To determine the gravitational force between two masses, you can use Newton's law of universal gravitation, which is formulated as follows:
[tex]\[ \vec{F} = G \frac{m_1 m_2}{r^2} \][/tex]
Given:
- [tex]\( G \)[/tex] (Gravitational constant) = [tex]\( 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \)[/tex]
- [tex]\( m_1 \)[/tex] (Mass 1) = [tex]\( 4.32 \, \text{kg} \)[/tex]
- [tex]\( m_2 \)[/tex] (Mass 2) = [tex]\( 163 \, \text{kg} \)[/tex]
- [tex]\( r \)[/tex] (Distance between the masses) = [tex]\( 83.0 \, \text{m} \)[/tex]
Step-by-step solution:
1. Identify the masses and the separation distance:
[tex]\[ m_1 = 4.32 \, \text{kg} \][/tex]
[tex]\[ m_2 = 163 \, \text{kg} \][/tex]
[tex]\[ r = 83.0 \, \text{m} \][/tex]
2. Substitute the values into the formula for the gravitational force:
[tex]\[
\vec{F} = 6.67 \times 10^{-11} \cdot \frac{4.32 \times 163}{83.0^2}
\][/tex]
3. Calculate the gravitational force:
- First, calculate the product of the masses:
[tex]\[ m_1 \times m_2 = 4.32 \times 163 = 704.16 \, \text{kg}^2 \][/tex]
- Next, calculate the square of the distance:
[tex]\[ r^2 = 83.0^2 = 6889.0 \, \text{m}^2 \][/tex]
- Now, substitute these intermediate results back into the formula:
[tex]\[
\vec{F} = 6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2 \cdot \frac{704.16 \, \text{kg}^2}{6889.0 \, \text{m}^2}
\][/tex]
- Simplify the division inside the parentheses:
[tex]\[ \frac{704.16}{6889.0} \approx 0.1022 \][/tex]
- Finally, multiply by the gravitational constant:
[tex]\[ 6.67 \times 10^{-11} \cdot 0.1022 \approx 6.82 \times 10^{-12} \, \text{N} \][/tex]
Thus, the gravitational force between the two masses is approximately:
[tex]\[ \vec{F} \approx 6.82 \times 10^{-12} \, \text{N} \][/tex]
