To find the change in the force of gravity between two planetary bodies when the distance between them is decreased by 14 meters, we need to follow several steps. Let's go through this methodically to understand the calculation:
1. Understand the Given Information:
- The mass of the Earth ([tex]\( m_1 \)[/tex]) is [tex]\( 5.972 \times 10^{24} \)[/tex] kg.
- The mass of the Moon ([tex]\( m_2 \)[/tex]) is [tex]\( 7.348 \times 10^{22} \)[/tex] kg.
- The initial distance ([tex]\( d \)[/tex]) between the Earth and the Moon is 384,400,000 meters (or 384,400 km).
- We want to calculate the gravitational force using Newton's law of gravitation:
[tex]\[
F = G \frac{m_1 m_2}{d^2}
\][/tex]
where [tex]\( G \)[/tex] is the gravitational constant, [tex]\( G = 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \)[/tex].
2. Calculate the Initial Force of Gravity:
We apply the gravitational formula using the given values:
[tex]\[
\text{Initial Force} = G \frac{5.972 \times 10^{24} \, \text{kg} \times 7.348 \times 10^{22} \, \text{kg}}{(384,400,000 \, \text{m})^2}
\][/tex]
This calculates to approximately [tex]\( 1.982110729079252 \times 10^{20} \)[/tex] N (newtons).
3. Calculate the New Force of Gravity (after decreasing the distance):
The new distance ([tex]\( d_{\text{new}} \)[/tex]) is:
[tex]\[
d_{\text{new}} = 384,400,000 \, \text{m} - 14 \, \text{m} = 384,399,986 \, \text{m}
\][/tex]
Using the new distance to find the gravitational force:
[tex]\[
\text{New Force} = G \frac{5.972 \times 10^{24} \, \text{kg} \times 7.348 \times 10^{22} \, \text{kg}}{(384,399,986 \, \text{m})^2}
\][/tex]
This calculates to approximately [tex]\( 1.982110873457773 \times 10^{20} \)[/tex] N.
4. Calculate the Change in Force:
The change in force ([tex]\( \Delta F \)[/tex]) is:
[tex]\[
\Delta F = \text{New Force} - \text{Initial Force}
\][/tex]
Substituting the values:
[tex]\[
\Delta F = 1.982110873457773 \times 10^{20} \, \text{N} - 1.982110729079252 \times 10^{20} \, \text{N}
\][/tex]
[tex]\[
\Delta F = 14437852119040 \, \text{N}
\][/tex]
Therefore, the change in the force of gravity when the distance between the two planets is decreased by 14 meters is [tex]\( \boxed{14437852119040} \)[/tex] newtons.
