Certainly! Let's solve this problem step-by-step.
1. Understanding the Scenario:
- The weight of the body on the surface of the Earth is [tex]\( 63 \, \text{N} \)[/tex].
- The height above the Earth's surface where we want to find the new gravitational force is [tex]\( 3200 \, \text{km} \)[/tex].
- The radius of the Earth is given as [tex]\( 6400 \, \text{km} \)[/tex].
2. Determine the Distances:
- Distance from the center of the Earth to the surface is the radius of the Earth: [tex]\( r_{\text{surface}} = 6400 \, \text{km} \)[/tex].
- Distance from the center of the Earth to the point 3200 km above the surface:
[tex]\[
r_{\text{above}} = r_{\text{surface}} + \text{height above surface} = 6400 \, \text{km} + 3200 \, \text{km} = 9600 \, \text{km}
\][/tex]
3. Using the Concept of Gravitational Force:
- The gravitational force varies inversely with the square of the distance from the center of the Earth. This relationship is given by:
[tex]\[
F_{\text{above}} = F_{\text{surface}} \left( \frac{r_{\text{surface}}}{r_{\text{above}}} \right)^2
\][/tex]
- Where:
[tex]\[
F_{\text{surface}} = 63 \, \text{N} \quad \text{(gravitational force at the surface)}
\][/tex]
4. Plug in the Values:
[tex]\[
F_{\text{above}} = 63 \, \text{N} \left( \frac{6400 \, \text{km}}{9600 \, \text{km}} \right)^2
\][/tex]
5. Calculate the Fraction:
- Simplify the fraction [tex]\( \frac{6400}{9600} \)[/tex]:
[tex]\[
\frac{6400}{9600} = \frac{2}{3}
\][/tex]
6. Square the Fraction:
[tex]\[
\left( \frac{2}{3} \right)^2 = \frac{4}{9}
\][/tex]
7. Calculate the Gravitational Force at the Height:
[tex]\[
F_{\text{above}} = 63 \, \text{N} \times \frac{4}{9} = 63 \, \text{N} \times 0.4444 = 28 \, \text{N}
\][/tex]
Therefore, the gravitational force on the body at a height of [tex]\( 3200 \, \text{km} \)[/tex] above the Earth's surface is [tex]\( 28 \, \text{N} \)[/tex].
