Net gravitational force of zero?

[tquiva]

Homework Statement

Imagine two spherical planets fixed on the x-axis, one with mass M at the origin, and the other with identical mass M at the position x = +d. (Assume that d is much greater than the radius of either planet.) At what position along the x-axis between the two masses could you position yourself so that you would experience a net gravitational force of zero?

Homework Equations

Fg = (Gm1m2) / r^2

The Attempt at a Solution

I know that
Fg = (Gm1m2) / r^2
and since both masses are identical, then
Fg = (GMM) / d^2 = (GM^2)/ d^2

I've tried plugging in numerous values for d in which both would cancel out, but I can't seem to find a value. Could someone please help me? Any help would be greatly appreciated...

 

[cepheid]

and since both masses are identical, then
Fg = (GMM) / d^2 = (GM^2)/ d^2

I've tried plugging in numerous values for d in which both would cancel out, but I can't seem to find a value.

Hopefully it is pretty obvious that plugging values into that particular equation is never going to get you a result of 0 for the force, not for any finite value of d. The error here is that you are calculating the force that each planet exerts on the other planet. What you need to calculate is the force that each planet exerts on you. This means there will be two equations, one for the force of planet 1 on you, and other for the force of planet 2 on you. Let's say that we're calling the planet that is at the origin 'planet 1.' Let's say also, for the sake of argument, that you are located a distance 'x' from the origin. This means that 'x' is your distance from planet 1. What is your distance from planet 2? Draw a picture.

You want to solve for the location 'x' that results in the force from planet 1 on you cancelling out the force from planet 2 on you. Remember that force has a direction, and in the case of gravity, the force always points towards the object that is causing it. This means that one of the forces will point to the right (which we could call the 'positive' direction), and the other force will point to the left (which we could call the 'negative' direction). Therefore, the forces will have opposite signs, so if you find x such that their magnitudes are equal, then they will cancel exactly.

 

[tquiva]

I finally came up with my value for x, which is x = d/2. However, part II of my problem asks for the same question except the mass at origin changes to 4M. I did the math, and it seems that it does not matter if both masses change, since the masses cancel out in the equation. So if I want a net gravitational force of zero with mass 4M, the value for x is also d/2 as in part I of the problem. Am I correct?

 

[cepheid]

I finally came up with my value for x, which is x = d/2. However, part II of my problem asks for the same question except the mass at origin changes to 4M. I did the math, and it seems that it does not matter if both masses change, since the masses cancel out in the equation. So if I want a net gravitational force of zero with mass 4M, the value for x is also d/2 as in part I of the problem. Am I correct?

Well, which one is it? Do both masses change, or just the one at the origin? If only one of the masses changes, then the answer will be different.

Intuitively, if both masses are the same, the balance point occurs at the location where they are equidistant from you. Makes sense, right? If the two masses are different, then the balance point will be closer to the lighter mass and farther from the heavier mass. Again, this makes sense, right?

 

[tquiva]

yes it does now.
thank you very much