Respuesta :
ANSWER:
a) 7614.11 m/s
b) 8.427 N/kg
STEP-BY-STEP EXPLANATION:
Given:
Height (h) = 5.0 x 10^5 m
Radius of the earth (r) = 6.38 x 10^6m
Mass of earth (M) = 5.98 x 10^24kg
Mass of satellite (m)
a)
We can calculate the speed of the satellite by taking into account the following:
[tex]\begin{gathered} F_g=G\cdot\frac{M\cdot m}{d^2} \\ \\ F_c=\frac{mv^2}{d} \end{gathered}[/tex]In this case, the gravitational force and the centripetal force are equal, therefore:
[tex]\begin{gathered} G\cdot\frac{M\cdot m}{d^2}=\frac{mv^{2}}{d} \\ \\ \frac{GM}{d}=v^2 \\ \\ v=\sqrt{\frac{GM}{d}} \\ \\ G=6.67\cdot10^{-11}\frac{m^3}{kg\cdot s^2} \\ \\ d=r+h=6.38\cdot10^6+5.0\cdot10^5=6380000+500000=6880000\text{ m} \\ \\ \text{ We replacing:} \\ \\ v=\sqrt{\frac{6.67\cdot10^{-11}\cdot5.98\cdot10^{24}}{6880000}}=7614.11\text{ m/s} \end{gathered}[/tex]b)
Now, we calculate the gravitational field at that height, like this:
[tex]\begin{gathered} g=\frac{GM}{d^2} \\ \\ \text{ we replacing } \\ \\ g=\frac{6.67\cdot10^{-11}\cdot5.98\cdot10^{24}}{6880000^2} \\ \\ g=8.427\text{ N/kg} \end{gathered}[/tex]