Planetary Motion HW: Orbit Radius & Speed Around Jupiter

[k-rod AP 2010]

Homework Statement

An explorer plans a mission to place a satellite into a circular orbit around Jupiter, the radius of the planned orbit would be R.

a)The explorer wants the satellite to be sychronized w/ Jupiter's rotation. Determine the required orbital radius in meters.
b) What must the orbital speed of this satellite be in order to maintain this orbital period?

Homework Equations

R_J=7.14x10⁷m
M_J=1.9x10²⁷kg
Jupter's rotational period=3.55x10⁴s
$$G\ =\ 6.673(10)\ \times\ 10^{-11}\ m^{3} kg^{-1} s^{-2}$$

The Attempt at a Solution

a) T=√(4π²R³/GM_J)
3.55x10⁴s=√(4π²R³/GM_J)
R=1.59374x10⁸ m

b)v=√(GM_J/R)
v=√(GM_J/1.59374x10⁸ m)
v=28207.7 m/s

arm I following the correct procedure on these problems? i am trying to get a handle on this planetary motion stuff and i am not sure if i have yet.

 

[rock.freak667]

Your method is correct.

 

[k-rod AP 2010]

ok, that's what i thought thanks for the help. and another question pertaining to this hw,...

if the orbital speed were mistakingly made slower than needed the orbit would be smaller than desired, and if it were faster than needed the orbit would be larger and more elliptical than wanted right?

 

[rock.freak667]

ok, that's what i thought thanks for the help. and another question pertaining to this hw,...

if the orbital speed were mistakingly made slower than needed the orbit would be smaller than desired, and if it were faster than needed the orbit would be larger and more elliptical than wanted right?

Not too sure how elliptical it would be in real life, but with the assumption of circular motion

F_c=mv²/r, if 'v' were smaller, the 'r' would need to be smaller to maintain the same F_c, and if 'v' were larger, 'r' would be larger as well. So yes you are correct, but once again, I am not too sure how elliptical the motions would become.

 

[k-rod AP 2010]

ok that's what i was thinking, thanks again for all your help with this problem