4. +-2 points SerPSET9 13.P016. My Notes A spacecraft in the shape of a long cyl

Question

4. +-2 points SerPSET9 13.P016. My Notes A spacecraft in the shape of a long cylinder has a length of 100 m, and its mass with occupants is 1 920 kg. It has strayed too close to a black holehaving a mass 90 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 100 m 10.0 km (a) Determine the total force on the spacecraft. (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 Need Help?-Read". Ivonen

Explanation / Answer

Solution:

(A)

Here;

The force on any slice of the spacecraft is
dF = GM*dm / (r + x)²
where,the mass of the slice

dm = *dx ,where is the linear mass density, m / L

=> dF = (GmM / L) dx / (r + x)²

Integrating over it's length:

=> F = (GmM/L) dx / (r + x)² = (GmM/L) * -1 / (r + x) {Evaluated from 0 to L}

=> F = (GmM/L) * [-1/(103 + 100)m + 1/103m] = (GmM/L) * 9.901e-7/m

=> F = (6.674*10^11N·m²/kg² * 1920kg * 90 * 1.99*10^30kg / 100m) * 9.901*10^-7/m

=> F = 2.27 * 10^17 N

(B)

As;

Acceleration = GM / d²

At the frontside of the rocktet, where d = 10000 m,

a = 1.43e14 m/s² (approx.)

At the rear end of the rocket, where d = 10100 m,

a = 1.41e14 m/s² (approx.)

difference is 2.87e12 m/s² (Approx.)

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