Finding the Rest Position of a Disk Suspended on a Wire

(due date October 2)

Imagine an elastic wire suspended between two poles of heights A and B. Imagine also, that a heavy disk of radius R is suspended on this wire on is edge. If the disk is so heavy that the wire stretches to the ground then the disk will come to rest at a position which minimizes the length of the wire. (This minimizes the energy stored in the wire.) Find this position.

Consult the diagram. Your first goal should be to define a function of the variable x which yields the length of the wire for each value of x. Do this in steps. Notice that the length of the wire is composed of two straight pieces and a circular piece. First find functions for LH and RH. Then find the lengths of the straight pieces of wire L and R. Then use this to find the length of the circular piece. Each individual function should be relatively simple; but their composition will not.


Note that the line labeled LH(x) and the radius that connects the center of the circle to the line labeled R(x) does not necessarily form a 180 degree angle. It just looks like it in this picture and we cannot assume that this will be true in order to calculate alpha(x) and beta(x). Let us enlarge the picture and we can see that in general those lines that pass through the center of the circle are not straight.

enlarged circle

Even enlarged these lines look as though they are straight but if they were then we must have that angle s = angle t. We can calculate that angle s = arctan(R(x)/radius) and angle t = arctan(L(x)/radius), but if angle s = angle t then R(x) = L(x) which is clearly not always true.

How do we find alpha(x) and beta(x)? We really only need to find alpha+beta. We know that alpha + beta + angle s + angle t + angle u = 2 Pi. We already have a formula for angle s and angle t , if we can find a formula for angle u then we can solve for alpha + beta. Draw a horizontal line that connects the wall labeled A, the wall labeled B and the center of the circle. This makes two more angles v and w (as in the picture below) and we can easily write down an equation for them because they each form an angle of a right triangle where two legs are known. Since we know that angle u + angle v + angle w = Pi, then we have solved for angle u and hence for alpha + beta.

enlarged circle with line


Indeed, the function will be so complicated that it will be too difficult to expect an algebraic solution to the problem. Instead, try a numeric approach. Substitute specific numeric values for the parameters, A, B and C and the radius. Look at the graph which yields the length of the wire as a function of the argument x . This should allow you to guess an interval which contain the minimum value. Then use fsolve to find a good approximation to the place where the derivative is zero in this interval, and, hence, where the minimum occurs.

Using this technique experiment with different values of the parameters and make conjecture about the qualitative behaviour of the system. Questions to consider are

  1. If the sides are equal, does the minimu occur in the middle as expected?
  2. If one side is twice as long as the other, does the disk come to rest twice as close to the short side as to the other?
  3. What if one side is three times as long as the other? Generalize?
  4. What happens if the radius is zero? Can you solve the problem algebraically in this case?
  5. Does the position of the disk change as the radius is made non-zero? How?
  6. Is the zero radius approximatiom to the problem a good one? What conditions should the parameters satisfy for the approximation to be very good? For what range is it not good?
When you use Maple to answer these questions you will have to substitute values for the parameters to get a solution, your answers however should reveal the patterns you discover for any values of A, B, C and r. Make sure that you answer the questions in a general manner using adjectives such as closer, taller, larger, smaller (not numerical answers since they depend on), but also make sure that your answers are precise and correct (what is true for one value of A, B and C is not true for all values).

Instructor

Mike Zabrocki
Email address: zabrocki@yorku.ca
Department of Mathematics and Statistics
York University
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