Hi Cassandra,
I have a policy of only answering one problem per post so I will answer problem 1 from your post:
(a) The gravitational force equation is: F = GMm/D^2, where G is a constant equal to 6.67 x 10^-11 Nm^2/kg^2, M and m are the masses of the objects in the question, and D is the distance between the objects in the question.
In this question:
M is the mass of the black hole: 4 million times the Sun's mass = 4,000,000(1.989 x 10^30 kg) = 7.956 x 10^36 kg.
m is the mass of the Earth, which is 5.972 x 10^24 kg.
For D, your question says that "the black hole is 2.4x10^20 away from Earth," so there is a word missing. I'll proceed assuming it's 2.4 x 10^20 meters. If it's some other unit of distance, such as miles or feet, then you'd first need to convert that to meters.
Plugging those values into the F = GMm/D^2 equation, we get:
(6.67 x 10^-11 Nm^2/kg^2)(7.956 x 10^36 kg)(5.972 x 10^24 kg)/[(2.4 x 10^20 meters)^2)] = 5.50 x 10^10 N.
(b) F = ma (force = mass x acceleration). To solve for a, divide both sides of the equation by m to get F/m = a.
We calculated F in part (a) as being 5.50 x 10^10 N. We know that m (the mass of the Earth) is 5.972 x 10^24 kg.
Plugging those values into the F/m = a. equation, we get:
(5.50 x 10^10 N)/ (5.972 x 10^24 kg) = a = 9.21 X 10^-15 m/s^2. [Units: note that one Newton (N) is equal to one (kg)(m)/s^2.]
(c) 9.21 X 10^-15 m/s^2 is a very small number (10^-15 is a millionth of a billionth!). So, the rate at which the Earth accelerates toward the black hole is negligible. Moreover, D is very large, i.e. the Earth is so far from the black hole that the earth would not accelerate quickly enough to reach the black hole in any of our lifetimes.
I hope you found this to be helpful! :)
Best,
Cynthia H.
