A spacecraft in the shape of a long cylinder has a length of 100 m, and its mass

Question

A spacecraft in the shape of a long cylinder has a length of 100 m, and its mass with occupants is 1 910 kg. It has strayed too close to a black hole having a mass 109 times that of the Sun. The nose of the spacecraft points toward the black hole, and the distance between the nose and the center of the black hole is 10.0 km.

(a) Determine the total force on the spacecraft.
N

(b) What is the difference in the gravitational fields acting on the occupants in the nose of the ship and on those in the rear of the ship, farthest from the black hole? (This difference in acceleration grows rapidly as the ship approaches the black hole. It puts the body of the ship under extreme tension and eventually tears it apart.)
N/kg

Explanation / Answer

(a)
According to Newton's Laws of Gravitation, the force is determined by the mass and by the radius. The radius is between the centres of two objects. So for part (a) the radius will be 10,000m + 50 m= 10,050m.

Mass of sun = 1.989 * 10^30 Kg
Mass of the black hole = 109 * Mass of sun
Mass of the black hole = 109 * 1.989 * 10^30 Kg
Mass of the black hole = 2.168 * 10^32 Kg

So the force will be
Fg = Gmm/r^2
= (6.67*10^-11)*(2.168*10^32) * (1910)/(10050^2)
Fg = 2.734 *10^17 N

(b)
The difference in the gravitational fields will be
delta g = Gm/(r1)^2-Gm/(r2)^2
delta g =(6.67*10^-11)(2.168 * 10^32)(1/(10000)^2 - 1/(10100)^2)
= 2.85 * 10^12 N/kg
The difference in the gravitational fields, = 2.85 * 10^12 N/Kg

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