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 p0, the average density of the original two worlds?
4.74
Density(total)=4.74Density(original)
To determine the average density of the combined world, we need to compare it to the original average density, p0.
The average density of an object is given by the mass divided by the volume. In this case, since we have two worlds combining into one, we need to consider the total mass and the total volume.
The total mass, M_combined, of the two worlds is simply twice the mass of one world, M_combined = 2M.
Now, let's calculate the total volume, V_combined, of the two worlds:
V_combined = (4/3πR^3) + (4/3πR^3) = (8/3πR^3)
Next, we need to find the average density of the combined world, ρ_combined, which is the total mass divided by the total volume:
ρ_combined = M_combined / V_combined = (2M) / (8/3πR^3)
To simplify this expression, we multiply the numerator and denominator by 3/8:
ρ_combined = (2M * 3/8) / ((8/3πR^3) * 3/8)
= (6M) / (πR^3)
Finally, to find the ratio of the average density of the combined world to the original average density, we divide ρ_combined by p0:
(ρ_combined / p0) = (6M) / (πR^3 * p0)
Therefore, the average density of the combined world, as a multiple of p0, is (6M) / (πR^3 * p0).
To find the average density of the combined world as a multiple of p0, we need to compare the masses and volumes of the two scenarios.
Let's start by considering the original two worlds. Each world has a mass of M and a radius of R. The volume of a sphere is given by the formula V = (4/3)πR³, so the volume of each world is (4/3)πR³.
Therefore, the total volume of the two original worlds is:
V0 = 2 * (4/3)πR³ = (8/3)πR³
Now, let's move on to the combined world. According to the information given, the combined world has a radius of (3/4)R. Using the volume formula, we can calculate the volume of the combined world:
V_combined = (4/3)π((3/4)R)³ = (27/64)πR³
Since mass is conserved in the process of coalescence, the mass of the combined world is equal to the sum of the masses of the original two worlds, which is 2M.
Now, let's calculate the average density, p, for both scenarios.
Original two worlds:
p0 = mass / volume = 2M / ((8/3)πR³) = 3M / (4πR³)
Combined world:
p_combined = mass / volume = 2M / ((27/64)πR³) = (128/27)M / (πR³)
To find the average density of the combined world as a multiple of p0, we divide p_combined by p0:
p_combined / p0 = ((128/27)M / (πR³)) / (3M / (4πR³))
Canceling out common terms:
p_combined / p0 = ((128/27) / 3) * (4/π)
Simplifying further:
p_combined / p0 = (512/81) * (4/π)
Therefore, the average density of the combined world is (512/81) * (4/π) times the average density of the original two worlds.