Explanation Given information: The radius of the circular orbit is r 0 . The velocity of the satellite is v 0 . The angle of the satellite is α . Calculation: The expression for the geocentric force acting on the satellite when it is on the surface of earth ( F ) as follows: F = G M m r 0 2 Here, G is the universal gravitational constant, M is the mass of the earth and m is the mass of the space vehicle. Show the figure with the velocities acting as in Figure (1). Show the figure of the circular orbit with the forces as in Figure (2). The expression for the normal acceleration ( a n ) as follows: a n = v 0 2 r 0 Calculate the velocity at the circular orbit ( v 0 ) by considering the force equilibrium using the relation: ∑ F n = m a n Substitute v 0 2 r 0 for a n and G M m r 0 2 for F n . G M m r 0 2 = m ( v 0 2 r 0 ) G M r 0 = v 0 2 v 0 = G M r 0 The velocity v 0 forms an angle α with the intended circular path. The expression or the principle of conservation of angular momentum at center of the earth to the point A as follows: m r 0 v 0 cos α = m r A v A v A = ( r 0 r A cos α ) v 0 The expression for the kinetic energy of the satellite at circular orbit ( T 0 ) as follows: T 0 = 1 2 m v 0 2 Here, m is the mass of the satellite. The expression for the gravitational potential energy of the circular orbit ( V 0 ) as follows: V 0 = − G M m r 0 The expression for the kinetic energy at point A ( T A ) as follows: T A = 1 2 m v A 2 The expression for the gravitational potential energy at point A ( V A ) as follows; V A = − G M m r A The expression for the principle of conservation of energy at the center of the earth and point A as follows: T 0 + V 0 = T A + V A Substitute 1 2 m v 0 2 for T 0 , − G M m r 0 for V 0 , 1 2 m v A 2 for T A , and − G M m r A for V A . 1 2 m v 0 2 − G M m r 0 = 1 2 m v A 2 − G M m r A v 0 2 2 − G M r 0 = v A 2 2 − G M r A v 0 2 2 − v A 2 2 = G M r 0 − G M r A v 0 2 − v A 2 2 = G M ( 1 r 0 − 1 r A ) v 0 2 − v A 2 = 2 G M r 0 ( 1 − r 0 r A ) Substitute ( r 0 r A cos α ) v 0 for v A and G M r 0 for v 0 2

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ISBN: 9781259633133

Author: BEER

Publisher: MCG

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Chapter 13.2, Problem 13.108P

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Find the maximum (rmax) and minimum (rmin) values of the distance from the center of the earth to the satellite.

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Chapter 13.2, Problem 13.107P

Chapter 13.2, Problem 13.109P

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Find the maximum ( r max ) and minimum ( r min ) values of the distance from the center of the earth to the satellite.

Solution Summary: The author calculates the distance from the center of the earth to the satellite by considering the force equilibrium.

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Find the maximum (rmax) and minimum (rmin) values of the distance from the center of the earth to the satellite.

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