Question

suppose two worlds, each having mass M and radius R, coalesce into a single world. Due to gravitational contraction, the combined world has a radius of only 3 4R. what is the average density of the combined world as a multiple of r0, the average density of the original two worlds

Answer 1

Not enough



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Answer 2

Answer: the average density of the combined world is 256/81 times the average density of the original two worlds.

Step-by-step explanation:

The total mass of the two original worlds is 2M, and their average density is given by:

ρ0 = 2M/(4/3 πR^3) = 3M/(2πR^3)

The final radius of the combined world is 3/4R, so its volume is:

V = 4/3 π(3/4R)^3 = 27/64 πR^3

The mass of the combined world is still 2M, so its density is:

ρ = 2M/V = 128M/(27πR^3)

The ratio of the average density of the combined world to that of the original worlds is:

ρ/ρ0 = (128M/(27πR^3)) / (3M/(2πR^3)) = 256/81