(a)The law of gravity states that the force of gravity is additive and the force between two bodies of mass m_1 and m_2 and separated by distance d, due to gravity will be,
In our case, let us use the following abbreviations
=mass of satellite *this will become irrelevant once we go further)
=mass of earth=6x10^24 kg
=mass of the sun=2x10^30 kg
L= distance between earth and sun=1.5x10^8m
h=distance of the satellite from earths surface
=radius of the earth=6.37x10^6m
G=universal gravitational constant=6.67x10^-11 Nm^2/kg^2
When the satellite is at a height of h from the surface of the earth, it is at a height of (h+r_e) from the center of the earth. So, the force of pull due to earths gravity will be
if acceleration due to gravity is g(h) then
The data and graph of h and g(h) is shown below.
(b)Continuing on the logic of the previous problem, when the satellite is at height h above the earth's surface, and is instantaneously on the line between the earth and the sun, it is (L-h-r_e) from the center of the sun. So, the total force on the satellite towards the sun will be
and the acceleration will be
The acceleration due to earths gravity will be (same as earlier arrived)
Substituting numbers,
Plotting, we get:
(c) To find out more, let us plot as a function of h.
From the above table, we see that the earths gravity is dominant at 2x10^4 km, where the sun's gravity is 1% of the earth's gravity. At 1.6x10^5 m, the suns gravity starts to exceed earths gravity. Assuming 1% error to be acceptable in approximations, the sun's gravity can be ignored till 2x10^4 km.
(d) By observing the previous table, we see that at 1.8x10^5 km, the suns gravity greater than the earths gravity, and at this point, it can be said that the satellite has broken from the earths gravity and is under the major influence of the sun.
2. Consider a system consisting of the Sun, Earth, and a satellite in a circular orbit...
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