(a) When an object is in a circular orbit of radius r around the earth (mass m, the period of the orbit is T, given by Eq. (12.12), and the orbital speed is v, given by Eq. (12.10). Show that when the object is moved into a circular orbit of slightly larger radius r + I1r, where I1r« r, its new period is T + I1T and its new orbital speed is v - I1v, where I1r, I1T, and I1v are all positive quantities and (Hint: Use the expression (1 + x)n = 1 + nx, valid for Ixl « 1.)
(b) The International Space Station (ISS) is in a nearly circular orbit at an altitude of 398.00 k/n above the surface of the earth. A maintenance crew is about to arrive on the space shuttle that is also in a circular orbit in the same orbital plane as the ISS, but with an altitude of 398.10 k/n. The crew has come to remove a faulty 125- m electrical cable, one end of which is attached to the ISS and the other end of which is floating free in space. The plan is for the shuttle to snag the free end just at the moment that the shuttle, the ISS, and the center of the earth all lay along the same line. The cable will then break free from the ISS when it becomes taut. How long after the free end is caught by the space shuttle will it detach from the ISS? Give your answer in minutes.
(c) If the shuttle misses catching the cable, show that the crew must wait a time t = T2/I1T before they have a second chance. Find the numerical value of t and explain whether it would be worth the wait.