Centrifugal force? Why does the Earth bulge at the equator?

[A.T.]

And you you just referenced a previous post that didn't address my question.

You asked to explain the bulge without inertial forces. Post #2 explains the bulge without inertial forces.

 

[A.T.]

Ok, let's assume the forces of the chemical bonds are negligible. Does the law of inertia and the force of gravity by themselves predict that a bulge would occur?

Under gravity and inertia alone the Earth would collapse and there would be no stable configuration anymore.

 

[paisiello2]

Assuming you're neither at the equator nor at the poles do the following thought experiment. Put a sphere on a level table so that it is a rest. From the point of view of a inertial observer that sphere is not at rest at all and is actually describing a circular motion. That means the net force acting on the sphere points towards the center of that circle. Also note that the center of the circle does not coincide with the center of the earth. But the force of gravity does point to the center of the Earth which means that the other force acting on the sphere - the normal force holding it up - cannot be parallel to the force of gravity, otherwise it would be impossible to build a net force pointing to the center of the circle. We just proved that a level table is not perpendicular to the line that passes by the center of the Earth and by the table so as the sphere moves from one end of the table to the other it slowly moves away from the center of the earth. put another table next to that one, and another, and so on all the way from the pole to the equator and the sphere moves away from the center of the Earth proving that the Earth's surface must bulge at the equator.

I think I follow what you are saying. Adding the table to the experiment seemed a little confusing but I get the gist: particles moving with the Earth's rotation and free to roam the Earth's surface are going to accelerate towards the equator.

I assume the reasoning would continue that if the Earth is made up of a large number of similar "spheres" or particles then their tendency will also be to accelerate towards the equator. But because they are constrained by one another they exert a force on each other instead. And the result of all these forces acting is to deform the Earth's shape into an oblate spheroid.

Would that be a valid extension of your reasoning?

 

[paisiello2]

You asked to explain the bulge without inertial forces. Post #2 explains the bulge without inertial forces.

My apologies then that I didn't understand your post.

 

[dauto]

I think I follow what you are saying. Adding the table to the experiment seemed a little confusing but I get the gist: particles moving with the Earth's rotation and free to roam the Earth's surface are going to accelerate towards the equator.

I assume the reasoning would continue that if the Earth is made up of a large number of similar "spheres" or particles then their tendency will also be to accelerate towards the equator. But because they are constrained by one another they exert a force on each other instead. And the result of all these forces acting is to deform the Earth's shape into an oblate spheroid.

Would that be a valid extension of your reasoning?

Whatever works for you.

 

[paisiello2]

I think it's not whatever works for me but rather whatever works for the physics.

Clearly your table analogy doesn't go far enough.

 

[dauto]

I think it's not whatever works for me but rather whatever works for the physics.

Clearly your table analogy doesn't go far enough.

This thread is much ado about nothing, in my opinion. If you don't like the tables, remove them and use the floor instead...

 

[paisiello2]

This thread is much ado about nothing, in my opinion. If you don't like the tables, remove them and use the floor instead...

Whatever works for you...

 

[BruceW]

Even if you pick the "fixed" stars as your inertial frame, you will still measure a bulge in the earth. So where did the force to cause this deformation come from? It can't be a fictitious force because they supposedly do not exist in an inertial frame.

then call it a lack of a force. think of each bit of matter as it spins around the Earth's axis. If there were more force on the matter at the equator, there would be less bulge. If the forces were suddenly zero, the bulge would be infinite, because the matter would just all fly outwards at it's current velocity.

 

[paisiello2]

then call it a lack of a force. think of each bit of matter as it spins around the Earth's axis. If there were more force on the matter at the equator, there would be less bulge. If the forces were suddenly zero, the bulge would be infinite, because the matter would just all fly outwards at it's current velocity.

But is this enough by itself to cause the deformation?

Based on the postings made by others I think now that the bulge is caused by the portion of the Earth outside the equatorial region pushing or squeezing against the portion of the Earth in the equatorial region. This squeeze is caused by the Earth's gravity trying to accelerate the moving portions towards the center of the earth.

 

[D H]

From the perspective of a frame that rotates with the Earth, the surface of the Earth is very well approximated as a surface of constant potential energy, where the potential from both the gravitation and centrifugal forces contribute to the total.

From the perspective of an inertial (non-rotating) frame, the surface of the Earth is very well approximated as a surface of constant total energy, where both the gravitation force and kinetic energy contribute to the total.

These two perspectives yield the same result.

That was wrong.

It's the difference between kinetic and potential energy that is minimized, not the sum. Nothing's moving from the perspective of an Earth-fixed frame, so there is no kinetic energy in this frame. There is however a potential due to the fictitious centrifugal force in this frame, $-\frac 1 2 \omega^2 r^2 \sin^2\theta$. There is no centrifugal force in an inertial frame, but things are moving in this frame. The specific kinetic energy $\frac 1 2 v^2$ can be rewritten as $\frac 1 2 \omega^2 r^2 \sin^2\theta$. Note well: The kinetic energy in the inertial frame and the centrifugal potential in the rotating frame are additive inverses.

What this means is that the Lagrangian L=T-V is the same whether one looks at things from the perspective of an Earth-fixed (rotating) frame or an inertial frame. The principal of least action mandates the presence of an equatorial bulge.