Gravitational Potential inside a mass

[Knightycloud]

I got an exam tomorrow and while i was studying, this popped in.
Is there a gravitational potential inside a mass (eg : 500 km deep from the Earth surface)?

We can calculate the Gravitational field intensity inside a mass by taking the distance from the center to that inner-point as x and calculating the mass within that radius; then taking that calculated mass into the G.I = -GM/x²

Can we do the same thing to calculate the potential? I googled and those graphs showed a similar behavior, we get as the potential inside a conductive sphere. Why can't us calculate it like that above method? I only understand GCE A/L by the way.

 

[D H]

Is there a gravitational potential inside a mass (eg : 500 km deep from the Earth surface)?

Of course.

Can we do the same thing to calculate the potential?

No.

Consider a uniform density sphere. Use $\Phi(r) = -GM(r)/r$ and take the gradient. This should yield the additive inverse of the force since $\vec F(\vec r) = -\nabla \Phi(\vec r)$. However, since $M(r)=4/3\pi \rho r^3$ for a uniform density sphere, this Φ(r) yields $F=+8/3\pi G \rho r \hat r$. The sign is wrong and the magnitude is twice as large as it should be. The potential must yield $F=-\partial \Phi(r)/\partial r = -4/3\pi G \rho r \hat r$.

 

[mfb]

Potentials are not uniquely defined - you can always add a constant. That is fine if you just consider a single potential, but it ruins the approach to combine the potential for each depth in that way.

You can calculate the force, and based on this force you can calculate the potential - that approach works.

 

[Knightycloud]

You can calculate the force, and based on this force you can calculate the potential - that approach works.

That makes sense. Because my level of knowledge is limited to the basic principles and i haven't used δf(x)/δx applications any where and that describes how few I know about this topic. :/

Thanks a lot for the support people.

 

[PJC01]

Yes, it is possible to calculate the gravitational potential inside a mass using the same method as calculating the gravitational field intensity. This is because the potential is directly related to the field intensity through the equation V = -GM/r, where V is the potential, G is the gravitational constant, M is the mass, and r is the distance from the center of the mass.

However, it is important to note that the potential inside a mass would not be constant, as it would vary depending on the distance from the center. This is due to the fact that the mass is distributed throughout the volume of the object, and the potential would decrease as you move closer to the center.

In addition, the potential inside a conductive sphere may appear similar to the potential inside a mass due to the fact that both are spherically symmetric. However, the potential inside a conductive sphere is due to the distribution of charges, while the potential inside a mass is due to the distribution of mass.

I hope this helps clarify your understanding. Good luck on your exam tomorrow!