Answer to Question #84571 in Mechanics | Relativity for sim

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

Calculate the average density by the relation as follows:

ρ=2M/((4π〖R'〗^3)/3)=6M/(4πR^'3 ).

Here, M Is the mass of the new world , R’ is the radius of new world.

Substitute R’= 3R/4

ρ=6M/(4π〖(3/4)〗^3 )=384M/(108πR^3 )=1.132 M/R^3

Find the ratio of ρ/r0.

ρ/ρ_0 =(1.132 M/R^3 )/(M/((4πR^3)/3))

ρ/(M/((4π〖r_0〗^3)/3))=(1.132 M/R^3 )/(M/((4πR^3)/3))=4.74

(4.18 ρ〖r_0〗^3)/M=4.74

ρ=1.33 M/〖r_0〗^3