12.78. Identify:
g
GM
R 2 , where M and R are the mass and radius of the planet.
Set Up: Let mU and RU be the mass and radius of Uranus and let g U be the acceleration due to gravity
8
at its poles. The orbit radius of Miranda is r  h  RU , where h  1.04 10 m is the altitude of Miranda
above the surface of Uranus.
Execute: (a) From the value of g at the poles,
2
7
g U RU2 11.1 m/s   2.556  10 m 

 1.09  1026 kg.
11
2
2
G
6.673

10
N

m
/kg


2
mU 
GmU /r 2  gU  RU /r   0.432 m/s2
2
(b)
.
2
2
(c) GmM /RM  0.080 m/s .
Evaluate: (d) No. Both the object and Miranda are in orbit together around Uranus, due to the
gravitational force of Uranus. The object has additional force toward Miranda.