Total Energy in Newtonian Cosmology

[BOYLANATOR]

Homework Statement

"Consider a perfectly homogeneous and isotropic universe filled with dust of uniform density ρ(t).
Let universe expand, dust is carried radially outward from origin.
Conservation of total energy E. E = K(t) + U(t)
K(t) = (1/2)*mv^2(t) and U(t) = -GMm/r(t)
where M = 4/3*∏r^3(t)ρ(t).
Let E = -1/2*mkc^2s^2 with k a constant and s labelling a shell"

Homework Equations

I am at loss as to where that last line came from.

The Attempt at a Solution

From the information given it is easy to show that v^2 - 8/3*∏Gρr^2 = -kc^2s^2 and therefore determine what values of k allow a bounded or unbounded universe but I just don't get that last conclusion in the introduction.

 

[anathema]

Thank you for your post. It seems that the last line may have been a typo or a mistake. It is important to carefully proofread and double check all equations and information before posting them in a forum or publication. However, from the information given, it is possible to determine the values of k that allow for a bounded or unbounded universe. As you mentioned, the equation v^2 - 8/3*∏Gρr^2 = -kc^2s^2 can be used to determine this. If k is positive, the universe is unbounded and if k is negative, the universe is bounded. This is because a positive value of k indicates a positive total energy, allowing for an unbounded universe, while a negative value of k indicates a negative total energy, resulting in a bounded universe. I hope this helps clarify the situation.