show me details In the future your advanced civilization makes a massive space s
ID: 2098515 • Letter: S
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In the future your advanced civilization makes a massive space station that looks like a long tube of mass M and Length 2L as shown. You are tiny space-walking mass some distance D above the center of the space station. While you float there, you decide to compute the net gravitational force between you and the space station. (Note: this makes use of integral calculus and closely follows an example problems done in class and in the book on integrating a mass distribution. It will be useful to know that dx/ (x2 + D2)3= x/D2 (x2+D2). Express the force as an integral of the force between the space walker and each little bit of mass, dm= (mu)dx dm= mu dx where mu (that is, mu) is the mass per unit length, M/2L and dx is a tiny bit of length along the tube. Express r, the distance to each bit of mass dm in terms of and given constants; note that the y-component forces cancel out by symmetry, and we only care about the x-component of the force, which can be expressed in terms of the given variables. )Explanation / Answer
..............................X..............................|
..............................|
..............................|
[====== L ====== A ====== L ======]
The dots are to try to space out the diagram evenly.
Set up an integral:
1) Let the x axis go along the length of the space station, so x goes from -L to +L
2) Assume the mass is distributed evenly along the length of the space station.
Therefore the mass m associated with a short length of space station, d, is m = M d / 2 L
3) The distance between a point x on the space station and the point where you are at is:
sqrt( D^2 + x^2)
4) The gravitational pull of a small piece of the space station of length d at point x is:
f = G m / r^2 = G M d / (2 L (D^2 + x^2))
5) However the gravity force vector has two components.
In the diagram the horizontal component is multiplied by x / sqrt(x^2 + D^2) and the vertical component is multiplied by D / sqrt(x^2 + D^2).
6) Therefore the vertical component of gravity is
f(vert) = G M d D / ( 2 L (D^2 + x^2)^(3/2) ) where d still represents a short length and x the location
7) Integrate that for x = -L to +L and d is dx. Calculus is very useful for future civilization!
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