Space visits for everyone? An alternative way to visit space has been proposed: A space station of mass \(m\) is tethered to one end of a long cable and the other end of the cable is attached to a point on the earth's equator, a distance \(R\) from the center of the earth. In the rotating frame of the earth, three forces act on the station: the centrifugal pseudoforce \(m r \omega^{2}\) outward, and earth's gravity \(G M m / r^{2}\) and the cable tension \(T\) inward, where \(r\) is the distance of the station from the center of the earth. As one goes to larger radii the centrifugal pseudoforce grows while gravity decreases, so there must be a radius \(r_{0}\) where these forces balance, so that the station will remain in place. Then one could ride an elevator up the cable and get some spectacular views and experience zero \(g^{\prime} s\) without using any rocket fuel.

(a) Assuming \(T=0\), find the distance \(r_{0}\) from the earth's center to the station where the forces balance, in terms of \(G, M\), and \(\omega\), the angular velocity of the earth's rotation.

(b) Of course the cable will require some tension \(T(r)\) if an elevator is to travel up and down along it. This might be achieved by placing the space station at a somewhat greater distance \(r_{0}+\Delta r_{0}\) from the earth's center, requiring a positive downward tension force for it to stay in place. Let the cable have uniform mass per unit length \(\lambda\). Then show that the tension \(T(r)\) obeys the equation

(c) At what radius is the tension in the cable a maximum?

(d) Find the tension \(T_{s}\) in the cable just where it is attached to the space station. Assume here that \(\Delta r_{0}

(e) Find a general expression for the tension \(T(r)\) anywhere along the cable, in terms of \(T_{s}, \lambda, \omega, r_{0}, G, M\), and \(r\).

(f) In particular, what is the cable tension at \(r=r_{0}\) ? (g) At what radius is the cable tension a minimum? (h) Find the minimum value of \(\Delta r_{0}\) required, in terms of other given parameters, so that the cable will never have a negative tension anywhere along its length, because that would cause it to buckle.

Transcribed Image Text:

dT = dr 1 (GM -rw)