Gravitational Potential at a Midpoint: What is the Solution?

[pyman999]

Homework Statement

Homework Equations

g = G*M / r^2, where g is the gravitational field strength, G is the gravitational constant, M is the mass of the attracting body, r is the distance from the center of mass of the body.
V = -G*M / r, where V is the gravitational potential.

The Attempt at a Solution

For the gravitational field strength, as P is at a midpoint, we can say that r is 1/2*r, and as the masses are also equal, G*M / 1/2*r^2 = G*M / 1/2*r^2 (as they are equal, and in opposite directions), and they will simply cancel to 0. Therefore, it can't be A or B.

For gravitational potential, again, the masses are equal, and r is 1/2*r, so -GM / 1/2*r = -GM / 1/2*r, they will again cancel to 0. However, the answer is apparently C, where gravitational potential is -4G*M / r? I can't see why.

 

[BvU]

In V = -G*M / r there are no vectors, only scalars. r is a distance in this formula.

(dividing by a vector is awkward...)

 

[pyman999]

In V = -G*M / r there are no vectors, only scalars. r is a distance in this formula.

(dividing by a vector is awkward...)

I see, so then you're left with -2GM / r + -2GM / r, my mistake.

 

[mfb]

The problem statement should clarify that the potential "at infinity" is defined as zero. Otherwise the question is ambiguous - you can always add a constant value to the potential without changing physics.
Using this, you can rule out answer (D) simply from the fact that an object there would need energy to escape, therefore the potential there cannot be zero.

 

[HalfLostAndSearching]

I would like to provide a response to this content by clarifying a few points. Firstly, the equations provided in the content are for the gravitational field strength and gravitational potential of a point mass. In order to accurately calculate the gravitational potential at a midpoint, we need to consider the gravitational potential due to two masses, which is given by the equation V = -G*(M1/r1 + M2/r2), where M1 and M2 are the masses and r1 and r2 are the distances from the midpoint to each mass.

In this case, since the masses are equal and the midpoint is equidistant from both masses, the equation simplifies to V = -2*G*M/r, where M is the mass of each body and r is the distance from the midpoint to either mass. This is the correct answer, which matches option C.

Furthermore, it is important to note that the gravitational potential is a scalar quantity, meaning it only has magnitude and no direction. Therefore, the statement "in opposite directions" is not applicable in this context.

Lastly, as a scientist, it is important to always double-check and verify the equations and assumptions being used in a solution. In this case, it seems that the incorrect application of the equations for a single point mass led to the incorrect answer. It is always important to carefully consider the given information and choose the appropriate equations for the problem at hand.