Halliday, Resnick & Walker — Gravitation: Problem 39

39 The radius \(R_h\) and mass \(M_h\) of a black hole are related by \(R_h = 2GM_h/c^2\), where \(c\) is the speed of light. Assume that the gravitational acceleration \(a_g\) of an object at a distance \(r_o = 1.001R_h\) from the center of a black hole is given by Eq. 13-11 (it is, for large black holes). (a) In terms of \(M_h\), find \(a_g\) at \(r_o\). (b) Does \(a_g\) at \(r_o\) increase or decrease as \(M_h\) increases? (c) What is \(a_g\) at \(r_o\) for a very large black hole whose mass is \(1.55 \times 10^{12}\) times the solar mass of \(1.99 \times 10^{30} \text{ kg}\)? (d) If an astronaut of height \(1.66 \text{ m}\) is at \(r_o\) with her feet down, what is the difference in gravitational acceleration between her head and feet? (e) Is the tendency to stretch the astronaut severe?