Net force is zero between two masses

[rpthomps]

Homework Statement

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 separated by distance r

Homework Equations

Netwon's Universal Law of Gravitation
Quadratic Formula

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?

 

[Chestermiller]

$$\frac{m_1}{x^2}=\frac{m_2}{(r-x)^2}$$
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

 

[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