Question

1. Dynamics and artificial gravity.

The space station has two thrusters pointed in opposite directions. They operate by expelling propellant at high speed. The effect is that there is a force of magnitude F₀ on the two ends of the space station in opposite directions, which causes the entire object to start rotating. The thrusters will stop firing when the artificial gravity created on the space station (described below) reaches the required value.

Upward force, magnitude F₀

Downward force, magnitude F₀

(a) If both thrusters operate at the same time, what is the net force F~ net on the space station, and how will this affect the translational motion of the center of mass of the space station?

(b) Do the two thrusters exert the same torque or different torques on the space station about the axis of rotation? Explain or show your work briefly.

(c) Determine the magnitude of the net torque on the space station due to both thrusters. Your answer should be in terms of R₀ and F₀.

(d) As the engineer, it is your job to determine how much time the thrusters need to fire. In order for the space station to reach an angular speed of ?, how long ?t do the thrusters need to fire? Assume the space station starts from rest. Your answer will be in terms of ?, M₀, R₀, and F₀.

(e) The quantity ?^₂r represents the centripetal acceleration of an object that is rotating with angular speed ? a distance r away from the rotation axis. The design specifications of the space station require that the center of capsule 1 has “1 g” of artificial gravity, meaning that the centripetal acceleration at that point is g, which is the acceleration due to gravity near the surface of Earth. This is done to allow the astronauts a region in which they can experience an Earth-like environment so that their muscles don’t atrophy, which is a real problem in space. For this requirement of 1 g to be met at the center of capsule 1, show that angular speed of the space station must be ? = square root of (g/ 7.5 R₀₎.

(f) With this value of ?, what would the value of artificial gravity be at the center of capsule 2 in “g”s?

(g) The design specifications state that M₀ = 10 000 kg = 1 × 10⁴ kg, R₀ = 5 m and F₀ = 200 N. If the space station is initially motionless, how long in seconds do you need to fire the thrusters to attain “1 g” of artificial gravity at the center of capsule 1?

(h) If the rate of propellant used by each thruster is 0.07 kg/s, how much propellant needs to be used during the rotation burn? (In case you’re curious, I got the thruster values from www.space-propulsion.com/ spacecraft-propulsion/ bipropellant-thrusters/ 200nbipropellant-thrusters.html which is pretty neat to look at.)

Answer 1

part a:

net foce=F0-F0=0

so there wont be any translational motion of center of mass.

part b:

torque exerted by each force is equal.

as torque=force*distance

=F0*R0

part c:

total torque=sum of torque by two forces

=2*F0*R0

part d:

as torque=moment of inertia*angular acceleration

moment of inertia of the system=M0*R0^2+M0*R0^2

=2*M0*R0^2

then angular acceleration=torque/total moment of inertia

=2*F0*R0/(2*M0*R0^2)

=F0/(M0*R0)

initial angular velocity=0

let time taken be t seconds

in order to reach angular speed of w,

time taken=(final angular speed-initial angular speed)/acceleratin

=w/(F0/(M0*R0))

=w*M0*R0/F0