Certainly! Let's find the gravitational force between the two masses step by step.
### Step 1: Understand the Problem
We are given:
- Mass 1 ([tex]\( m_1 \)[/tex]) = 912 kg
- Mass 2 ([tex]\( m_2 \)[/tex]) = 878 kg
- Distance between the masses ([tex]\( r \)[/tex]) = 25.4 meters
- Gravitational constant ([tex]\( G \)[/tex]) = [tex]\( 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \)[/tex]
We need to find the gravitational force ([tex]\( \vec{F} \)[/tex]) between the two masses.
### Step 2: Use the Formula for Gravitational Force
The formula to calculate the gravitational force between two masses is:
[tex]\[ \vec{F} = G \frac{m_1 m_2}{r^2} \][/tex]
### Step 3: Plug in the Given Values
Substitute the given values into the formula:
[tex]\[ \vec{F} = 6.67 \times 10^{-11} \frac{912 \times 878}{25.4^2} \][/tex]
### Step 4: Evaluate the Expression
Let's break it down into parts:
1. Calculate the numerator ([tex]\( m_1 \times m_2 \)[/tex]):
[tex]\[ 912 \times 878 = 800,736 \][/tex]
2. Calculate the square of the distance ([tex]\( r^2 \)[/tex]):
[tex]\[ 25.4^2 = 645.16 \][/tex]
3. Now, calculate the fraction:
[tex]\[ \frac{800,736}{645.16} \approx 1240.970957 \][/tex]
4. Finally, multiply by the gravitational constant:
[tex]\[ \vec{F} = 6.67 \times 10^{-11} \times 1240.970957 \approx 8.278425693 \times 10^{-8} \, \text{N} \][/tex]
### Step 5: Express the Force in Scientific Notation
We have calculated the gravitational force as [tex]\( 8.278425693 \times 10^{-8} \, \text{N} \)[/tex]. This can be written in the format:
[tex]\[ \vec{F} = 8.278425693 \times 10^{-8} \, \text{N} \][/tex]
To summarize, the gravitational force between the two masses is:
[tex]\[ \vec{F} \approx 8.278425693 \times 10^{-8} \, \text{N} \][/tex]
So, the final answer is:
[tex]\[
\begin{array}{c}
\vec{F} \approx 8.278425693 \times 10^{-8} \, \text{N} \\
\end{array}
\][/tex]
