To determine the gravitational force between two masses [tex]\( m_1 \)[/tex] and [tex]\( m_2 \)[/tex] separated by a distance [tex]\( r \)[/tex], we use Newton's law of universal gravitation, which states:
[tex]\[
\vec{F} = G \frac{m_1 m_2}{r^2}
\][/tex]
where:
- [tex]\( \vec{F} \)[/tex] is the gravitational force,
- [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]\( 55.0 \, \text{kg} \)[/tex]),
- [tex]\( m_2 \)[/tex] is the mass of the second object ([tex]\( 1.38 \, \text{kg} \)[/tex]),
- [tex]\( r \)[/tex] is the distance between the centers of the two masses ([tex]\( 0.435 \, \text{m} \)[/tex]).
### Step-by-Step Solution
1. Substitute the given values into the formula:
[tex]\[
\vec{F} = 6.67 \times 10^{-11} \times \frac{55.0 \times 1.38}{0.435^2}
\][/tex]
2. Calculate the product of the masses:
[tex]\[
55.0 \times 1.38 = 75.9 \, \text{kg}^2
\][/tex]
3. Calculate the square of the distance:
[tex]\[
0.435^2 = 0.189225 \, \text{m}^2
\][/tex]
4. Calculate the ratio:
[tex]\[
\frac{75.9}{0.189225} \approx 401.28
\][/tex]
5. Combine with the gravitational constant:
[tex]\[
\vec{F} = 6.67 \times 10^{-11} \times 401.28
\][/tex]
6. Calculate the final gravitational force:
[tex]\[
\vec{F} \approx 2.6754022988505743 \times 10^{-8} \, \text{N}
\][/tex]
### Scientific Notation
To express the answer in the form [tex]\( \vec{F} = a \times 10^b \)[/tex]:
- Here, [tex]\( a = 2.675402298850574 \)[/tex]
- And [tex]\( b = -8 \)[/tex]
So, the gravitational force between the two masses is:
[tex]\[
\vec{F} = 2.675402298850574 \times 10^{-8} \, \text{N}
\][/tex]
In summary, the gravitational force between the two masses of 55.0 kg and 1.38 kg separated by a distance of 0.435 meters is approximately:
[tex]\[
\vec{F} = 2.675402298850574 \times 10^{-8} \, \text{N}
\][/tex]
