A spacecraft n the shape of a long cylinder has a length of loo m, and its mass
ID: 1462716 • Letter: A
Question
A spacecraft n the shape of a long cylinder has a length of loo m, and its mass with occupants is 1 270 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. Black hole (a) Determine the total force on the spacecraft. A black hole can be treated in the same manner as any other point mass and used in the equation for the gravitational force as long as you are outside the Schwarzschild radius. N (b) What s 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.)Explanation / Answer
We know that the gravitational constant, G = 6.67384 * 10-11
Mass of the sun = 1.9891 * 1030 kg
Assume that the space craft has a uniform mass along its length. The mass is 1270/100
= 12.7 kg/m
F = integral (10000,10100) (G * 1.09* 1.9891 * 1032 * 12.7/ r2) dr
F= (10000,10100) (-1.83* 1023 / r)
F= 1.83* 1023 * (1/10000 - 1/10100)
F= 1.83 * 1018 * (10 - 9.9010)
F= 1.83 * 1017 * 0.99010
F= 1.818 * 1017 N
b)Force on 1 kg at the nose of the ship= G * 1 * 1.9891 * 1.08 * 1032 / 104
Force on 1 kg at the tail of the ship= G * 1 * 1.9891 * 1.08 * 1032 / (1.01 * 104)
Difference over the length of the ship= G * 2.14823 * 1028 / (1 - 0.99099)
= 1448.0 * 1017N/kg
= 1.448 * 1020 N/kg
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