Answer :
Answer:
a). F = 3.376 N, θ = 59.18°
b). W = 1.3x [tex]10^{9}[/tex] J
Explanation:
We know
Gravitational constant, G = 6.673 x [tex]10^{-11}[/tex] N-[tex]m^{2}[/tex]/[tex]kg^{-2}[/tex]
Mass of the earth, M = 5.97 x [tex]10^{24}[/tex] kg
mass of the moon, m = 7.35 x [tex]10^{22}[/tex] kg
Mass of the satellite, [tex]m_{s}[/tex] = 1250 kg
Distance between the objects, r = 3.84 x [tex]10^{5}[/tex] km
= 3.84 x [tex]10^{8}[/tex] m
Now
The force on the satellite due to moon
[tex]F_{m}= \frac{G\times m\times m_{s}}{r^{2}}[/tex]
[tex]F_{m}= \frac{6.673\times 10^{-11}\times 7.35\times 10^{22}\times 1250}{(3.84\times 10^{8})^{2}}[/tex]
[tex]F_{m}[/tex] = 0.0415 N ( in the positive x direction )
The force on the space craft due to the earth
[tex]F_{m}= \frac{G\times M\times m_{s}}{r^{2}}[/tex]
[tex]F_{m}= \frac{6.673\times 10^{-11}\times 5.97\times 10^{24}\times 1250}{(3.84\times 10^{8})^{2}}[/tex]
[tex]F_{m}[/tex] = 3.377 N ( at 60° to x axis )
Now component of force of earth along x axis
[tex]F_{e_{x}} = F_{e}\times cos 60[/tex]
= 3.377 x 0.5
= 1.6885 N
Now component of force of earth along y axis
[tex]F_{e_{y}} = F_{e}\times sin 60[/tex]
= 3.377 x 0.86
= 2.90 N
∴ Net force on the space craft due to earth and moon along x axis
[tex]F_{x}[/tex] = [tex]F_{e}[/tex] cos 60+[tex]F_{m}[/tex]
= 1.3885+0.0415
= 1.73 N
Net force on the space craft due to earth and moon along y axis
[tex]F_{x}[/tex] = [tex]F_{e_{y}}[/tex]
= 2.90 N
Therefore, total force F = [tex]\sqrt{(F_{x}^{2})+(F_{y}^{2})}[/tex]
F = [tex]\sqrt{(1.73^{2})+(2.90^{2})}[/tex]
F = 3.376 N
∴ Magnitude of the net gravitational force on the space craft is 3.376 N
Direction of net force on the space craft is given by
[tex]\Theta = \arctan \left (\frac{F_{y}}{F_{x}}\right )[/tex]
[tex]\Theta = \arctan \left (\frac{2.90}{1.73}\right )[/tex]
[tex]\Theta = 59.18[/tex]°
Therefore this direction is 59.18° from the line joining earth and the space craft.
b).
∴ Gravitational potential energy of the space craft is given by
[tex]E = \frac{G.M.m_{s}}{r}+\frac{G.m.m_{s}}{r}[/tex]
[tex]E = \frac{G\times m_{s}\left ( M+m \right )}{r}[/tex]
[tex]E = \frac{6.673\times 10^{-11}\times 1250\left ( 5.97\times 10^{24}+7.35\times 10^{22} \right )}{3.84\times 10^{8}}[/tex]
E = 1312769385 J
E = 1.3 x [tex]10^{9}[/tex] J
Therefore minimum work done is 1.3x [tex]10^{9}[/tex] J
