46 By what percent the energy of the satellite has to be increased to shift it from an orbit of - Physics - Gravitation

Question

46 By what percent the energy of the satellite has to be increased
to shift it from an orbit of radius 'r' to 3r/2 --------
47 At what height from the surface of earth, the total energy of
satellite is equal to its potential energy at a height 2R from the
surface of earth (R=radius of earth)-------

Somnath physics · Accepted Answer

46
Let mass of the satellite be mTotal energy of the satellite = 12Iω2+(−GMmr) for a mass moving in a circle, MI about its center I=mr2 =>E1=12mr2ω2−GMmr −−−1
 Again,Satellite's centripetal force must balance the gravitational pullmω2r=GMmr2=>ω2=GMr3 −−−2
From eqn  1 and eqn
 2E1=12mr2GMr3−GMmr=−12GMmrAt  1
5rE2=−GMm3rΔE=E2−E1=−GMm3r−(−12GMmr)=>ΔE=GMm6rΔE|E|%=GMm6rGMm2r×100=1003=33
3% ​(approx) will have to be increased

47
 Height 2R from surface of the earth = 3R from center of the earth
Potential energy of the satellite= −GMm3RTotal energy of a satellite at a distance r from the center = −12GMmrA/q −12GMmr=−GMm3R=>r=32RFrom the surface of the earth the distance will be r−R32R−R=R2

46.By what percent the energy of the satellite has to be increased to shift it from an orbit of radius 'r' to 3r/2 --------47.At what height from the surface of earth, the total energy of satellite is equal to its potential energy at a height 2R from the surface of earth (R=radius of earth)-------

46.By what percent the energy of the satellite has to be increased to shift it from an orbit of radius 'r' to 3r/2 --------
47.At what height from the surface of earth, the total energy of satellite is equal to its potential energy at a height 2R from the surface of earth (R=radius of earth)-------