How Can We Deform a Material to Maximize Its Gravitational Field at a Point?

[ShayanJ]

Consider a piece of a deformable material with mass m and constant density $\rho$. To what shape should we deform it so that its gravitational field is maximum in a given point?

 

[mfb]

This looks a bit like homework, so I'll start with a hint: for the best shape, every point on the surface will contribute the same to the gravitational field strength (I guess field strength is meant, not potential).

 

[ShayanJ]

This isn't a homework, because I have no idea what to do about it!
I should say that just naming a shape isn't enough, the equation of the shape should be derived somehow!

 

[A.T.]

https://www.physicsforums.com/showthread.php?t=543287

 

[pmacias]

I would approach this question by first understanding the concept of gravitational field and its relationship to mass and density. The gravitational field is a measure of the strength of the gravitational force at a given point in space, and it is directly proportional to the mass of an object and inversely proportional to the square of the distance from the object.

In this scenario, we are dealing with a deformable material with a constant density, which means that the mass of the material will be directly proportional to its volume. This means that the shape of the material will not affect its mass, and therefore, it will not have a direct impact on the strength of its gravitational field.

However, the shape of the material can indirectly affect the gravitational field if it is deformed in a way that changes the distance between the material and the given point. For example, if the material is stretched or compressed in a way that brings it closer to the given point, the gravitational field at that point would increase.

Therefore, to maximize the gravitational field at a given point, we would need to deform the material in a way that brings it as close as possible to that point without changing its mass. This could potentially be achieved by shaping the material into a cone or pyramid-like structure with the given point at its apex, as this would minimize the distance between the material and the point. However, it is important to note that the exact shape needed to maximize the gravitational field would depend on the specific properties and dimensions of the material and the given point.

In summary, the shape of a deformable material does not directly affect its gravitational field, but it can indirectly impact it by changing the distance between the material and a given point. To maximize the gravitational field at a given point, the material should be deformed in a way that minimizes this distance while maintaining its mass.