Gravitational force of particles

[sweetrose]

I was working on some gravitational force problems, and this one was particularly challenging to me:

"Two particles are located on the x axis. Particle 1 has a mass m and is at the origin. Particle 2 has a mass 2m and is at x=+L. Where on the x-axis should a third particle be located so that the magnitude of the gravitational force on BOTH particles 1 and particle 2 doubles? Express your answer in terms of L. Note that there are two answers."

I don't know where to get started on this problem, I know I will probably have to cancel out variables using the equation F=Gm1m2/r^2. I can picture the problem in my mind, but I don't know what I can solve for. Can anybody enlighten me by explaining to me where they would start?

*I started out simply setting up an equation for the force acting between particle 1 and particle 2. My diagram looks a little bit like this right now:


-----p1--------p2
...<--L------>

I know that if I place p3 in the somewhere between p1 and p2, the force on p1= F(between p1 and p3) + F(between p1 and p2), and the force on p2 would be F(between p2 and p3) + F(between p1 and p2).

When I try to picture the particle either on behind p1 or pass p2, it becomes a bit more difficult for me to picture, but I still get that you're going to have to subtract the forces, instead of adding them. The mass for p3 is not given, so I'm thinking that it cancels out somewhere in the problem. That's all I have for this problem. :shy:

 

[andrevdh]

The forces on the two original masses are equal in magnitude. If that needs to be doubled by a third mass it implies that the additional force caused on the two masses by the new mass must also be equal for both. Use this and write the distances in the equation in terms of x and L - x.

 

[quicknote]

I would approach this problem by first reviewing the concept of gravitational force and the equation F=Gm1m2/r^2. This equation tells us that the force between two particles is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

In this problem, we are given the masses of particle 1 and particle 2, and we are asked to find the location of a third particle that would result in the gravitational force on both particles doubling. This means that the force between particle 1 and particle 3, as well as the force between particle 2 and particle 3, must each be twice the original force between particle 1 and particle 2.

To start, I would set up an equation for the force between particle 1 and particle 3, using the given information and the equation F=Gm1m3/r^2. Similarly, I would set up an equation for the force between particle 2 and particle 3. These two equations can be set equal to each other, since we want the forces to be equal.

Solving for the distance r in terms of L, we can find two possible locations for particle 3, one between particle 1 and particle 2, and one beyond particle 2. This is because the force between two particles is always attractive, so the distance cannot be negative.

I would also recommend drawing a diagram to help visualize the problem and see how the forces are acting on each particle. This can also help in understanding why there are two possible solutions for the location of particle 3.

Overall, this problem requires a good understanding of the concept of gravitational force and the ability to set up and solve equations. With practice and a solid understanding of the concept, solving problems like this can become easier.