78. A launching vehicle carrying an artificial satellite of mass m is set for launch on the surface of the earth of mass M and radius R. If the satellite is intended to move in a circular orbit of radius 7R, the minimum energy required to be spent by the launching vehicle on the satellite is (Gravitational constant= G)
B. - 13 G M m14 R The energy of artificial satellite at the surface of the earth E1 = - G M mR When the satellite is intended to move in a circular orbit of radius 7R, then energy of artificial satellite E2 = - 12G M m7R The minimum energy required E = E1 - E2 = - G M mR + 12 G M m7R = - 14 G M m + G M m14 R E = -13 G M m14R The following four statements about circular orbits are equivalent. Derive any one of them from first principles.
Circular orbits arise whenever the gravitational force on a satellite equals the centripetal force needed to move it with uniform circular motion. Fc = Fg Substitute this expression into the formula for kinetic energy. K = ½m2v2 Note how similar this new formula is to the gravitational potential energy formula. K = −½Ug The kinetic energy of a satellite in a circular orbit is half its gravitational energy and is positive instead of negative. When U and K are combined, their total is half the gravitational potential energy. E = K + Ug E = −½Ug + Ug E = ½Ug The gravitational field of a planet or star is like a well. The kinetic energy of a satellite in orbit or a person on the surface sets the limit as to how high they can "climb" out of the well. A satellite in a circular orbit is halfway out (or halfway in, for you pessimists). Magnify Determine the minimum energy required to place a large (five metric ton) telecommunications satellite in a geostationary orbit. Start by determining the radius of a geosynchronous orbit. There are several ways to do this (which includes looking it up somewhere), but the traditional way is to start from the principle that the centripetal force on a satellite in a circular orbit is provided by the gravitational force of the Earth on the satellite. Combine this with the formula for the speed of an object in uniform circular motion. The algebra is somewhat tedious and has been condensed in the derivation below.
rf = 4.223 × 107 m (geostationary orbit) Next, use the virial theorem to determine the total energy of the satellite in orbit. This will be the final energy of the system.
Ef = −2.357 × 1010 J Ef = −23.57 GJ (geostationary orbit) To satisfy the minimum energy requirements of this problem the satellite should be launched from someplace on the equator where the speed of rotation (and thus the kinetic energy) is a maximum.
The initial energy of the satellite is the gravitational potential energy it has on the Earth's surface plus the kinetic energy it has due to the Earth's rotation. (Remember, gravitational potential energy is negative.) Ei = Ki + Ui
Ei = −3.120 × 1011 J Ei = −312.0 GJ (on the equator) Subtract the initial and final energies to finish the problem.
A satellite of mass m is in orbit about the Earth, which has mass M and radius R. (State all answers in terms of the given quantities and fundamental constants.)
Locate the L1, L2, and L3 Lagrange points for the Earth-Sun system using energy considerations. State your answers as distances…
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