To find the gravitational force between the two masses, we will use Newton's law of universal gravitation, which is expressed by the formula:
[tex]\[
\vec{F} = G \frac{m_1 m_2}{r^2}
\][/tex]
where:
- [tex]\(\vec{F}\)[/tex] is the gravitational force between the masses,
- [tex]\(G\)[/tex] is the gravitational constant [tex]\(6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2\)[/tex],
- [tex]\(m_1\)[/tex] is the mass of the first object ([tex]\(912 \, \text{kg}\)[/tex]),
- [tex]\(m_2\)[/tex] is the mass of the second object ([tex]\(878 \, \text{kg}\)[/tex]),
- [tex]\(r\)[/tex] is the distance between the centers of the two masses ([tex]\(25.4 \, \text{m}\)[/tex]).
Now, let's go through the steps to calculate the force:
1. Substitute the given values into the formula:
[tex]\[
\vec{F} = \left(6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2\right) \times \frac{(912 \, \text{kg}) \times (878 \, \text{kg})}{(25.4 \, \text{m})^2}
\][/tex]
2. Calculate the product of the masses [tex]\(m_1\)[/tex] and [tex]\(m_2\)[/tex]:
[tex]\[
912 \times 878 = 801,936 \, \text{kg}^2
\][/tex]
3. Square the distance [tex]\(r\)[/tex]:
[tex]\[
(25.4)^2 = 645.16 \, \text{m}^2
\][/tex]
4. Calculate the fraction:
[tex]\[
\frac{801,936}{645.16} = 1,243.109
\][/tex]
5. Multiply by the gravitational constant [tex]\(G\)[/tex]:
[tex]\[
\vec{F} = 6.67 \times 10^{-11} \times 1,243.109 = 8.278425692851386 \times 10^{-8} \, \text{N}
\][/tex]
So, the gravitational force between the two masses in scientific notation is:
[tex]\[
\vec{F} = 8.278425692851386 \times 10^{-8} \, \text{N}
\][/tex]
If we express the force in the form [tex]\([\text{magnitude}] \times 10^{[\text{exponent}]}\)[/tex]:
[tex]\[
\vec{F} \approx 8.278 \times 10^{-8} \, \text{N}
\][/tex]
Thus, the gravitational force between the two masses is:
[tex]\[
\boxed{8.278 \times 10^{-8} \, \text{N}}
\][/tex]
