(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

The radius R_{h}and mass M of a black hole are related by R_{h}= [itex]\frac{2GM}{c^{2}}[/itex], where c is the speed of light. Assume that the gravitational acceleration ag of an object at a distance r_{o}= (1 + ε)R_{h}from the center of a black hole is given by a_{g}= [itex]\frac{GM}{r^{2}}[/itex], where ε is a small positive number. (This formula is valid for certain values of ε. Exactly how large or small ε must be depends on the size of the black hole.) (a) What is a_{g}at r_{o}for a mass M black hole, to first order in ε? (b) If an astronaut with a height of H is at r_{o}with her feet toward this black hole, what is the difference in gravitational acceleration between her head and her feet? Assume H << R_{h}. Express your answers in terms of M, G, H, c, and ε.

2. Relevant equations

a_{g}= [itex]\frac{GM}{r^{2}_{g}}[/itex]

R_{h}= [itex]\frac{2GM}{c^{2}}[/itex]

r[itex]_{0}[/itex] = (1+[itex]\epsilon[/itex])R[itex]_{h}[/itex]

3. The attempt at a solution

a[itex]_{0}[/itex] = [itex]\frac{GM}{r^{2}_{0}}[/itex]

= [itex]\frac{GM}{[(1+\epsilon)^{2}R_{h}]^{2}}[/itex]

= [itex]\frac{GM}{(1+\epsilon)^{2}(\frac{2GM}{c^{2}}^{2}}[/itex]

=[itex]\frac{c^{4}}{4(1+\epsilon)^{2}GM}[/itex]

part b)

a_{g}= [itex]\frac{GM}{r^{2}_{g}}[/itex]

differentiating gives da_{g}= [itex]\frac{-2GM}{r^{3}}[/itex]dr

= [itex]\frac{-2GM}{r^{3}}[/itex]dr

=[itex]\frac{-2GM}{[(1+\epsilon)R_{h}]^{3}}[/itex]

=[itex]\frac{-2GMH}{(1+\epsilon)^{3}\frac{2GM}{c^{2}}^{3}}[/itex]

=[itex]\frac{-Hc^{6}}{4(1+\epsilon)^{3}GM^{2}}[/itex]

I'm not sure why my answers are wrong, but my hint says

The gravitational acceleration depends on the mass of the black hole and the distance from its center. Here the distance is a not fixed value but a certain multiple of the radius Rh. Use Taylor series or the binomial expansion to determine the leading order dependence on ε.

Which doesn't help me. I'd appreciate any hints/explanations :)

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Homework Help: Accceleration due to gravity of a Black Hole

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