Point between earth and moon where net gravitational force is zero

[tv2le]

A) If the moon of mass m_M has radius R_M and the distance between the centers of the Earth and the moon is R_EM, find the total gravitational potential energy of the particle-earth and particle-moon systems when a particle with mass m is between the Earth and the moon, and a distance r from the center of the earth. Take the gravitational potential energy to be zero when the objects are far from each other. Take the mass of Earth as m_E.

B)There is a point along a line between the Earth and the moon where the net gravitational force is zero. Use the expression derived in part (a) to find the distance of this point from the center of the Earth in meters.


The attempt at a solution: In Part A, I was able to find that U= -Gm_Em/r - Gm_Mm/(R_EM-r)
For Part B, do I just set the equation for potential energy equal to 0? I tried that, and got stuck, because I end up with 2 unknown variables (distance from Earth and distance from moon), so what would the second equation be to help solve for both of these variables?

 

[Char. Limit]

Aren't the distance from the Earth and the distance from the moon related? That is, isn't their sum constant?

 

[tv2le]

I ended up with r = ((m_E/m_M)*R_EM)/(1-(m_E/m_M)) Is that right?

 

[D H]

No, it's not.

You did (at least) two things wrong here.
1. You have the wrong expression for potential energy. Your expression is correct for points between the Earth and Moon, but not for points beyond.

2. You (apparently) solved for the point where energy is equal to zero. You are supposed to find where the force is zero.

 

[tv2le]

I think I got it. Thank you.