical languages have "professional" routines to inte- grate differential equations. These can be very sophisticated. For example, they can adapt their time step, taking large steps in regions where the solution does not change much and slowing down (small time steps) in regions of rapid varia- tion. For simple problems such as these, it would probably take you more time to learn to use those routines properly (they have a lot of set up). Here, please use the simple way - it is easier to get to work and much more educational, even if other methods are ultimately more "efficient."
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Transcribed Image Text:1. Coffee cup and mass (Ex. 6.27), 20 marks.
(a) (10 marks) A coffee cup of mass M is connected to a mass m by a string. The
coffee cup hangs over a frictionless pulley of negligible size, and the mass m is
initially held with the string horizontal, as shown below. The mass m is then
released. Find the equations of motion for r (the length of string between m and
the pulley) and (the angle that the string to m makes with the horizontal).
Assume that m somehow does not run into the string holding the cup up.
(b) (10 marks) The coffee cup will initially fall, but it turns out that it will reach a
lowest point and then rise back up. Write a program (see, for example, Morin,
Section 1.4) that numerically determines the ratio of the r at this lowest point to
the r at the start, for a given value of m/M. To check your program, a value of
m/M=1/10 yields a ratio 0.208. At what time t is this minimum achieved?
What is the corresponding value of (t*)?
You can write your program in Maple, as suggested in the text (widely available
at SFU) or in any other language that you may know (Python, Mathematica,
Matlab, R, C, Java, Igor, etc.). Please write the program to use an explicit loop,
in the manner suggested by Morin 1.4.
If you have not yet programmed, please allow enough time for this part. Being
able to do simple programming tasks is an essential "life skill" for a physicist. If
you follow the model in Morin 1.4, this should not be a hard assignment. Ask
your TA or instructor for assistance, if you are stuck on the programming part.
Ask your friends for help in getting one of these languages set up on your own
computer (or on accessing an installation on a University machine).
m
M
Finally, note that most technical languages have "professional" routines to inte-
grate differential equations. These can be very sophisticated. For example, they
can adapt their time step, taking large steps in regions where the solution does
not change much and slowing down (small time steps) in regions of rapid varia-
tion. For simple problems such as these, it would probably take you more time
to learn to use those routines properly (they have a lot of set up). Here, please
use the simple way - it is easier to get to work and much more educational, even
if other methods are ultimately more "efficient."
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