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 1 210 kg. It has strayed too close to a black hole having a mass 102 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.

(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) to get the total force acting on the craft, we have to divide it into slices of thickness "dr" so that the mass of one slice is dm = (mass of craft / length of craft) dr = 10.6 dr

Now...

force on one slice = G M dm / r^2 where M is mass of black hole and

total force on craft = integral of ( G M dm / r^2) = GM * integral of (10.6 dr/r^2) from r = 10000 m to r = 10100 m.

or

total force = 10.6 GM (1/10000 - 1/10100) =

= 10.6 * 6.67x10^-11 * 102 * 1.99x10^30 * (1/10000 - 1/10100) =

= 1.42 x 10^17 Newtons

(b) strength of field at front of craft =

= G M / r^2 = 6.67x10^-11 * 102 * 1.99x10^30 / 10000^2 =

= 1.3539 x 10^14 m/s^2

at back of craft =

= G M / r^2 = 6.67x10^-11 * 102 * 1.99x10^30 / 10100^2 =

= 1.3272 x 10^14 m/s^2

difference? 1.3539 - 1.3272 = 2.67 x 10^12 m/s^2   

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