The radius R h and mass M h of a black hole are related by R h = 2 GM h / c 2 ,
ID: 1305749 • Letter: T
Question
The radius Rhand mass Mh of a black hole are related by Rh = 2GMh/c2, where c is the speed of light. Assume that the gravitational acceleration agof an object at a distance ro= 1.001Rh from the center of a black hole is given by ag = G M / r2 (it is, for large black holes). (a) What is ag at rofor a very large black hole whose mass is 1.67 1014 times the solar mass of 1.99 1030 kg? (b) If an astronaut with a height of 1.72 m is at rowith her feet toward this black hole, what is the difference in gravitational acceleration between her head and her feet?
Explanation / Answer
M = 1.67*10^14*Msun
= 1.67*10^14*1.99*10^30
= 3.3233*10^44 kg
Rh = 2*G*M/c^2
= 2*6.67*10^-11*3.3233*10^44/(3*10^8)^2
= 4.925869*10^17 m
a) ro = 1.001*Rh = 4.93*10^13 m
a = G*M/ro^2
= 6.67*10^-11*3.3233*10^44/(4.93*10^17)^2
= 0.09117 m/s^2 <<<<<<<<<<-Answer
b)
ah = G*M/ro^2
= 6.67*10^-11*3.3233*10^44/(4.93*10^17)^2
= 0.09117 m/s^2
at = G*M/(ro 1.72)^2
= 6.67*10^-11*3.3233*10^44/(4.93*10^17 + 1.72)^2
= 0.09117 m/s^2
ah - at = 0 (approximately)<<<<<<<<<<-Answer
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