Solve Universal Gravitation Homework: Find Distance between Moon & Satellite

[mtayab1994]

Homework Statement

Let (S) be a satellite between the Earth and the moon such that (S) is at a distance dm from the moon. All forces on the the satellite are null (equal zero). Find the distance dm .

Given: Distance between moon and Earth is 38*10^4km and the mass of Earth is 81 times the mass of the moon.

The Attempt at a Solution

Well it's been a long time since I've done any universal gravitation. It would be nice if someone can just give me an idea on how to start and I'll go from there.

 

[SammyS]

Homework Statement

Let (S) be a satellite between the Earth and the moon such that (S) is at a distance dm from the moon. All forces on the the satellite are null (equal zero). Find the distance dm .

Given: Distance between moon and Earth is 38*10^4km and the mass of Earth is 81 times the mass of the moon.

The Attempt at a Solution

Well it's been a long time since I've done any universal gravitation. It would be nice if someone can just give me an idea on how to start and I'll go from there.

Start with Newton's law of universal gravitation. Link

If the distance between the moon & the satellite is d_m , then what is the distance between the satellite & the Earth ?

 

[mtayab1994]

Start with Newton's law of universal gravitation. Link

If the distance between the moon & the satellite is d_m , then what is the distance between the satellite & the Earth ?

The distance between the Earth and the satellite is d=D-dm and Newton's law of gravitation says that F=(G*m1*m2)/(r^2)

 

[mtayab1994]

Let's name the distance from Earth to the satellite by D. So D=d-dm.

And by using g=(GM)/D^2 we get that the distance from the Earth is 6.36*10^3 km.

So that means the distance from the satellite to the moon is

d-D=3.8*10^5-6.36*10^3=3.74*10^5 km. Is that correct??

 

[SammyS]

Let's name the distance from Earth to the satellite by D. So D=d-dm.

And by using g=(GM)/D^2 we get that the distance from the Earth is 6.36*10^3 km.

So that means the distance from the satellite to the moon is

d-D=3.8*10^5-6.36*10^3=3.74*10^5 km. Is that correct??

It's not clear what all you put together to get that result.

Please fill in and explain some steps.


My approach would be set the magnitudes of the following two forces equal to each other.

The force exerted on the satellite by the moon.

The force exerted on the satellite by the earth.​

 

[mtayab1994]

By setting the forces equal to one another i got in the end:

Me/Mm=dm^2/de^2 and we know that the mass of the Earth is 81 times the mass of the moon so we get 81Mm/Mm=(dm^2)/(de^2) and then we cancel with the mass of the moon and square root both sides and we are left with. √81=dm/de and we know that the distance from the satellite and the Earth is d-dm. and by doing some simple algebra we find that the distance to the moon is d/10 the total distance between the moon and Earth so that gives us
3.8*10^4 km.

 

[SammyS]

By setting the forces equal to one another i got in the end:

Me/Mm=dm^2/de^2 and we know that the mass of the Earth is 81 times the mass of the moon so we get 81Mm/Mm=(dm^2)/(de^2) and then we cancel with the mass of the moon and square root both sides and we are left with. √81=dm/de and we know that the distance from the satellite and the Earth is d-dm. and by doing some simple algebra we find that the distance to the moon is 9/10 times the total distance between the moon and Earth so that gives us
3.42*10^5 km. Is that correct now??

It's quite difficult to try to figure out what all you have done to get your answer. Having youe text all packed together so tightly doesn't help either.

\displaystyle G \frac{M_m\, m}{{d_m}^2}=G \frac{M_e\, m}{(d-d_m)^2}\,,\

where M_m is the moon's mass, M_e is the Earth's mass, m is the satellite's mass, d_m is the satellite's distance from the moon, and d is the monn's distance from earth.

M_e = 81 M_m. Plugging this in & doing some manipulation gives:

\displaystyle \frac{M_m}{{d_m}^2}=\frac{81M_m}{(d-d_m)^2}\ \displaystyle\quad\to\quad d-d_m=d_m\sqrt{81}=9d_m\ \displaystyle\quad\to\quad d_m=\frac{d}{10}\

Well, yes, your answer is correct.