   Chapter 9, Problem 78AP

Chapter
Section
Textbook Problem

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 4 R . What is the average density of the combined world as a multiple of ρ0, the average density of the original two worlds?

To determine
The average density of the combined world as multiple of ρ0 , the average density of the original two worlds.

Explanation

The density of an object in general is defined as ρ=M/V where M is mass of the object and V is volume of that object.

Write the average density of each world of two before they coalesce.

ρ0=MV

• ρ0 is the average density of original world
• M is the mass of object
• V is the volume of the object

Rewrite the expression in terms of radius.

ρ0=M4πr3/3

• r is the radius

Express the average density of the combined world.

ρ=MtotalV'

• ρ is the average combined density
• Mtotal is the combined mass
• V' is the total volume

Substitute 2M for Mtotal to simplify

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