Centrifugal force created by the Earth's rotation

[Kyrisch]

How much centrifugal force is created by the Earth's rotation, if any? It is obviously not enough to counteract the Earth's gravity, and send us all flying into space, but is there still some acting on the bodies on the Earth?

[MetaFrizzics]

In Newtonian gravity theory, the bulge and mass allows you to calculate the rotation relative to Absolute space. However, independant experiments seem to confirm that Absolute Space and the backdrop of fixed stars are more or less synonymous. This means that you can go the other way and calculate the mass of the earth by its bulge and the speed of its rotation (relative to the backdrop of stars).

 

In General Relativity, there is a similar argument, but it is based upon the gravitational field (geometry of spacetime) rather than 'Absolute Space'.

[swansont]

How much centrifugal force is created by the Earth's rotation, if any? It is obviously not enough to counteract the Earth's gravity, and send us all flying into space, but is there still some acting on the bodies on the Earth?

 

It wouldn't send you flying into space in any event - the astronauts in orbit have the condition you seek. They remain in orbit and feel weightless, as they are in perpetual freefall.

 

Find the centripetal force of an object on the surface of the earth (mv²/r), and that will tell you how much of the gravitational force is "being used" to keep you moving in a circle, or is the centrifugal force of an object in a rotating reference frame. The "leftover" amount is what you feel as weight.

[mikeknight37]

Centrifugal force distributes a small proportion of itself to make us rotate with the earth and a large proportion of it becomes the gravity.

[swansont]

Centrifugal force distributes a small proportion of itself to make us rotate with the earth and a large proportion of it becomes the gravity.

 

 

Ummm, no.

 

"Centrifugal" force is a pseudoforce present in a rotating coordinate system - it's not a real force. It looks like it's there because you want Newton's laws to hold.

 

The reality is that an object moving in a circle feels a centripetal force - toward the center of the circle. And the centripetal force is not a new or extra force - it has to be one of the fundamental forces acting on the object. In the astronaut example, the centripetal force is the gravitational force They are the same, and are thus set equal when solving for the equation of motion to find speed or orbital period, etc.

[Daecon]

Ummm' date=' no.

 

"Centrifugal" force is a pseudoforce present in a rotating coordinate system - it's not a real force. It looks like it's there because you want Newton's laws to hold.

 

The reality is that an object moving in a circle feels a centripetal force - toward the center of the circle. And the centripetal force is not a new or extra force - it has to be one of the fundamental forces acting on the object. In the astronaut example, the centripetal force [i']is[/i] the gravitational force They are the same, and are thus set equal when solving for the equation of motion to find speed or orbital period, etc.

 

If I were to swing a bucket of water around my head on a lentgh or rope, what keeps the water from spilling out? I always thought that's what centrifugal meant...

[mezarashi]

How much centrifugal force is created by the Earth's rotation, if any? It is obviously not enough to counteract the Earth's gravity, and send us all flying into space, but is there still some acting on the bodies on the Earth?

 

As swansont has corrected, centrifugal force is virtual. It exists only to fulfill Newton's third law of force pairs. The force that keep's an object in circular motion is called the "centripetal force", and this force may be gravity in planetary orbits or a string if you spin in circles a rock. The rock is subjected to a centripetal force and thus centripetal acceleration. Your hand feels the centrifugal force tugging on the string (the force that appears to cause the rock to pull on the string), which is in fact only an equal and opposite force since you are using the string to pull the rock into circular motion.

 

On the original question. A better question would be, what portion of the Earth's gravity contributes to the centripetal force that keeps us in circular motion about the Earth's surface. Obviously, if all the gravity contributed to keeping us in rotation around Earth, then we would feel weightless as astronauts do in space. The numbers will vary depending on where you live on earth, but let's take for example the equator. We know that the Earth has a circumference of about 40,000km, which we transverse in 24 hours, leading to speeds of 462 meters/second. According to a = v^2 / r, the centripetal acceleration required to keep us on the surface of the Earth is 0.03557 m/s^2 while the gravitation acceleration created by the earth's gravity is 9.79m/s^2.

[swansont]

If I were to swing a bucket of water around my head on a lentgh or rope, what keeps the water from spilling out? I always thought that's what centrifugal meant...

 

 

If you are spinning it fast enough, the water is falling out, but the bucket moves fast enough to be perpetually catching it. There is no force directed outward. If the rope broke, the motion would initially be tangent to the circle, not outward (and then whatever path is dictated by gravity), in accordance with Newton's laws.

[Daecon]

That's what's confusing. I would have imagined the water to "spill" out of the bucket, not downward towards the earth - but out from behind the direction of the bucket's travel, like a comet's tail... so how could the bucket continue to catch the water?

 

I was sure that the water looked to be 'level' inside the bucket.

 

I hope that made sense. I'm having trouble picturing how all the forces interact with each other in my mind...

[mezarashi]

That's what's confusing. I would have imagined the water to "spill" out of the bucket' date=' not downward towards the earth - but out from behind the direction of the bucket's travel, like a comet's tail... so how could the bucket continue to catch the water?

 

I was sure that the water looked to be 'level' inside the bucket.

 

I hope that made sense. I'm having trouble picturing how all the forces interact with each other in my mind...[/quote']

 

Yes, the water should generally be level. What you have to fundamentally keep in mind to understand the forces involved here is that, objects, including water molecules, if left alone (no forces acting on them), will continue moving in a straight line. Now, circular motion is definitely not a straight line is it.

 

Take the circling bucket system. At any instant, the water molecules in the bucket want to speed out in a straight line tangent to the circular motion. However, the bucket is pushing the water in, so that it may not do what it would naturally do. The bucket is exerting a centripetal force to keep the water going in circular motion, and thus the water is undergoing centripetal acceleration (acceleration towards the center of the circle). Without the bucket (and if the bucket were to at any instant disappear), the water would speed out in a straight line tangent to the circling action, and then subject to the projectile motion laws as in normal kinematics.

 

Now in order to fulfill Newton's third law: every force has an equal and opposite reaction, we have this virtual force called the centrifugal force. Where is this centrifugal force? Because as stated, the bucket is exerting a centripetal force on the water to keep it in circular motion, the water must then exert and equal and opposite force on the bucket, and that is what is in this case the centrifugal force. Then water pushes the bucket, the bucket pulls on your arm. The faster you spin it, the more force you will feel the bucket pulling.

 

The last part about gravity. When performing this experiment on earth, with the rotational plane perpendicular to the Earth's surface, you will have gravity involved. Because gravity affects everything (on Earth), it will also affect the water molecules that you are swinging in the bucket. If you notice, although you are swinging the bucket at a constant rotational velocity, the bucket feels lighter at the top of the cycle and heavier at the bottom. This is because at the top, gravity becomes part of the centripetal force that is required to keep the water in circular motion. At the top, the centripetal force would need to point downwards, which happens to be the direction of gravity. At the bottom cycle, gravity adds an additional weight, because not only do you need to exert a force on the bucket to keep the water in circular motion, but you need to exert an additional force equal to the weight of the water in the bucket like you would if you were just holding the bucket still.

 

I hope that in a way or another helped clarify things for you.

[Daecon]

Hey thank you! That made sense and I get it now

 

*is satisfied*

[MetaFrizzics]

There is no force directed outward. If the rope broke, the motion would initially be tangent to the circle, not outward (and then whatever path is dictated by gravity), in accordance with Newton's laws.

Interestingly, this statement is too vague to be precise: There in fact is always an outward component in the velocity, regardless of reference frame, and the water/bucket will always increase its distance from the centre of rotation. The motion will only be tangent (in a straight line) to the circle in one unique reference frame (or set of connected frames), which is under key constraints:

 

First it must not be moving relative to the centre of rotation, unless the motion is actually along the tangent (defined in a moment).

 

Second, it cannot be rotating around this point either, relative to the rest frame that gives the true absolute velocity of the swinging bucket.

 

Looked at differently, we must find a frame of reference that is not changing its displacement relative to the centre of rotation, unless it is along a line parallel to the 'tangent' line at the point of departure. The point of departure is defined as the point in space and time at which the radial displacement (distance from bucket to centre of rotation) changes (increases when the rope breaks). This frame of reference cannot be rotating relative to absolute space around the centre of rotation in the plane of rotation either. (Strictly speaking, we could choose a reference frame that is moving with constant velocity relative to absolute rest, but this requires establishing absolute rest first, and adding a needless layer of complication.)

 

But its a gimmicky process: Since we can choose a frame of reference, we can always choose one that makes the bucket of water flying off appear to be moving in a straight line. We do this by first placing our frame of reference that stays a fixed distance from the point of rotation (for convenience we pick the centre of rotation!). Then we simply have it rotate backward at just the right velocity to straighten the trajectory of escaping bucket when measured, and this will also appear to be at a tangent to the orbit of rotation. This we declare by fiat as an inertial frame for the rotation, since now the First Law (translational motion of constant velocity) appears to hold.

 

The point in all this is that 'centripetal' force is no more an explanation of rotational motion than 'centrifugal' force is. We only prefer the 'centripetal' viewpoint because it allows us to treat rotational motion with the same Newtonian Law(s) we use for translational motion (or at least in an analogical manner). There actually is no real explanation for the force, other than in certain reference frames it can be seen as an outcome of the First Law (Translational Motion of Constant Velocity in the absence of forces).

 

The genius and coherence of the viewpoint of 'centripetal' force is its simplification of the treatment of rotational motion, due to Newton's definition of momentum as the product of mass and velocity. As an 'explanation' of either centripetal or centrifugal forces (which appear and disappear with frame of reference) it is a glorious failure.

 

One of the reasons why the physical scenario is underdetermined, and allows for the appearance and disappearance of various 'forces' depending upon reference frame, is that it is simply not possible to properly define a velocity vector in space from the centre of rotation (CR) alone. With just an observer at the CR, all that can be measured is the distance of the receding bucket and its movement away. The observer can always add rotation to himself so that the bucket stays in view, but there is no way to determine the absolute rotation of the observer at the CR, and hence the actual velocity or straightness of the bucket's trajectory. That requires at least one more viewpoint, or measurement point out in 'space'.

 

In fact, as far as absolute distances and time go, the bucket will always appear to be accelerating away from the centre of rotation, since its line of motion changes angle from it with distance. Thus more and more of the motion of the bucket is being applied directly to increasing the distance. The bucket will appear to be accelerating at a decreasing rate, approaching the absolute velocity as a limit. To keep the bucket in view, the observer will have to rotate with it, but gradually brake his rotation to almost nothing. Without constantly adjusting his motion via a force, the bucket will appear to recede backward in the direction opposite of its 'true' (inertial) motion!

[J.C.MacSwell]

Interestingly' date=' this statement is too vague to be precise: There in fact is [b']always[/b] an outward component in the velocity, regardless of reference frame, and the water/bucket will always increase its distance from the centre of rotation. The motion will only be tangent (in a straight line) to the circle in one unique reference frame (or set of connected frames), which is under key constraints:

 

There is no outward component to the velocity prior to release. This outward component will start at 0 and increase although this "pseudo acceleration" is not linear and this outward velocity will eventually approach the total velocity.

 

The motion will be tangent (in a straight line) to the circle in any inertial frame, gravitational effects aside. How it appears to an observer at the origin of the inertial frame does not change this.

[swansont]

The point in all this is that 'centripetal' force is no more an explanation of rotational motion than 'centrifugal' force is. We only prefer the 'centripetal' viewpoint because it allows us to treat rotational motion with the same Newtonian Law(s) we use for translational motion (or at least in an analogical manner). There actually is no real explanation for the force' date=' other than in certain reference frames it can be seen as an outcome of the First Law (Translational Motion of Constant Velocity in the absence of forces).

[/quote']

 

 

Despite your attempts to mystify, there is an explanation, and it's quite simple. In an inertial frame, linear moion at v with a constant acceleration toward the center of v²/r will yield circular motion. Just a solution to a particular condition using the definitions of the terms. The math isn't difficult.

 

The centripetal force isn't a new type of force, however, as I stated previously.

[DQW]

How much centrifugal force is created by the Earth's rotation, if any? It is obviously not enough to counteract the Earth's gravity, and send us all flying into space, but is there still some acting on the bodies on the Earth?

At the equator, [imath]v^2/Rg \approx 0.3 [/imath]%.