Dr. Doom on Planet IPhOO
The evil Dr. Doom seeks to destroy his enemy, the Intergalactic Federation, and has devised a plan to despin the Federation's space station. The hoop-shaped space station of mass and radius rotates once every hours to maintain artificial gravity equal to that on IPhOO. Dr. Doom plans to mount two thruster rockets, one rocket on opposite sides of the space station, to stop its rotation. Dr. Doom must accomplish his crime within a time to avoid getting caught. How much force should each rocket deliver in order to despin the Federation's space station in 1 hour? Express your answer in Newtons rounded to 4 significant figures.
Note : A more general version of this problem was proposed by Kimberly Geddes for the IPhOO .
The answer is 4.545.
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1 solution
The inital velocity of the space station is ω = 8 6 4 0 0 2 π ( r a d / s )
⇒ The needed deceleration to stop the station is α = 8 6 4 0 0 2 π ⋅ 3 6 0 0 1 ( r a d / s 2 )
The inertia of a hoop is I = m r 2 = 4 . 5 ⋅ 1 0 1 1 ( k g ⋅ m 2 )
The torque is τ = I α = F r
⇒ F = r I α = 4 . 5 ⋅ 1 0 1 1 ⋅ 8 6 4 0 0 2 π ⋅ 3 6 0 0 1 ⋅ 1 0 3 1 = 9 . 0 9 0 2
There are 2 rockets, so the force is 9 . 0 9 0 2 / 2 = 4 . 5 4 5
Not sure if I was right. I think it's a bit "too" straightforward.
Btw, it seems that I can't post another solution once I deleted the first one.