A team is chosen to design an orbiting space station that is shaped like a wheel 5o meters in diameter with is mass of 4*10^5 kg contained in the rim portion of the wheel. The space station will rotate at a rate such that a person working in the wheel will experience a simulated gravitational force equal to that of earth, meaning that the normal force of the floor on an a person will be equal to g=9.8m/s^2. When the station is finished, it will have to be set at a required rotation rate to produce this “artificial gravity”. This is accomplished with the firing of four small thruster rockets attached symmetrically to the outer wall of the wheel. The rockets will fire for 1 hour to bring the staion “up to speed”. I have to determine the thrust, or force, each rocket must have to produce the required rate in this particular amount of time. Assume the station starts rotating from rest.
Where to start?
You want the space statiion to rotate at angular rate w such that
R w^2 = g, where R = 25 m.
Therefore w = sqrt(g/R).
Treat the space station as a wheel with all of its mass at distance R from the center. Its moment of inertia wil them be
I = M R^2
The "torque impulse" applied by the rockets in 1 hour must be enough to provide the required angular momentum of the space station, which is I w.
4 T R *(3600 s) = I w = M R^2 w
Solve for the thrust, T. The number 4 comes from the number of thrusters
To determine the thrust or force each rocket must have to produce the required rate of rotation in a specific amount of time, we can start by using the principle of conservation of angular momentum.
The principle of conservation of angular momentum states that the total angular momentum of a system remains constant if no external torques act on it. In this case, we can assume that no external torques act on the space station, so the initial angular momentum of the system should be equal to the final angular momentum.
Since the space station starts rotating from rest, the initial angular momentum of the system is zero. The final angular momentum is given by the formula:
Angular momentum = Moment of inertia x Angular velocity
The moment of inertia of a solid wheel rotating around its central axis is given by:
Moment of inertia = 0.5 x Mass x Radius^2
Substituting the given values, we can calculate the moment of inertia of the wheel.
Next, we need to determine the final angular velocity of the space station. We know that the gravitational force experienced by a person in the wheel should be equal to the acceleration due to gravity (g = 9.8 m/s^2). This gravitational force is provided by the centripetal force acting on the person, which is given by:
Centripetal force = Mass x Gravity
The centripetal force is also equal to the net force acting on the person, which can be expressed as the product of mass and tangential acceleration (mass x tangential acceleration). The tangential acceleration can be calculated using the equation:
Tangential acceleration = Radius x Angular acceleration
The angular acceleration is the change in angular velocity divided by the time taken. In this case, the time taken is given as 1 hour, so we need to convert it to seconds before using it in the calculations.
Once we have the tangential acceleration, we can solve for the net force acting on the person and determine the final angular velocity by dividing this net force by the mass, and then dividing by the radius.
Finally, since there are four thruster rockets firing symmetrically, the total thrust or force required to produce this rate of rotation would be four times the force calculated above. Divide this total force by 4 to determine the force that each rocket must have.
Remember to use consistent units during the calculations.