Question

Constants
Black hole gravitational force A black hole exerts a 60-
N gravitational force on a spaceship. The black hole is
1014 m from the ship.
Part A
What is the magnitude of the force that the black
hole exerts on the ship when the ship is one-half
that distance from the black hole? (Hint: One-half of
1014 m is not 10 m].
Express your answer to two significant figures
and include the appropriate units.
F
12.10-12 N

Answer 1

The magnitude of the gravitational force becomes 240 N

Explanation:

The magnitude of the gravitational force between two objects is given by:

$F=G\frac{m_1 m_2}{r^2}$

where

$G=6.67\cdot 10^{-11} m^3 kg^{-1}s^{-2}$ is the gravitational constant

m1, m2 are the masses of the two objects

r is the separation between them

We notice that the gravitational force is inversely proportional to the square of the distance, $F\propto \frac{1}{r^2}$. Since G, m1 (the mass of the black hole) and m2 (of the spaceship) in our problem are constant, we can rewrite the equation as

$F=\frac{k}{r^2}$

Which can be rewritten as

$Fr^2=k$ (1)

At the beginning, when the distance is

$r_1=10^{14} m$

The force is

$F_1=60 N$

and later the distance is halved, so

$r_2 = \frac{r_1}{2}$

We can rewrite eq. (1) as

$F_1 r_1^2 = F_2 r_2^2$

where F2 is the magnitude of the force when the distance is halved. Substituting,

$F_2 = \frac{F_1 r_1^2}{r_2^2}=\frac{F_1 r_1^2}{(r_1/2)^2}=4F_1 = 4(60 N)=240 N$

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