I-A:      Work out the last assigned problem in the Classical Mechanics section of the Kent State Physics Candidacy Exam of Tuesday September 1, 1998.

I-B:      The name Beanstalk has been applied to a "Space Tower" scheme devised in 1960 by a Russian scientist, Yuri Artsutanov. Follow the link to find out what is the general idea and form of a Beanstalk.

1. Find the expression for escape velocity at a distance R away from a large body of mass M, and the radius of the circular orbit corresponding to a period T.
2. Suppose that the beanstalk counterweight has mass m, and the whole system orbits synchronously about the large body with period T. (T is of course one day if we consider a beanstalk on earth). Given that we require the counterweight to move at escape velocity, find the expression for the tension in the cable at the point of connection to the counterweight.
3. Using a qualitative or quantitative argument, explain why a tapered cable as suggested by Jerome Pearson makes sense, and why the thickness must be greatest at the radius of the geosynchronous orbit.
4. The most likely scenario for a practical beanstalk is for transportation to and from space colonies in independent orbit about the sun. Suppose that the colonies are structures of radius 1 km, rotating so as to simulate terrestrial gravity at this radius, and also suppose that launch speeds up to 3 km/s are needed. How long should the beanstalk be?

I-C:      In 1901, Allo Diavolo introduced the circus act depicted below. Assuming a circular loop of radius R, and taking the approximation that Allo's pedaling just compensates for friction, (1) find the expression for the minimum height h' needed at the top of the ramp. (2) What is Allo's horizontal component of velocity on the ramp as a function of his horizontal distance s from the bottom, given that the ramp height is described by a known function h(s)? (3) Starting from this expression, find the time taken to reach the bottom of the ramp if   h(s)  =  a s , where a is a constant.