To solve this problem, let's follow these steps:
1. Identify the given values:
- Gravitational constant, [tex]\( G = 6.67 \times 10^{-11} \, \text{N} \cdot \text{m}^2 / \text{kg}^2 \)[/tex]
- Mass of the Earth, [tex]\( M_{\text{Earth}} = 5.97 \times 10^{24} \, \text{kg} \)[/tex]
- Mass of the satellite, [tex]\( m_{\text{satellite}} = 100 \, \text{kg} \)[/tex]
- Initial orbit radius, [tex]\( r_1 = 7.5 \times 10^6 \, \text{m} \)[/tex]
- Final orbit radius, [tex]\( r_2 = 7.7 \times 10^6 \, \text{m} \)[/tex]
2. Calculate the initial gravitational force [tex]\( F_1 \)[/tex] between the Earth and the satellite using Newton's law of universal gravitation:
[tex]\[
F_1 = \frac{G \cdot M_{\text{Earth}} \cdot m_{\text{satellite}}}{r_1^2}
\][/tex]
By substituting the given values:
[tex]\[
F_1 = \frac{(6.67 \times 10^{-11}) \times (5.97 \times 10^{24}) \times 100}{(7.5 \times 10^6)^2}
\][/tex]
This evaluates to:
[tex]\[
F_1 \approx 707.909 \, \text{N}
\][/tex]
3. Calculate the final gravitational force [tex]\( F_2 \)[/tex] when the satellite is at the new orbit radius:
[tex]\[
F_2 = \frac{G \cdot M_{\text{Earth}} \cdot m_{\text{satellite}}}{r_2^2}
\][/tex]
By substituting the given values:
[tex]\[
F_2 = \frac{(6.67 \times 10^{-11}) \times (5.97 \times 10^{24}) \times 100}{(7.7 \times 10^6)^2}
\][/tex]
This evaluates to:
[tex]\[
F_2 \approx 671.612 \, \text{N}
\][/tex]
4. Determine the change in gravitational force [tex]\( \Delta F \)[/tex]:
[tex]\[
\Delta F = F_2 - F_1
\][/tex]
By substituting the calculated values:
[tex]\[
\Delta F = 671.612 - 707.909
\][/tex]
This evaluates to:
[tex]\[
\Delta F \approx -36.297 \, \text{N}
\][/tex]
Thus, the approximate change in gravitational force from Earth as a result of this change in the satellite's orbit is [tex]\(-36 \, \text{N}\)[/tex]. Hence, the correct answer is:
D. [tex]\(-36 \, \text{N}\)[/tex]
