To find the gravitational force [tex]\(\vec{F}\)[/tex] between two masses, we use Newton's Law of Universal Gravitation, given by the formula:
[tex]\[ \vec{F} = G \frac{m_1 m_2}{r^2} \][/tex]
where:
- [tex]\( G \)[/tex] is the gravitational constant, [tex]\( G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \)[/tex],
- [tex]\( m_1 \)[/tex] is the mass of the first object,
- [tex]\( m_2 \)[/tex] is the mass of the second object,
- [tex]\( r \)[/tex] is the distance between the centers of the two masses.
Given:
- [tex]\( m_1 = 84.2 \, \text{kg} \)[/tex],
- [tex]\( m_2 = 28.4 \, \text{kg} \)[/tex],
- [tex]\( r = 4.62 \, \text{m} \)[/tex].
Step-by-step solution:
1. Substitute the given values into the formula:
[tex]\[
\vec{F} = 6.67 \times 10^{-11} \cdot \frac{84.2 \cdot 28.4}{4.62^2}
\][/tex]
2. Calculate the product of the masses:
[tex]\[
m_1 \cdot m_2 = 84.2 \, \text{kg} \times 28.4 \, \text{kg} = 2391.28 \, \text{kg}^2
\][/tex]
3. Calculate the square of the distance:
[tex]\[
r^2 = 4.62 \, \text{m} \times 4.62 \, \text{m} = 21.3444 \, \text{m}^2
\][/tex]
4. Substitute these results back into the formula:
[tex]\[
\vec{F} = 6.67 \times 10^{-11} \, \frac{2391.28}{21.3444}
\][/tex]
5. Compute the division inside the parentheses:
[tex]\[
\frac{2391.28}{21.3444} \approx 112.037
\][/tex]
6. Finally, multiply by the gravitational constant:
[tex]\[
\vec{F} \approx 6.67 \times 10^{-11} \, \times 112.037
\][/tex]
7. Calculate the result:
[tex]\[
\vec{F} \approx 7.472609958583983 \times 10^{-9} \, \text{N}
\][/tex]
Therefore, the gravitational force [tex]\( \vec{F} \)[/tex] between the two masses is approximately:
[tex]\[
\vec{F} \approx 7.47 \times 10^{-9} \, \text{N}
\][/tex]
So, we have:
[tex]\[
\vec{F} \approx 7.47 \times 10^{-9} \, \text{N}
\][/tex]
