Calculating the resting position of a hanging mass
Consider a mass hanging from a string and pulley
arrangement shown in the accompanying diagram. The laws of physics decree
that that the mass will come to rest at a position which minimizes its
energy. For the configuration being considered, this simply means that
the rest configuration will be that configuration in which the mass is
as low as possible. What will the angle a be in this rest position?
Begin your analysis of this problem by considering cases of specific values
for A and B. For example, start with
A = 2 and B
= 3. Why should you be especially careful in considering the case
A
= 1 and B = 2? Use the observations you have made in studying
these specific cases to deal with the following problems:
-
What is the angle a when the configuration is at rest if A =
2?
-
What is the angle a when the configuration is at rest if A =
2B?
-
What is the angle a when the configuration is at rest when A
and
B
are arbitrary?
Make an effort to present your answer to the final question in an understandable
form. The most obvious approach to a problem is not always the one that
results in the most direct path to a solution. For example, explore the
results you achieve by using different choices of
independent variable. For example, instead of using the angle a as
the independent variable, try x = cos(a). Instead of the parameters A and
B try using A and r= B/A instead.
Instructor
Juris
Steprans
email address: steprans@mathstat.yorku.ca
Department of Mathematics and Statistics.
Ross 624 South, ext. 33952
York University
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