At what distance is the gravitational pull balanced

[a97e]

Homework Statement

On December 25, 2004, the Huygens probe separated from the Cassini spacecraft orbiting Saturn and began a 22 day journey to Saturn's giant moon Titan (see the figure below (Figure 1) ), on whose surface it landed. It is useful to know that Titan is 1.22×106 km from the center of Saturn and has a mass of 1.35×1023 kg and a diameter of 5150 km . At what distance from Titan should the gravitational pull of Titan just balance the gravitational pull of Saturn?

Homework Equations

Ms=1.35*10^23
R=2575km=2575000m
Dt(Distance from titan)=?
Ds(Distance from saturn)=1.22*10^6

G*M/r^2=G*M/r^2

The Attempt at a Solution

G*(1.35*10^23)/(2575000)=

 

[berkeman]

Homework Statement

On December 25, 2004, the Huygens probe separated from the Cassini spacecraft orbiting Saturn and began a 22 day journey to Saturn's giant moon Titan (see the figure below (Figure 1) ), on whose surface it landed. It is useful to know that Titan is 1.22×106 km from the center of Saturn and has a mass of 1.35×1023 kg and a diameter of 5150 km . At what distance from Titan should the gravitational pull of Titan just balance the gravitational pull of Saturn?

Homework Equations

Ms=1.35*10^23
R=2575km=2575000m
Dt(Distance from titan)=?
Ds(Distance from saturn)=1.22*10^6

G*M/r^2=G*M/r^2

The Attempt at a Solution

G*(1.35*10^23)/(2575000)=

Welcome to the PF.

Don't stop now! Keep on going...

 

[a97e]

G*(1.35*10^23)/(2575000)^2=G*(mass of saturn)/(d)^2 ??
I'm not sure what to do with the distance from saturn to the moon or if it is multiplied by the mass of saturn. I'm really confused

 

[berkeman]

G*M/r^2=G*M/r^2

This is the key equation, but use different variables for the two different masses and distances. Try again?

 

[a97e]

G*M/r^2= G*m/(d)^2 where M is the mass of the moon, r is the radius of the moon, m is the mass of Saturn and d is the distance between them?
So,
G's cancel
(1.35*10^23)/(2575000)^2= (5.68*10^26)/(d)^2
d^2= (5.68*10^26)*(2575000)^2 /(1.35*10^23)
d= sqrt(2.78977*10^16)
d=167026167.8

 

[a97e]

G*M/r^2= G*m/(d)^2 where M is the mass of the moon, r is the radius of the moon, m is the mass of Saturn and d is the distance between them?
So,
G's cancel
(1.35*10^23)/(2575000)^2= (5.68*10^26)/(d)^2
d^2= (5.68*10^26)*(2575000)^2 /(1.35*10^23)
d= sqrt(2.78977*10^16)
d=167026167.8

Oh wait, is it d+radius of the moon?

 

[haruspex]

Dt(Distance from titan)=?
Ds(Distance from saturn)=1.22*10^6

I assume you are defining Dt as the distance from Titan to the point where the fields balance. The given distance 1.22*10^6km is from Titan to Saturn. In terms of these, how far is it from Saturn to where the fields balance?

When you have answered that, what is the field due to each at that point?

 

[a97e]

I assume you are defining Dt as the distance from Titan to the point where the fields balance. The given distance 1.22*10^6km is from Titan to Saturn. In terms of these, how far is it from Saturn to where the fields balance?

When you have answered that, what is the field due to each at that point?

So it's d+1.22*10^6 instead of d?

 

[haruspex]

So it's d+1.22*10^6 instead of d?

Draw a diagram. Show the centre of Saturn, S, the centre of Titan, T, a circle around Titan radius r, and a point where the two gravitational fields balance, P. Let the distance ST be Ds.
Where, roughly speaking, is P in relation to ST?
If the distance SP is x, what is the distance TP?