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

[Silversonic]

I'm trying to understand why a superdeformed nucleus may be represented as bulging perpendicular to the axis of rotation, and I'm guessing this is akin to why the Earth does so too. I've gone through secondary school and 3 years of University to have professors/teachers snigger every time they hear the word centrifugal force. But whenever I look up the explanation for bulging, centrifugal force is always mentioned.

I honestly can't recall a situation where I've had to consider anything called a centrifugal force and I can't with any confidence even say what such a force even is. If I were to guess, in the non-inertial rotating frame of reference it's a force that appears to exist due to the Coriolis effect - throw a ball and it will seem to move from its original trajectory, as if a force was being applied. However to a stationary observer outside the rotating frame we of course just see the ball being thrown in a straight line - no force is being applied to anything.

Is there any way to understand why the Earth bulges at the equator due to its rotational motion, without having to delve into the concept of centrifugal forces?

 

[A.T.]

Is there any way to understand why the Earth bulges at the equator due to its rotational motion, without having to delve into the concept of centrifugal forces?

On the poles all the gravity acts to compress the planet. On the equator some of the gravity is used up for centripetal acceleration.

 

[Cosmobrain]

Is there any way to understand why the Earth bulges at the equator due to its rotational motion, without having to delve into the concept of centrifugal forces?

I don't know how, but I'll try. A point on the equator is inert. It wants to keep moving forward forever, but the rest of the planet is gravitationally curving it. At the same time, the point is also pulling the planet and trying to make it go forward with it, that's why the Earth bulges.
I do hope someone can explain this better

cb

 

[dauto]

There is nothing wrong with using the centrifugal force and in this case it is the easiest way to understand what's going on. just keep in mind that the centrifugal is an inertial force that appears because the rotating frame isn't an inertial frame. You also should know that the centrifugal force and the Coriolis force are two completely separate phenomena. Both appear in rotating reference frames but one does not create the other.

 

[FactChecker]

If I were to guess, in the non-inertial rotating frame of reference it's a force that appears to exist due to the Coriolis effect

Centrifugal force and Coriolis force are not directly related. Either can be 0 while the other is not. Both are "pseudo" forces to allow F=mA in a rotating frame but the Coriolis force is at right angles to the centrifugal force and is caused by a different velocity. It is zero if there is no change in the distance from the object of interest to the axis of rotation, whereas centrifugal force can be nonzero.

Is there any way to understand why the Earth bulges at the equator due to its rotational motion, without having to delve into the concept of centrifugal forces?

Yes. The bulge is because objects want to continue in a straight line tangent to the circle they are on. But you might as well give it a name and introduce centrifugal force.

 

[AlephZero]

If I were to guess, in the non-inertial rotating frame of reference it's a force that appears to exist due to the Coriolis effect

Almost, but not quite. If the non-inertial frame is rotating at constant angular velocity, the centrifugal force on a particle depends on its position in the rotating frame, and is a fictitious force of $m r \omega^2$ acting radially outwards.

The Coriolis force depends on the velocity of the particle in the rotating frame (which presumably is zero for your question).

For a rotating object in an inertial frame, there is a real force $m r \omega^2$ acting radially inwards on a parrticle, creating the centripetal acceleration.

I'm not sure why you are your professors are making such a big deal about one formulation rather than the other.

 

[A.T.]

Coriolis force is at right angles to the centrifugal force and is caused by a different velocity. It is zero if there is no change in the distance from the object of interest to the axis of rotation

Coriolis force is perpendicular to the velocity (in the rotating frame), not to the centrifugal force. If the velocity is purely tangential the Coriolis force is radial, not zero.

 

[FactChecker]

Coriolis force is perpendicular to the velocity (in the rotating frame), not to the centrifugal force. If the velocity is purely tangential the Coriolis force is radial, not zero.

I stand corrected. Coriolis has more to it than I visualized. Thanks.

 

[jtbell]

Is there any way to understand why the Earth bulges at the equator due to its rotational motion, without having to delve into the concept of centrifugal forces?

Imagine whirling an object around over your head at the end of a string. At a fixed angular speed (radians per second or revolutions per minute or whatever), the larger the radius of the object's path, the more centripetal force you have to exert on the string (and the string has to exert on the object).

All parts of the Earth rotate at the same angular speed. As the latitude decreases, moving towards the equator, the radius of revolution (at the Earth's surface) increases. Therefore the parts closer to the equator need larger centripetal forces to keep them revolving. To get those larger forces, the material of the Earth must "stretch out" more, similar to a spring whose tension force increases as you stretch it.

 

[bcrowell]

BTW, superdeformation doesn't always have to be stabilized by rotation. Superdeformation was discovered way back in the 60's, in nuclei in the uranium/plutonium region, where it exists at zero spin. The term to google for is "fission isomer." What *is* needed in all cases in order to stabilize superdeformed shapes is a gap in the single-particle states at superdeformed shape, i.e., a superdeformed shell closure.

 

[yuiop]

As an (hopefully) interesting side note, relativity predicts that if the Earth was a perfect rotating sphere, clocks at the Equator would tick slower than clocks at the poles due to time dilation caused by the tangential velocity at the equator. Relativity also predicts that time dilation if a function of distance from the gravitational mass and for a bulging Earth, the reduction in time dilation due the increased distance from the centre of the Earth at the equator cancels out the increased time dilation due to tangential velocity at the equator. This means that all clocks tick at the same rate at sea level anywhere on the Earth.

This is more than just a coincidence. The time dilation due to the combined factors of gravitational potential and tangential velocity create an effective potential and matter flows from a high effective potential to low effective potential and this shaped the Earth into its bulged shape. There is of course much more to it all than mentioned here, but this is just an aside (and definitely not a classical physics explanation).

 

[paisiello2]

... the centrifugal is an inertial force that appears because the rotating frame isn't an inertial frame.

Both are "pseudo" forces to allow F=mA in a rotating frame...

Can you give an example of an inertial frame where these forces do not appear and the bulge can still be explained?

 

[A.T.]

Can you give an example of an inertial frame where these forces do not appear and the bulge can still be explained?

See post #2

 

[D H]

Can you give an example of an inertial frame where these forces do not appear and the bulge can still be explained?

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.

 

[paisiello2]

And what exactly is the inertial reference frame you are considering? The rotating Earth is not an inertial reference frame.

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.

Suppose you had two planets of equal mass and density but you only could measure their diameters. You would have to infer that one of them was rotating if you measured a bulge on one of them. Yet what external real force was causing the bulge?

 

[A.T.]

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.

How exactly do you compute that total energy, which is constant over the surface in the inertial frame?

 

[A.T.]

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?

See post #2

Yet what external real force was causing the bulge?

There are no external forces, just gravity versus pressure.

 

[A.T.]

To get those larger forces, the material of the Earth must "stretch out" more, similar to a spring whose tension force increases as you stretch it.

The Earth is under radial tension?

 

[paisiello2]

See post #2


There are no external forces, just gravity versus pressure.

See post #15.

 

[A.T.]

See post #15.

I did, and answered it in post #17.

 

[paisiello2]

I did, and answered it in post #17.

.
And I responded in my post #19

 

[A.T.]

And I responded in my post #19

No, you didn't address my answer. You just referenced your question again, which was answered in post #2 before you even asked it.

 

[FactChecker]

Can you give an example of an inertial frame where these forces do not appear and the bulge can still be explained?

The rotating frame is not an inertial frame. As far as the inertial frame goes, the bulge is because objects in motion want to continue in a straight line. That is a law, not a force. The only force involved is the one that stops it from flying apart and limits it to a bulge.

 

[paisiello2]

No, you didn't address my answer. You just referenced your question again, which was answered in post #2 before you even asked it.

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

 

[paisiello2]

The rotating frame is not an inertial frame. As far as the inertial frame goes, the bulge is because objects in motion want to continue in a straight line. That is a law, not a force. The only force involved is the one that stops it from flying apart and limits it to a bulge.

It is not clear to me however that this law by itself would cause a bulge. Would a perfectly round rotating sphere also not satisfy this law?

 

[AlephZero]

You have to make some realistic assumptions about the material properties of the object. Obviously a perfectly rigid infinitely strong sphere would stay perfectly spherical at any rotational speed (ignoring relativistic effects), but no such thing exists.

In the case of the earth, a reasonable assumption is that below the relatively thin outer crust, the material behaves like a fluid, considering the length of time available for it to reach its current shape.

 

[paisiello2]

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?

 

[dauto]

It is not clear to me however that this law by itself would cause a bulge. Would a perfectly round rotating sphere also not satisfy this law?

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.

 

[D H]

How exactly do you compute that total energy, which is constant over the surface in the inertial frame?

The exact same way you would compute the potential energy in a rotating frame. The centrifugal potential in a rotating frame is equivalent to the kinetic energy in an inertial frame.

Whether you choose to look at things from the perspective of an inertial frame or an Earth-fixed frame, the potential due to centrifugal force (rotating frame perspective) / kinetic energy (inertial frame perspective) is not the hard part of the calculation. The hard part of the calculation is the gravitational potential energy. That's why space agencies around the world have poured so many resources into satellites that measure the Earth's gravitational field.

 

[russ_watters]

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?

Yes.

 

[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.