Certainly! Let's go through the steps to solve this problem in detail.
### Step-by-Step Solution:
We are given the following conditions:
1. [tex]\( m \propto \frac{1}{r^3} \)[/tex], which means [tex]\( m \)[/tex] is inversely proportional to the cube of [tex]\( r \)[/tex].
2. When [tex]\( r = 2 \)[/tex], [tex]\( m = 1 \)[/tex].
First, let's express the relationship mathematically as:
[tex]\[ m = \frac{k}{r^3} \][/tex]
where [tex]\( k \)[/tex] is the constant of proportionality that we need to determine.
#### Step 1: Determine the constant [tex]\( k \)[/tex]
Using the given values [tex]\( m = 1 \)[/tex] and [tex]\( r = 2 \)[/tex]:
[tex]\[ 1 = \frac{k}{2^3} \][/tex]
[tex]\[ 1 = \frac{k}{8} \][/tex]
Solving for [tex]\( k \)[/tex], we get:
[tex]\[ k = 1 \times 8 \][/tex]
[tex]\[ k = 8 \][/tex]
So the constant [tex]\( k \)[/tex] is 8.
#### Step 2: Find [tex]\( m \)[/tex] when [tex]\( r = 4 \)[/tex]
Now that we have determined the constant [tex]\( k \)[/tex], we can use the relationship to find [tex]\( m \)[/tex] for any given [tex]\( r \)[/tex]. We need to find [tex]\( m \)[/tex] when [tex]\( r = 4 \)[/tex]:
[tex]\[ m = \frac{k}{r^3} \][/tex]
[tex]\[ m = \frac{8}{4^3} \][/tex]
[tex]\[ m = \frac{8}{64} \][/tex]
[tex]\[ m = 0.125 \][/tex]
Therefore, [tex]\( m \)[/tex] when [tex]\( r = 4 \)[/tex] is:
[tex]\[ m = 0.125 \][/tex]
### Final Answer:
When [tex]\( r = 4 \)[/tex]:
[tex]\[ m = 0.125 \][/tex]
