Concept explainers
A 500-kg spacecraft first is placed into a circular orbit about the earth at an altitude of 4500 km and then is transferred to a circular orbit about the moon. Knowing that the mass of the moon is 0.01230 times the mass of the earth and that the radius of the moon is 1737 km, determine (a) the gravitational force exerted on the spacecraft as it was orbiting the earth, (b) the required radius of the orbit of the spacecraft about the moon if the periodic times (see Prob. 12.83) of the two orbits are to be equal, (c) the acceleration of gravity at the surface of the moon.
(a)
Find the gravitational force exerted on the spacecraft as it was orbiting the earth.
Answer to Problem 12.85P
The gravitational force exerted on the spacecraft as it was orbiting the earth is
Explanation of Solution
Given information:
The mass
The altitude
The mass of the moon is 0.01230 times the mass of the earth
The radius
Calculation:
Write the general equation of weight (W).
The radius
Find the altitude
Substitute
Consider the Newton’s law of universal gravitation:
Write the equation for mass of planet:
Substitute Equation (2) in Equation (1).
Substitute 500 kg for m,
Thus, the gravitational force exerted on the spacecraft as it was orbiting the earth is
(b)
Find the required radius of the orbit of the spacecraft about the moon if the periodic times of the two orbits are to be equal.
Answer to Problem 12.85P
The required radius of the orbit of the spacecraft about the moon if the periodic times of the two orbits are to be equal is
Explanation of Solution
Calculation:
Write the equation of acceleration
Write the Equation of attraction force between earth and satellites.
Here, m is the mass of satellite.
Find the mass of earth (M):
Consider the Newton’s law of universal gravitation:
Here, F is the attraction force, G is the universal constant, M is the mass of earth, and m is the mass of satellite.
Substitute Equation (3) in Equation (4).
Substitute Equation (1) in Equation (4).
Write the equation of velocity of earth:
Substitute Equation (8) in Equation (7).
Find the radius of moon
The periodic time of two orbit is equal.
Substitute
Substitute
Thus, the required radius of the orbit of the spacecraft about the moon if the periodic times of the two orbits are to be equal is
(c)
Find the acceleration of gravity at the surface of the moon.
Answer to Problem 12.85P
The acceleration of gravity at the surface of the moon is
Explanation of Solution
Calculation:
The radius of the moon
Find the acceleration of gravity at the surface of the moon
Substitute
Substitute
Thus, the acceleration of gravity at the surface of the moon is
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Chapter 12 Solutions
Connect 2 Semester Access Card for Vector Mechanics for Engineers: Statics and Dynamics
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