Question

Futuristic space stations are often shown to spin to generate "artificial gravity". Consider a space station made up of a 3 km diameter ring (that you can treat as thin) with 4 radial spokes from the center (that you can also treat as thin). Ignore the effect of the "hub" at the center where the spokes meet. The ring and spoke pieces have circular cross sections. The ring's cross sectional diameter is 50 m, and that of the spokes is 20 m. The average density of each piece is 800 kg/m 3 . 1.3a What is the moment of inertia of the space station in kg−m 2 ? 9. 1.3b At what angular velocity must it rotate in rad/ s to generate an artificial gravitational acceleration of 9.8 m/s 2 ? 10. 1.3c At what distance from the center will the artificial gravity be one half this value? Give your answer in m. 11. 1.3d If two of SpaceX's second stage Falcon 9 rockets were attached to the sides and run at full thrust of 934kN, for how many hours would they need to run to reach the angular velocity calculated in part b?

Answer 1

To calculate the moment of inertia of the space station, we need to consider the individual moments of inertia for the ring and the spokes.

Step-by-step explanation:

To calculate the moment of inertia of the space station, we need to consider the individual moments of inertia for the ring and the spokes. For the ring, we can use the equation I = (1/2) * m * r^2, where I is the moment of inertia, m is the mass, and r is the radius. Substituting the given values, we have I_ring = (1/2) * (800 kg/m^3) * (pi * (25 m)^2) * (50 m). For the spokes, we can use the same equation, but with a different radius. Substituting the given values, we have I_spokes = (1/2) * (800 kg/m^3) * (pi * (10 m)^2) * (20 m) * 4 (since there are four spokes). Adding the moments of inertia of the ring and the spokes, we get the total moment of inertia of the space station.

Answer 2

To create artificial gravity, calculations involve finding the moment of inertia of the space station and determining the necessary angular velocity to simulate Earth's gravity, with adjustments for half gravity conditions and estimating the time needed for rockets to impart required angular velocity.

Step-by-step explanation:

Calculating Moment of Inertia and Angular Velocity

The concept artificial gravity in a rotating space station relies on centripetal force to simulate gravitational effects. Calculating the moment of inertia for the space station involves summing the individual moments of inertia for the ring and the spokes. The ring can be approximated as a thin-walled hollow cylinder with a moment of inertia I_ring = mR2, where m is the mass and R the radius. Similarly, each spoke can be treated as a thin rod rotating about one end with a moment of inertia I_spoke = (1/3)mL2. To find the angular velocity (ω) that produces an acceleration equal to g, the centripetal acceleration equation can be used: ac = ω2R, solving for ω gives us ω = √(g/R).

For the artificial gravity to be half the value, we adjust the equation to ac = (1/2)g, which then gives ω = √((1/2)g/R), and we solve for the new R. To accelerate the space station to the desired angular velocity with SpaceX Falcon 9 rockets, we use the torque produced by the rockets and the equation τ = Iα, where α is angular acceleration. With the total time T and the desired angular velocity, we solve T = ω/α.