Calculating the Maximum Mass of a Black Hole at a Given Distance

In summary: I appreciate it. I see, i would go with the other method to avoid calculating errors, but thank you very much. I appreciate it.
  • #1
i_hate_math
150
2

Homework Statement


The radius Rh of a black hole is the radius of a mathematical sphere, called the event horizon, that is centered on the black hole. Information from events inside the event horizon cannot reach the outside world. According to Einstein's general theory of relativity, Rh = 2GM/c2, where M is the mass of the black hole and c is the speed of light.

Suppose that you wish to study a black hole near it, at a radial distance of 43Rh. However, you do not want the difference in gravitational acceleration between your feet and your head to exceed 10 m/s2 when you are feet down (or head down) toward the black hole. (a) Take your height to be 2.0 m. What is the limit to the mass of the black hole you can tolerate at the given radial distance? Give the ratio of this mass to the mass MS of our Sun. (b) Is the ratio an upper limit estimate or a lower limit estimate?

Homework Equations


Rh = 2GM/c^2
Fg=GmM/R^2
Fg=m*a

The Attempt at a Solution


Using the two expressions of the gravitational attraction: a=GM/(R+∂R)^2=10
and we have R=43*Rh=86GM/c^2
and my height is ∂R=2 metres
now sub the expression for R in, and do the massive calculation I got M=1.0125*10^32 kg
the sun has mass Msun=1.99*10^30 kg
thus the ratios is M/Msun=50.879...
This is not however the correct answer
Please let me know where I might have gone wrong
 
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  • #2
i_hate_math said:
a=GM/(R+∂R)^2
Note the first word below:
i_hate_math said:
difference in gravitational acceleration
 
  • #3
haruspex said:
Note the first word below:
Oh I see, so it should be a=GM/(∂R)^2 instead of the whole thing
 
  • #4
i_hate_math said:
Oh I see, so it should be a=GM/(∂R)^2 instead of the whole thing
No, it's the difference between two accelerations. Find the expressions for the accelerations and THEN take the difference.
 
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  • #5
haruspex said:
No, it's the difference between two accelerations. Find the expressions for the accelerations and THEN take the difference.
a(head)=GM/(R+∂R)^2
a(feet)=a=GM/(R)^2
and a(difference)=10=GM/(R+∂R)^2-GM/(R)^2
does that look right?
 
  • #6
i_hate_math said:
a(head)=GM/(R+∂R)^2
a(feet)=a=GM/(R)^2
and a(difference)=10=GM/(R+∂R)^2-GM/(R)^2
does that look right?
Yes, except there's a good chance you have a sign error.
 
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  • #7
haruspex said:
Yes, except there's a good chance you have a sign error.
Thanks a lot. I just found another way to do this(should've read the textbook thru thoroughly), they used the differential eq, a=GM/(R)^2 becomes da=-GM/(R)^3 dR where dR=h=2 and subbing in everything this is much easier
 
  • #8
i_hate_math said:
Thanks a lot. I just found another way to do this(should've read the textbook thru thoroughly), they used the differential eq, a=GM/(R)^2 becomes da=-GM/(R)^3 dR where dR=h=2 and subbing in everything this is much easier
The next step from post #5 (after fixing the sign error) would have been to use the binomial expansion of (R+∂R)-2 and make the approximation for small ∂R/R. That comes to the same as taking the differential.
 
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  • #9
haruspex said:
The next step from post #5 (after fixing the sign error) would have been to use the binomial expansion of (R+∂R)-2 and make the approximation for small ∂R/R. That comes to the same as taking the differential.
I see, i would go with the other method to avoid calculating errors, but thank you very much.
 

Related to Calculating the Maximum Mass of a Black Hole at a Given Distance

1. What is a black hole?

A black hole is a region in space where the gravitational pull is so strong that nothing, including light, can escape from it. This is due to the immense amount of mass packed into a small space.

2. How is a black hole formed?

Black holes are formed when a massive star dies and collapses under its own gravity. As the star runs out of fuel, it can no longer produce energy to counteract the force of gravity and it collapses in on itself.

3. How does gravity work in a black hole?

In a black hole, gravity is incredibly strong due to the concentration of mass. As objects get closer to the black hole, the gravitational pull increases exponentially. At the event horizon, the point of no return, the gravitational pull is so strong that not even light can escape.

4. Can anything escape a black hole?

Once an object crosses the event horizon of a black hole, it cannot escape. However, there are certain theories that suggest that particles may be able to escape through quantum effects, but this has not been proven.

5. How do black holes affect the space around them?

Black holes have a strong gravitational pull, which can distort the fabric of spacetime. This can cause the space around a black hole to become warped and create intense gravitational waves, which can be detected by advanced technology.

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