Maximize Gravitational Field at Point P: Homework Solution

[neelakash]

Homework Statement

Given a point P in space and a given a piece of malleable material of constant density, how should you shape and place the material in order to create maximum gravitational field at P??

Homework Equations

The Attempt at a Solution

Possibly I can see that the shape should be spherical and the mass should be placed so that the CM differes from P only infinitesimally.

Please help me with the quntitative approach

 

[D H]

Do you know what the gravitational force due to a spherical shell is inside the shell? Knowledge of this answer might well change your answer to the original question.

 

[neelakash]

I know.It is zero.But,here we are not dealing with spherical shell.It's a given mass that I have to configure.
I was talking about a solid sphere and its CM
By the way,I might be wrong.What is the answer and why?

 

[D H]

Think of your solid sphere as a set of nested spherical shells.

 

[neelakash]

Ok,then,you are saying that the Intensity due to the spherical solid may be rejected.Obvously!I was gone off my head.The electrostatic analogy was:
(rho*r/3 epsilon).Thank you.
Please tell me what should be the configuration.

 

[D H]

I can't just give you the answer. That would involve breaking the rules. More importantly, it would also mean that you wouldn't learn anything.

The one answer you now know is wrong is the center of a solid sphere. I can give you some options to investigate.

• A spherical mass, with the test point somewhere inside the sphere (but obviously not at the center).
• A spherical mass, with the test point somewhere outside the sphere.
• Some other shape, such as a disk (where you would put the test point?)

 

[neelakash]

I did not ask you to provide me the solution...But,I needed the options you gave.
The problem is that there are many configurations available.You gave three.I may ask you why did you left a cylindrical volume or a conical volume.Are you sure they will not be?How do you know.

The configurations you suggested may be readily solved for the intensity...But that would not be an exact method...trial and error...

 

[neelakash]

Intensity at P must be eqal to the force exerted by the particle at P on the "system of particle"-the body you are referring to.Consider every pair of forces,superpose and then maximize...That might give some relationship between the particles' relative positioning.