Gravitational Force Calculation w/ Superposition Principle

[Ranku]

Consider an object which is constituted of two different materials, which are inseparably mixed together, but which have different magnitude of masses. To calculate the gravitational force of the object upon another object, can the superposition principle be applied to the two constituent materials/masses of the object, as in considering the constituent masses as distinct, even though they have the same centre of gravity?

 

[Ibix]

Yes. This is a standard trick for computing the gravitational field of a hollow sphere. Compute the gravitational field of a solid sphere and add the gravitational field of the solid sphere that would exactly fill the hollow but has negative density.

 

[PeroK]

Yes. This is a standard trick for computing the gravitational field of a hollow sphere. Compute the gravitational field of a solid sphere and add the gravitational field of the solid sphere that would exactly fill the hollow but has negative density.

I never understand this idea of negative density. Instead:

Gravity of solid sphere = gravity of smaller sphere + gravity of hollow sphere

Then, rearrange that equation as required.

 

[A.T.]

I never understand this idea of negative density. Instead:

Gravity of solid sphere = gravity of smaller sphere + gravity of hollow sphere

Then, rearrange that equation as required.

The rearranging gives you the negative gravity term, which mathematically corresponds to what negative density would result in.

 

[A.T.]

Consider an object which is constituted of two different materials, which are inseparably mixed together, but which have different magnitude of masses. To calculate the gravitational force of the object upon another object, can the superposition principle be applied to the two constituent materials/masses of the object, as in considering the constituent masses as distinct, even though they have the same centre of gravity?

They don't even need to have the same center of gravity. The mixture can have varying proportions, so each component has a different distribution and center of mass.

Using @Ibix's example of the hollow sphere, but with the spherical cavity not concentric with the shell, you can use the superposition principle to show that the gravitational field in the cavity is non-zero and uniform.

 

[PeroK]

The rearranging gives you the negative gravity term, which mathematically corresponds to what negative density would result in.

Physically there is no such thing.

 

[jbriggs444]

Physically there is no such thing.

Agreed.

Similarly, a straight rod does not have a physical x component nor a physical y component. Nonetheless, if we find one laying on a piece of grid ruled paper, we are happy to apply the Pythagorean theorem.

 

[A.T.]

Physically there is no such thing.

It's no different than using negative gauge pressures. The density of the cavity is negative compared to the density of the filled part.

 

[PeroK]

It's no different than using negative gauge pressures. The density of the cavity is negative compared to the density of the filled part.

Why bother inventing a non-existent property when you don't need to?

 

[A.T.]

Why bother inventing a non-existent property when you don't need to?

No property is being invented. An existing property is merely expressed in relative terms, as we do for many other properties.

 

[PeroK]

No property is being invented. An existing property is merely expressed in relative terms, as we do for many other properties.

The concept of a physical body with negative density is being invented.

I don't understand why you don't accept that.

 

[A.T.]

The concept of a physical body with negative density is being invented.

No, as already explained the density is just negative in relative terms:

The density of the cavity is negative compared to the density of the filled part.

Similarly, using negative gauge pressures for gases doesn't imply that they have negative absolute pressures.

And using negative amounts of money in accounting doesn't imply that there are coins and notes with negative values on them.

 

[PeroK]

No, as already explained the density is just negative in relative terms:

And using negative amounts of money in accounting doesn't imply that there are coins and notes with negative values on them.

Money is invented in any case. I can owe you a physical object. That physical object cannot, however, have negative density.

 

[A.T.]

That physical object cannot, however, have negative density.

A physical object cannot have negative absolute density, just like a gas cannot have negative absolute pressure.

But both properties can be negative relative to some base value.

 

[DaveC426913]

Why bother inventing a non-existent property when you don't need to?

I believe imaginary numbers are very useful in electronics.
They vastly implify the mathematics, even though , technically, the mathematics can be solved (albeit onerously) without them.

Another example: Coriolis Force. Technically, it's a fictional force, but it's a very useful one in the right frame of reference.

PF-intelligentsia: please correct me if I'm wrong.