Gravitational Potential of Scooped Sphere System

[devvaibhav]

1. From a solid sphere of radius R and mass M, a small sphere of radius R/4 is scooped out and placed on the surface of the original sphere on a diametrically opposite point as shown in the figure. The gravitational potential of the resulting system at a point P(as shown in the figure) is (Take potential to be zero at infinity).

2. The gravitational potential due to a uniform solid sphere at a distance d from its coaxial axis is given by -GM/d

3. If the density of smaller sphere is ρ, the density of the scooped part is -ρ and the density of the sphere up on the bigger sphere is ρ or -ρ(i am confused here). However, i have not used it here. I have done my attempt by taking the density of the small sphere which is on bigger one to be +ρ.

Do i just need to add them or there is something else?
Thanks

 

[D H]

You pretty much need to assume that the original sphere has a uniform density. You won't get a definitive answer without this assumption. This is a fairly reasonable assumption for an introductory level physics problem.

With this assumption, you should be able to calculate the mass of the scooped-out sphere in terms of the mass of the original sphere. What is it?

You can calculate the potential due to that smaller, scooped out sphere that is atop the larger one using Φ=-GM/d. You cannot directly use this to calculate the potential due to the sphere with the scooped out mass.

You can use this expression of you take advantage of a little trick. You are treating this problem as two masses. Treat it as three masses: The original uniform sphere of mass M, another uniform sphere of mass m, and a third uniform sphere of mass -m. Potential energy is subject to the superposition principle: it is additive. So just calculate these three potentials and add them together.

 

[devvaibhav]

The Gravitational potential due to m and -m mass spheres will cancel out. So we are left only with the gravitational potential due to larger sphere left part whose mass is 63M/64. Hence the net gravitational potential becomes

[ATTACH=full]148920[/ATTACH]

.
Is my answer correct? or is there something still left?
Thanks

 

[D H]

No scooped out mass. You have two objects, a small sphere of mass m and a larger sphere of mass M less that scooped-out mass. The trick is to treat those two objects as three objects: The original sphere of mass M, a sphere of negative mass -m, and a sphere of positive mass +m. As you noted, the potential due to the small sphere of negative mass -m and that due the small sphere with positive mass +m cancel along the reference line. All that is left is the original sphere of mass M.

 

[devvaibhav]

No scooped out mass. You have two objects, a small sphere of mass m and a larger sphere of mass M less that scooped-out mass. The trick is to treat those two objects as three objects: The original sphere of mass M, a sphere of negative mass -m, and a sphere of positive mass +m. As you noted, the potential due to the small sphere of negative mass -m and that due the small sphere with positive mass +m cancel along the reference line. All that is left is the original sphere of mass M.

Yes exactly. I think i have treated them as three different objects and the potential due to +m and -m gets canceled. Is my thinking correct?

 

[D H]

Your thinking regarding the cancellation of potential is correct, at least along the reference line (it does not cancel in general). Your thinking that led you to that factor of 63/13 is incorrect.

 

[devvaibhav]

Your thinking regarding the cancellation of potential is correct, at least along the reference line (it does not cancel in general). Your thinking that led you to that factor of 63/13 is incorrect.

Well i calculated it like this.

The mass of volume 4∏R[SUP]3[/SUP]/3 → M
The mass of volume 1 → 3M/4∏R[SUP]3[/SUP]
Hence the mass of volume 4∏/3(R[SUP]3[/SUP]- R[SUP]3[/SUP]/64) → 3M/4∏R[SUP]3[/SUP] X 4∏/3(R[SUP]3[/SUP]- R[SUP]3[/SUP]/64 = 63M/64

What else do i need to do? Please tell me.

 

[D H]

As soon as you scoop that mass out you can no longer use -GM/r to compute the potential. That is only valid for point masses and masses with a spherical distribution.

You can however take advantage of the fact that gravitational potential is subject to the superposition principle.

 

[D H]

Another way to look at it: Conservation of mass. When you partition some composite object (your composite sphere with a scooped out mass plus the scooped out mass) into sub objects, the total mass of the sub objects must equal the mass of the composite object.

Here the composite object has a mass of M. When you split this into a large sphere of some mass, a smaller sphere of mass +m, and another smaller sphere of mass -m, what must the mass of that larger sphere be for the mass to total M?

 

[devvaibhav]

Another way to look at it: Conservation of mass. When you partition some composite object (your composite sphere with a scooped out mass plus the scooped out mass) into sub objects, the total mass of the sub objects must equal the mass of the composite object.

Here the composite object has a mass of M. When you split this into a large sphere of some mass, a smaller sphere of mass +m, and another smaller sphere of mass -m, what must the mass of that larger sphere be for the mass to total M?

Mass of larger sphere( means the one that is left after scooping out?) = M
Are you saying about this sphere?

But that is hard to digest. Means -m mass don't makes sense practically..It seems absurd. The mass of sphere(before scooping out) was also M and after scooping out is too M. Please but i am not able to feel it.
Thanks

 

[gneill]

Mass of larger sphere( means the one that is left after scooping out?) = M
Are you saying about this sphere?View attachment 42617
But that is hard to digest. Means -m mass don't makes sense practically..It seems absurd. The mass of sphere(before scooping out) was also M and after scooping out is too M. Please but i am not able to feel it.
Thanks

Yes, the mass of the original, unscooped large sphere is left as it is. The scooped-out version of the sphere is obtained by superimposing the small sphere of mass -m upon it. Essentially it is a mathematical trick.

To draw an analogy using natural numbers, suppose you have the number 6 and you want to remove 1 from it. You know that the result should be 6 - 1 = 5. Now, by extending the natural numbers to integers, thus allowing negative numbers to exist, you can then write the operation as a sum: 5 = 6 + (-1). Note that -1 is not a natural number; what does -1 of something "really mean"? It doesn't matter to us here! It is sufficient to know that mathematically 6 - 1 is equivalent to 6 + (-1).

So here we take the sphere of mass M and superimpose a small sphere of mass -m upon it. The "mathematical trick" of its negative mass exactly cancels the positive mass that it overlays, achieving precisely the same effect as scooping out that region of mass from the large sphere. The great advantage of this is that the masses involved are individually entirely spherical so that you can apply to them all the usual Newtonian formulas that apply to spherical masses.

 

[vela]

If the negative mass bothers you, you can think of it this way as well. Think of the big sphere as consisting of two pieces: the piece with the hole in it and the scooped out piece. Then by due to superposition, you can say that

potential of sphere = potential of big mass with hole + potential of scooped out piece

Then you're just rearranging things algebraically:

potential of big mass with hole = potential of sphere - potential of scooped out piece

 

[devvaibhav]

Thanks a lot @gneill and @vela
@vela You said that

potential of big mass with hole = potential of sphere - potential of scooped out piece

When we calculate the potential of big mass with hole = potential of sphere - potential of scooped out piece(which is at the top?). The potential which we would get will be more as potential of sphere - (-Gm/r). - and - = + . The result will be more(if - included as it will go towards 0) and less in magnitude(means if - in result is not included). My point is that the resultant potential which we will get will be different from that of M if we say so. Am i correct or not?
Thanks...

 

[vela]

Thanks a lot @gneill and @vela
@vela You said that

potential of big mass with hole = potential of sphere - potential of scooped out piece

When we calculate the potential of big mass with hole = potential of sphere - potential of scooped out piece(which is at the top?).

No, it's not at the top. You still need to account for that piece separately.

The potential which we would get will be more as potential of sphere - (-Gm/r). - and - = + . The result will be more(if - included as it will go towards 0) and less in magnitude(means if - in result is not included). My point is that the resultant potential which we will get will be different from that of M if we say so. Am i correct or not?
Thanks...

I'm not sure what you're getting at here.

 

[devvaibhav]

No, it's not at the top. You still need to account for that piece separately.I'm not sure what you're getting at here.

Well ok. So the net result of this discussion is
Potential_net at P = Potential of sphere + Potential of scooped out piece(that has been superimposed, we mean to say we assume it to be scooped and then superimpose it) + Potential of scooped out piece(which is at the top)
The last two terms cancel out as one has mass m and other -m. So the answer becomes

Is that right?
Thanks...

 

[D H]

That's it. Congrats!