# Net force is zero between two masses

1. Nov 24, 2014

### rpthomps

1. The problem statement, all variables and given/known data

The gravitational field strength between two objects is the sum of two vectors pointing in opposite directions. Somewhere between the objects, the vectors will cancel, and the total force will be zero. Determine the location of zero force as a fraction of the distance r between the centres of two objects of mass

Mass 1 and Mass 2 seperated by distance r

2. Relevant equations

Netwon's Universal Law of Gravitation

3. The attempt at a solution

Here is my work...

my problem is that when m1=m2 the equation blows up rather than be x=½ of r, which it should be.

Any thoughts on why this is so?

2. Nov 24, 2014

### Staff: Mentor

$$\frac{m_1}{x^2}=\frac{m_2}{(r-x)^2}$$
Taking the square root of both sides:
$$\frac{\sqrt{m_1}}{x}=\frac{\sqrt{m_2}}{r-x}$$
Note that only the positive square root makes sense physically. So,
$$\frac{x}{r}=\frac{\sqrt{m_1}}{(\sqrt{m_1}+\sqrt{m_2})}$$
In your final equation, only the negative term is applicable. So:

$$\frac{x}{r}=\frac{m_1-\sqrt{m_1m_2}}{m_1-m_2}=\frac{\sqrt{m_1}(\sqrt{m_1}-\sqrt{m_2})}{m_1-m_2}=\frac{\sqrt{m_1}}{(\sqrt{m_1}+\sqrt{m_2})}$$

Chet

3. Nov 24, 2014

### rpthomps

oooh...very nice. You came up with a nice short solution and then debugged my own. Thanks for your time and your insight.

Ryan