5. A space station consists of three modules, connected to form an equilateral t

Question

5. A space station consists of three modules, connected to form an equilateral triangle of side length 82.0 m. Suppose 100 people, with an average mass of 75.0 kg each, live in each capsule and the mass of the modules is negligible compared to the mass of the people. If everyone went to the top left module for a parade what would be the change in position of (a) the center of mass of the station and (b) top left module, during the parade (Assume the actual station's mass is negligible)? Suppose now that everyone is back to their respective module and artificial gravity is simulated by rotating the station. (c) How fast would one of the modules have to be moving to simulate gravity on earth? (d) If the station does not change mass or interact with an outside agent, could it actually go from a state of not rotating to a state of rotating? Explain. Suppose that the station is rotating at the artificial Earth gravity speed and the moment of inertia of the station without the people is 2.00 x 10 kg m. If the distance between each module is reduced to half, what will be the new (e) tangential velocity and () radial acceleration in units of g?

Explanation / Answer

a)

center of mass initially = cnetroid of triangel

= (2/3)*root(3) / 2 * length of side

= 82 / root(2)

= 47.342722 m .. from any of the capsule...(i.e. from any of the vertex )

final center of mass = vertex ..
as this is in space ... so the center of mass will not change it's position.. instead the spacecraft will align itself such that the final center of mass is always at the same position..

so... a) so.. shift in cm = 0
so.. we know that the final center of mass will be at the top left module....

so.. b) change in positon of top left module = 47.342722 m

c)

Let the angular speed be w ..
so.. w^2 * radius = 9.8
so.. w^2 * 47.342722 = 9.8

so.. angular speed required = w = 0.4549738 rad/sec

d)

no.. it cannot go without being in contact with an outsid agent or loosing some mass.. This is because, according to newton third law.. every action has an equal and opposite reaction...
so.. if the body starts rotating, then it must impart it's opposite rotation on some external agent or by loosing some of its mass

e)

initital total inertia = inertia of spacecraft + inertia of people

= 2*10^9 + 75*100* 47.342722^2

= 2016809999.9477696

finao inertia of the craft = 2*10^9 / 4

= 5 * 10^8

initital total inertia = inertia of spacecraft + inertia of people

= 5*10^8 + 75*100* 47.342722^2

= 516809999.94776963

conserving angular momentum
I1 * w1 = I2 * w2
so.. 2016809999.9477696 * 0.4549738 = 516809999.94776963 * w2
so.. w2 = 1.775499 rad/sec

so.. final tangential velcity = w2 * 47.342722/2

= 42.028481 m/sec

f)

acceleraion = v^2 / r = 42.028481^2 / ( 47.342722/2)

= 74.62153 m/sec2

= 7.614442 * 9.8

= 7.614442 g

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