Exercício de Dinamica Aplicada (273)

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434 
 
PROBLEM 12.85 
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 Problem 12.83) of the two orbits are to be equal, (c) the acceleration of gravity at the 
surface of the moon. 
 
SOLUTION 
First note that 66.37 10 mER = × 
Then 
6 6
6
(6.37 10 4.5 10 ) m
10.87 10 m
E E Er R h= + = × + ×
= ×
 
(a) We have 
2
[Eq. (12.28)]
GMm
F
r
= 
and 2GM gR= [Eq. (12.29)] 
Then 
2
2
2
m R
F gR W
rr
 = =  
 
 
For the earth orbit, 
26
2
6
6.37 10 m
(500 kg)(9.81 m/s )
10.87 10 m
F
 ×=   × 
 
or 1684 NF =  
(b) From the solution to Problem 12.78, we have 
 
2
31 2
M r
G
π
τ
 =  
 
 
Then 
3/22 r
GM
πτ = 
Now 
3/2 3/22 2E M
E M
E M
r r
GM GM
π πτ τ=  = (1) 
or 
1/3
1/3 6(0.01230) (10.87 10 m)M
M E
E
M
r r
M
 
= = × 
 
 
or 62.509 10 mMr = × 2510 kmMr = 