During Friday's Late Show with Steven Colbert broadcast, host Steven Colbert asked astrophysicist Neil DeGrasse Tyson who would win the force of the black hole in the center of our galaxy or the force of you family asking/guilting you to join them during the holidays. Deciding on the winner is a tough one, but at minimum we can compare the impact of gravity's force on you as a person. According to Stephen's comment the black hole has a mass of 8.2^1036 kg, this is a number that doesn't quite sound right to me, seeing as 8.2^1036 would have hundreds of zeros in it. According to wikipedia, the black hole at the center of our galaxy is estimated to mass in at somewhere between 1.0*10^6 and 4.1*10^6 solar masses or, 1 million to 4.1 million solar masses. A solar mass is about 2*10^30 kg, using multiplication, (2.0*10^30)*(1.0*10^6) the lowest estimated mass is 2.0*10^36 kg at the upper bound 8.2*10^36 kg. It looks like some staffer didn't bother to properly note the math for Mr. Colbert. Now the next question is how far away is this super massive blackhole, well I am lazy and I am going to source wikipedia, where the distance between us and the black hole at the center of our galaxy is about 26,000 lightyears. 26,000 light years works out to about 2.46 *10^17 kilometers.
Now we need to do some math about the amount of gravitational force that your family applies to you. This is a bit harder to account for as I don't want to speak for everyone's family situation. Instead I am going to set up the problem symbolically and we can see what potential family situations would have the family's gravitational force on you is greater than the black hole at the center of our galaxy. Calculating the force of gravity between two objects can be solved using the equation, Force is equal to the Gravitational constant multiplied by the mass of the two objects, this result is then divided by the distance between the objects squared. (F= G*m1*m2/[r1^2]) In total we have two equations, one for the force of the black hole and the other for the family. The mass of you the reader is reasonably constant, we will designate that as m1, m2 is the mass of the blackhole (8.2*10^36kg), m3 is the mass of your family an undefined value. radius 1 (r1) is the distance between you and the black hole (2.46*10^20 m), and finally r2 is the distance between you and your family (while I can't know the exact distance between you and your family we have a comfortable range to limit ourselves to, it is unlikely that you are closer than 0.1 meters to a family member and unless someone you know works on the ISS the furthest they can be away from you is 12,742,000 meters (read the diameter of the Earth). First I want to estimate if there is a mass and distance where your family exerts a gravitational force roughly equal to that of the black hole. To do this we can set the equations to be equal. ( G*m1*m2/[r1^2])= G*m1*m3/[r2^2]) this can be simplified to m2/[r1^2]= m3/[r2^2] next we can put the mass of the family onto one side. m3=[r2^2]*m2/[r1^2]
for reference again
m2 is the mass of the black hole 8.2E36kg
r1 is the distance between the Earth and the black hole is 2.46E20 meters
r1^2 is then 6.0516E40 m^2
r2 has a range between 0.1m and 12.742E6m
r2^2 has a range between 1 E-2 m^2 and 1.62 E14m^2
m3 is the unknown mass
Now we can solve for an actual numerical answer.
m3=8.2E36kg*{(1E-2m^2)<->(1.62E14m^2)}/6.0516E40m^2
for the closest radius, read you guys are basically touching, your family would only need to have a mass of 0.000001355 kg, basically if your aunt is an actual ant, walking next to you her gravitational attraction is greater than the black hole
at the maximum possible distance of 12,742 kilometers, your family would need to weigh just under 22 billion kilograms, assuming the average person weighs 70 kilograms, your family would be rather massive with roughly 315 million members, otherwise known as the population of the United States.
As I am a red blooded 'Murican I do believe that Murican's are my family, in one way or another, therefore the gravitational attraction of the black hole is weaker than the bond I have with my fellow Murican's
the end detail is meant to be silly, but the overall math seems to work out.
If your family lives within 10 kilometers of you, 6 miles for those of us who have landed on the moon.
(8.2E36kg)*(1E8m^2)/(6.0516E40m^2)=m3
m3 is equal to 13550 kilograms, or the mass of 200 of our "average" humans
Now we need to do some math about the amount of gravitational force that your family applies to you. This is a bit harder to account for as I don't want to speak for everyone's family situation. Instead I am going to set up the problem symbolically and we can see what potential family situations would have the family's gravitational force on you is greater than the black hole at the center of our galaxy. Calculating the force of gravity between two objects can be solved using the equation, Force is equal to the Gravitational constant multiplied by the mass of the two objects, this result is then divided by the distance between the objects squared. (F= G*m1*m2/[r1^2]) In total we have two equations, one for the force of the black hole and the other for the family. The mass of you the reader is reasonably constant, we will designate that as m1, m2 is the mass of the blackhole (8.2*10^36kg), m3 is the mass of your family an undefined value. radius 1 (r1) is the distance between you and the black hole (2.46*10^20 m), and finally r2 is the distance between you and your family (while I can't know the exact distance between you and your family we have a comfortable range to limit ourselves to, it is unlikely that you are closer than 0.1 meters to a family member and unless someone you know works on the ISS the furthest they can be away from you is 12,742,000 meters (read the diameter of the Earth). First I want to estimate if there is a mass and distance where your family exerts a gravitational force roughly equal to that of the black hole. To do this we can set the equations to be equal. ( G*m1*m2/[r1^2])= G*m1*m3/[r2^2]) this can be simplified to m2/[r1^2]= m3/[r2^2] next we can put the mass of the family onto one side. m3=[r2^2]*m2/[r1^2]
for reference again
m2 is the mass of the black hole 8.2E36kg
r1 is the distance between the Earth and the black hole is 2.46E20 meters
r1^2 is then 6.0516E40 m^2
r2 has a range between 0.1m and 12.742E6m
r2^2 has a range between 1 E-2 m^2 and 1.62 E14m^2
m3 is the unknown mass
Now we can solve for an actual numerical answer.
m3=8.2E36kg*{(1E-2m^2)<->(1.62E14m^2)}/6.0516E40m^2
for the closest radius, read you guys are basically touching, your family would only need to have a mass of 0.000001355 kg, basically if your aunt is an actual ant, walking next to you her gravitational attraction is greater than the black hole
at the maximum possible distance of 12,742 kilometers, your family would need to weigh just under 22 billion kilograms, assuming the average person weighs 70 kilograms, your family would be rather massive with roughly 315 million members, otherwise known as the population of the United States.
As I am a red blooded 'Murican I do believe that Murican's are my family, in one way or another, therefore the gravitational attraction of the black hole is weaker than the bond I have with my fellow Murican's
the end detail is meant to be silly, but the overall math seems to work out.
If your family lives within 10 kilometers of you, 6 miles for those of us who have landed on the moon.
(8.2E36kg)*(1E8m^2)/(6.0516E40m^2)=m3
m3 is equal to 13550 kilograms, or the mass of 200 of our "average" humans
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