Question

As a think tanker, you are given the task of evaluating the feasibility of building a space city with a simulated gravitational acceleration of g = 9.8 m/s². For a quick evaluation, you consider a donut-shaped space city where the simulated gravitation is created by rotating the city. The general shape of the city is shown in figure below.



The space city can be built by linking a large number of modular blocks. As indicated in the figure above, the size of each block is 20 m ⨯ 100 m ⨯ 50 m, which is approximately the size of the international space station currently in Earth orbit. When the space city is rotating, a radially outward centrifugal acceleration is exerted on the outer surface (in red), which is the simulated gravitational acceleration.

Considering that the mass of each block is 50000 kg, you calculate the linear mass density of the space city: ρ = 50000/20 kg/m = 2500 kg/m.

(a) Assume that the radius R of the space city is much larger than the size of a single block. For the space city to have a ground area of 7.7⨯10⁶m² , what should be the radius R of the space city?

(b) What should be the angular speed ω for the space city to have a simulated gravitational acceleration of 9.8 m/s²?

Answer 1

a) The lienar mass density of the spaceship is taken by dividing the mass of each block by 20.

This means that the 20 m side of the spaceship is placed along the shape of the circle.

The space city will be in doughnut shape.

Let R be the radius of the shape and dR be the thickness or width of the shape.

Then, the area of the floor is given by

We can take 50 m as the height of the saceship, such that 100 m shde will be the ground

dR = 100 m

Here, the needed A = 7.7*10^6 m^2

So,

b) The centrifugal Force is given by

Using F = ma,

The acceleration is

Here,

So,