(1) 2 particles of equal mass are fixed at X=0 and another +ve point on the x axis.
No other grav influences are in the system. (they can be ignored)
I have to derive an expression for Vgrav at a general position x, on the x axis.
I believe this is related to V=-GM/r, but can't see how to develop it for 2 masses.
(2) I have to say what the significance of the point where dV/dx=0 is, and what it's coords are in the system.
i believe the derivative is where there is no slope, so no V. Am I right?
I believe the point is on the x axis at infinity, and y=0
Any steers appreciated. Thanks
To derive an expression for the gravitational potential (Vgrav) at a general position x on the x-axis, given two particles of equal mass fixed at X = 0 and another positive point on the x-axis, you can use the principle of superposition.
Assuming the two particles are fixed at positions x1 = 0 and x2 = d, where "d" represents the distance between the two particles, we can calculate the gravitational potential at position x.
The gravitational potential at a point due to a single mass can be expressed as V = -GM/r, where G is the gravitational constant, M is the mass of the particle creating the potential, and r is the distance between the particle and the point at which potential is to be calculated.
Let's consider the potential due to the first particle at x1 = 0. At this position, the distance between this particle and the point x is r1 = x.
Therefore, the potential at position x due to the first particle is given by V1 = -GM/r1 = -GM/x.
Now, let's consider the potential due to the second particle at x2 = d. The distance between this particle and the point x is r2 = d - x.
Therefore, the potential at position x due to the second particle is given by V2 = -GM/r2 = -GM/(d - x).
Since the gravitational potential obeys the principle of superposition, the total potential at position x due to both particles is the sum of the potentials calculated above:
V = V1 + V2 = -GM/x - GM/(d - x).
Regarding your second question...
The significance of the point where dV/dx = 0 is that it represents the equilibrium or stable points in the system, where the gravitational potential energy is at its minimum or maximum. At these points, there is no net force acting on an object, and it would remain at rest.
To find the coordinates of this point, we can find the value of x for which dV/dx = 0. Taking the derivative of the potential expression V with respect to x and setting it to zero:
dV/dx = GM/x^2 - GM/(d - x)^2 = 0.
Solving this equation will give you the value of x for the equilibrium point(s).