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.
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.
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