Cen-something-al Force...or is it?

[King Author]

Okay, here's what I know, because I'm a human being capable of learning from his experiences...

1) If I turn a bucket of water upside down, it spills, because there's nothing stopping the water from doing so.
2) If I swing a bucket in a circle by its handle, even when it goes upside down, it doesn't spill.
3) When I get my blood tested, they use a machine that spins my blood around super-fast, separating out the heavier from the lighter phases.

Here's what I don't know, because the science taught in American classrooms is laughable and also because nerds all-the-times be fightin' over their definitions...

1) What stops the water from spilling out of the bucket when I swing it?
2) Is the thing that separates out the parts of my blood the same thing that stops the water from spilling out of the bucket?
3) WTF do you call the above force(s)?

[JBJ]

1) What stops the water from spilling out of the bucket when I swing it?
The bottom and sides of the bucket.

2) Is the thing that separates out the parts of my blood the same thing that stops the water from spilling out of the bucket?
Yes...? Depends on the frame of reference. See next question.

3) WTF do you call the above force(s)?
Centripetal and Centrifugal. Centripetal means to the center and centrifugal means leave the center.
When and how to use it depends on your frame of reference. When looking at a whole system, you really only figure the centripetal force, which is the force on the string, arm, or whatever member is pulling the object toward the center, i.e., the tension. From the reference point of the water inside the bucket, the force pressing against the bottom of the bucket is the centrifugal force.

So, long answer short, centripetal if you are measuring the whole system, centrifugal if you are inside the system. Of course, I could be mistaken, but that's how I have come to understand it.

[LaserGuy]

1) What stops the water from spilling out of the bucket when I swing it?

Have you ever been on one of those carnival rides called a gravitron? You go into a big machine, and it starts spinning really fast. When this happens, everybody pretty much gets plastered against the walls. They can even drop the floor away, because the friction between you and the wall is enough to keep you suspended. These examples are all pretty much work under the same principle: inside the bucket/centrifuge/gravitron, the contents will experience a force driving it away from the centre of rotation. This force is proportional to the square of the velocity, and inversely proportional to the distance from the centre. So the faster you spin, the stronger the force. This force is called a centrifugal force.

As JBJ correctly points out, there is an equivalent formulation that can be constructed for someone looking at the system from an outside point of view. In this case, there is a net force towards the centre. This is called the centripetal force. Which formulation that you use depends on what (or perhaps more accurately, where) you are trying to measure from. Depending on what you actually care about, you may choose to look at the problem from one point of view or the other.

[legend]

There is no such thing as centrifugal force in an inertial frame of reference. There is however a centripetal force (e.g. the force from your arm acting on the bucket) acting inwards. Because the net force is thus not zero the bucket is accelerated (rotating).
You can however transform the whole system into the reference frame of the bucket (a rotating, non-inertial reference frame). In this reference frame the bucket isn't moving, thus the net force is zero and so there must be an additional force acting on the bucket. This force is called centrifugal force and it's acting outwards "compensating" the centripetal force.
Now one can argue that there is no such thing as centrifugal force because you only have it in non-inertial frames. But by using the same argument you can also deduce that there is no such thing as gravity because it also only appears because we have no global inertial frames of reference.

[Meteoric]

1) What stops the water from spilling out of the bucket when I swing it?

Before we too technically talk about the various forces and etc, let's just approach this intuitively: swinging the bucket in a certain way doesn't suddenly make it immune to the effects of gravity. Gravity is still pulling down on that bucket and the water in it as you swing it over your head. However, by that point the water and bucket are already moving with some speed in a direction other than where gravity is pulling, and the bucket doesn't stay upside-down long enough for gravity to completely reverse that motion. If you swing the bucket too slowly, the water does spill out.

Also, even at the top of the arc, when the bucket and water are for a moment no longer moving upward at all, and both gravity and your arm are pulling straight down, the water still doesn't spill out. This is because the bucket is also being pulled by gravity and (possibly) your arm, so even though the water is being pulled downward by gravity and (possibly) pushed downward by the bottom of the bucket, it will still move along the same path as the bucket, and not spill. The two (possibly) parts there depend on how fast you're swinging the bucket; if you swing it just barely fast enough for the water to not spill, your arm should be basically slack at the top of the arc; if you swing faster than that, there will be tension on your arm even at the top of the arc.

[eSOANEM]

Now one can argue that there is no such thing as centrifugal force because you only have it in non-inertial frames.

This is the main argument levied against centrifugal force. It is also an unhelpful one.

Yes inertial reference frames are generally nicer to work with but a lot of the time, your measurement apparatus is not in an inertial frame and so you need some way to map between these two frames. The only way to do this is by introducing inertial forces.

These inertial forces all ultimately come down to the idea that, from the reference frame of a car accelerating, the fluffy dice hanging from its rear view mirror experience some force in the opposite direction of acceleration.

This explains trivially why roller coasters throw you from side to side as they turn but it is slightly less clear how it produces circular motion.

If we have a ball on a string spinning in a circle, in order for the ball to move in a circle it must always be accelerating inwards (there's a good video on youtube somewhere, I can't seem to find it though, of a guy demonstrating this by making a bowling ball move in an approximate circle by skating alongside it with a croquet mallet and hitting it towards the centre). Now, we know that inertial forces act in the opposite direction to the acceleration so, in a rotating reference frame, an inertial force (called centrifugal force), appears.

[Twistar]

Imagine someone is swinging the bucket in a vertical counterclockwise circle. (I'm too lazy to make a picture, sorry.)
Now think about the path the water is taking. It is moving in a circle. Now think about the velocity of the water at each point along the path. it is tangent to the circle, so at the right part of the path the velocity is upwards. At the top it is leftwards at the bottom it is rightwards etc.
At this point we have to talk about Newton's laws (in an inertial reference frame, I find non-inertial reference frames hard and confusing to think about without math.)
The first law says that an object in motion will remain in motion unless acted on by a force. This means that if something (water in a bucket) has a given velocity (that I talked about earlier) it will keep that velocity unless a force acts on it.
This means that if the person is swinging the bucket around and at the top of the arc the bucket suddenly disappears but the water doesn't the water will fly off in whatever direction it's velocity was before the bucket disappears. This means that water will actually fly to the LEFT rather than downwards if the bucket disappears at the top of the arc. That is the really non-intuitive part about this problem. In fact, This shows that the question shouldn't be "what keeps the water from falling down out of the bucket" but rather that the question should be "what keeps the water from flying to the left and instead keeps it moving in a circle.
This is where Newtons second law comes in:
Forces cause acceleration. In other words, Forces change the velocities (either their magnitude or their direction) of objects.
At the top of the circle the velocity is to the left, but at the left part of the circle the velocity is down. This means that there was a change in velocity! What caused that change in velocity? It has to have been some force. It turns out that this force is generated by the combination of gravity and the contact force between the bucket and the water. If you add up these two forces it turns out that the net result is a centripetal force (center seeking) which is directed exactly towards the center of the circle. This force acts to continuously tilt the velocity vector of the water to keep the water moving in a perfect circle.
This helps explain what meteoric was explaining about the tension in your arm while you spin the bucket. At the top of the arc gravity is doing a lot of the work to generate this centripetal force so you don't need to do much, but at the bottom the arc gravity is putting a force AWAY from the center of the bucket so your arm needs to overcome gravity PLUS provide the centripetal force so you need to work much harder.

[King Author]

So in xkcd123, the hat guy was right? "No such thing as centrifugal force" is, as he put it, overzealous? Or perhaps pedantic, would be a better word.

Have you ever been on one of those carnival rides called a gravitron? You go into a big machine, and it starts spinning really fast. When this happens, everybody pretty much gets plastered against the walls. They can even drop the floor away, because the friction between you and the wall is enough to keep you suspended. These examples are all pretty much work under the same principle: inside the bucket/centrifuge/gravitron, the contents will experience a force driving it away from the centre of rotation. This force is proportional to the square of the velocity, and inversely proportional to the distance from the centre. So the faster you spin, the stronger the force. This force is called a centrifugal force.

I've never been on one of those rides, no, but I'm with you 100%, I totally understand everything you just said.

As JBJ correctly points out, there is an equivalent formulation that can be constructed for someone looking at the system from an outside point of view. In this case, there is a net force towards the centre. This is called the centripetal force. Which formulation that you use depends on what (or perhaps more accurately, where) you are trying to measure from. Depending on what you actually care about, you may choose to look at the problem from one point of view or the other.

This is where you lose me. Maybe I just don't understand frames of reference. Correction; I definitely don't "understand" frames of reference, being a layperson, but I thought I had a nice grip on them considering.

How could a net force towards the center stop the water from spilling out? That doesn't make any sense whatsoever to me.

There is no such thing as centrifugal force in an inertial frame of reference. There is however a centripetal force (e.g. the force from your arm acting on the bucket) acting inwards. Because the net force is thus not zero the bucket is accelerated (rotating).
You can however transform the whole system into the reference frame of the bucket (a rotating, non-inertial reference frame). In this reference frame the bucket isn't moving, thus the net force is zero and so there must be an additional force acting on the bucket. This force is called centrifugal force and it's acting outwards "compensating" the centripetal force.
Now one can argue that there is no such thing as centrifugal force because you only have it in non-inertial frames. But by using the same argument you can also deduce that there is no such thing as gravity because it also only appears because we have no global inertial frames of reference.

See, this is what gets me. Regardless of how we construct our explanations of things, some thing is objectively happening. Whether we look at my arm or the bucket or the water doesn't change what's physically happening in the world, it only changes the names and mathematics we use. That's what trips me up. You and I and a seven-year-old all know perfectly well why a swung bucket of water doesn't spill, we know this intuitively from our experiences. The confusion comes from the nomenclature. Whether we say centrifugal force is acting on the water or centripetal on, uh, my arm or whatever, the same thing is happening in reality.

It just always seemed to me that when someone sanctimonously says "there's no such thing as centrifugal force, you plebian" they're trying to say that reality is something different than what it obviously is. I feel like they're saying the physical force itself doesn't exist, rather than the more-acceptable but still-obnoxious "you're looking at things from the wrong perspective and using the wrong names for things."

Now one can argue that there is no such thing as centrifugal force because you only have it in non-inertial frames.

This is the main argument levied against centrifugal force. It is also an unhelpful one.

Yes inertial reference frames are generally nicer to work with but a lot of the time, your measurement apparatus is not in an inertial frame and so you need some way to map between these two frames. The only way to do this is by introducing inertial forces.

These inertial forces all ultimately come down to the idea that, from the reference frame of a car accelerating, the fluffy dice hanging from its rear view mirror experience some force in the opposite direction of acceleration.

This explains trivially why roller coasters throw you from side to side as they turn but it is slightly less clear how it produces circular motion.

If we have a ball on a string spinning in a circle, in order for the ball to move in a circle it must always be accelerating inwards (there's a good video on youtube somewhere, I can't seem to find it though, of a guy demonstrating this by making a bowling ball move in an approximate circle by skating alongside it with a croquet mallet and hitting it towards the centre). Now, we know that inertial forces act in the opposite direction to the acceleration so, in a rotating reference frame, an inertial force (called centrifugal force), appears.

@__@
That's so counter-intuitive.

Imagine someone is swinging the bucket in a vertical counterclockwise circle. (I'm too lazy to make a picture, sorry.)
Now think about the path the water is taking. It is moving in a circle. Now think about the velocity of the water at each point along the path. it is tangent to the circle, so at the right part of the path the velocity is upwards. At the top it is leftwards at the bottom it is rightwards etc.
At this point we have to talk about Newton's laws (in an inertial reference frame, I find non-inertial reference frames hard and confusing to think about without math.)
The first law says that an object in motion will remain in motion unless acted on by a force. This means that if something (water in a bucket) has a given velocity (that I talked about earlier) it will keep that velocity unless a force acts on it.
This means that if the person is swinging the bucket around and at the top of the arc the bucket suddenly disappears but the water doesn't the water will fly off in whatever direction it's velocity was before the bucket disappears. This means that water will actually fly to the LEFT rather than downwards if the bucket disappears at the top of the arc. That is the really non-intuitive part about this problem. In fact, This shows that the question shouldn't be "what keeps the water from falling down out of the bucket" but rather that the question should be "what keeps the water from flying to the left and instead keeps it moving in a circle.

Actually, to me, this is all very intuitive so far. I was just being lazy saying "what stops the water from falling." I should've really said flinging all over the place.

This is where Newtons second law comes in:
Forces cause acceleration. In other words, Forces change the velocities (either their magnitude or their direction) of objects.
At the top of the circle the velocity is to the left, but at the left part of the circle the velocity is down. This means that there was a change in velocity! What caused that change in velocity? It has to have been some force. It turns out that this force is generated by the combination of gravity and the contact force between the bucket and the water. If you add up these two forces it turns out that the net result is a centripetal force (center seeking) which is directed exactly towards the center of the circle. This force acts to continuously tilt the velocity vector of the water to keep the water moving in a perfect circle.

...aaand lost again. How could a force towards the center stop the water from spilling?

I guess that's my big hangup with centriwhatever force -- intuitively, I think of a force pushing the water away from the center of the rotation. But apparently that's wrong, and I can't understand how.

[yurell]

Let's say I'm a woman watching the bucket be spun. I can see the only forces being applied to the bucket are gravity and some force towards the centre of the circle it's tracing (called 'centripetal force'). This is an inertial frame of reference — I am not accelerating.

Now, imagine I'm in the bucket. I am now in a reference frame that's accelerating (i.e. a non-inertial one). I see that there is gravity pulling on me, but there appears to be another force pushing me away from the centre — this I call centrifugal force.
What I am calling centrifugal force in this frame is actually my inertia in the original frame — my resistance to being accelerated by the bucket. However, in the bucket's frame it doesn't look like I'm accelerating, and so the explanation is that there exists another force.

And yes, black hat guy was correct.

[King Author]

Ah! Inertia! That helps. But I still completely don't see how you, the woman standing off to the side watching me swing a bucket like a weirdo, sees a force going towards the center. Intuitively, I'd think you'd be seeing gravity and a force pushing away from the center.

I realize my intuition means nothing here, I'm just saying, it's keeping me from understanding this concept of an inward force.

[yurell]

Ah, okay, let's ignore the water in the bucket for a moment and just look at how objects spin, that seems to be the sticking point. The easiest thought is swinging around a yo-yo, and imagine it's tracing an anti-clockwise circle, which means at the top its velocity is to the left.
Now, you know from Newton's First Law that "an object at rest or in a state of rectilinear motion will continue in that state unless acted upon by an external force" i.e. the yo-yo wants to keep going left. What's stopping it though? The string — the string pulls on the yo-yo, and the only direction it can pull it is towards the centre.

Do you know how to use vectors? Let's say y velocity was {-,0} (i.e. I have some speed to the left (shown as a negative sign), and none down), and I accelerate towards the centre by {0,-} (i.e. I'm adding no new leftwards velocity, but some downwards). My new velocity is {-,-} i.e. at an angle to the centre.
A circle is where I keep changing this velocity so that my inertial tendency to travel in a straight line keeps getting pulled off-course by some force pulling me towards the centre i.e. the centripetal force.

[gmalivuk]

I realize my intuition means nothing here, I'm just saying, it's keeping me from understanding this concept of an inward force.

The walls of the bucket are the inward force, acting against inertia.

And there's nothing strange about forces looking different for different people, because force is proportional to acceleration, which like velocity can depend on your frame of reference.

[++$_]

Look at the situation where the bucket is directly over your head.

If there were no force on the water, it would move in a straight line at constant velocity forever. But instead, it's moving in a circle. Why? Because there is a force pushing it towards the center of the circle.

"No," you protest, "there can't be a force pushing it towards the center of the circle. There must be a force pushing it away from the center, otherwise it would end up at the center." That is incorrect. Look at the difference between the trajectories with and without force:

centripetal.jpg (5.99 KiB) Viewed 1750 times

Now, notice that a force is causing the trajectory to bend. Which way is this force pointing? Do you still think the force is outward? If so, how do you explain the fact that the trajectory of the water bends inward, if the force is outward?

In fact, we can calculate the amount of the inward force, using Newton's Second Law, to be mv²/r, where m is the mass of the water, v is the velocity, and r is the radius of the circle. Just for fun, let's make up some numbers. Let's say the bucket is going around a circle with a 1 meter radius, taking 1.5 seconds to do so, and there are 2 kilograms of water in the bucket. Now, if it takes 1.5 seconds to go around a circle with a 1 meter radius, the water is traveling at a velocity of 4.2 meters per second. Plugging everything into the formula we get F = (2 kg)*(4.2 m/s)²/(1 m) = 35.3 N. That is how much force we need just to keep the water moving in a circle. Remember, if you had less force than that, it would instead move in a way more like the straight line depicted in the diagram.

Now, the force due to gravity on the water is 19.6 N. This force is directed towards the center of the circle. Great -- that supplies 19.6 N out of the 35.3 N that we need to keep the water moving in a circle. But we still need an additional 15.7 N of downward force just to keep the water moving in a circle (otherwise it will fly out in more of a straight-line direction).

So, as you can see, the question is not "why doesn't the water fall?" The water is falling! In fact it is not falling enough! Gravity is helping to bend the path of the water from the straight line it would follow without force, to the curved line that it is actually following. But gravity isn't completely sufficient; as the calculation showed, we need another 15.7 N of force to do this. This extra force is provided by the bottom of the bucket.

[Qaanol]

A force in the direction you’re already moving will cause you to speed up or slow down. A force perpendicular to the direction you’re moving will cause you to turn. Similarly, a force at an angle from the direction you’re already moving will cause you to turn somewhat and also speed up or slow down somewhat. That should agree with your intuition.

For the case of uniform circular motion at constant speed, you never speed up or slow down, so the force must be entirely perpendicular to the direction you’re moving at any moment. Since you’re moving along the circumference of a circle, the direction you are currently moving points along a tangent line to the circle. That means the perpendicular direction must point along a radial line, namely toward the center of the circle.

Intuitively, imagine tying a string to a rock, and setting the rock on the table. To move the rock, you can pull on the string, and the rock will start moving toward your hand. You can only pull on a string, you can’t push on a string. The only direction you can make the rock move is toward your hand by pulling on the string.

If you swing the rock-on-a-string around in a circle, it will move just like the bucket of water, and it will feel just the same to you. But now it’s obvious that the force on the rock is pointed toward the center of the circle, because the only direction you can pull on the string is “inward”.

Now imagine you let go of the string at some angle, like David aiming at Goliath. The rock is moving really fast, and it will fly through the air. It’ll go up for a while, then fall back down, moving sideways the whole time. Indeed, the rock ends up going much higher than the top of the circle you were swinging it around.

So the real questions are “Why didn’t the rock fly that high earlier, if it was moving upward so fast?” and “Why didn’t the rock hit Goliath earlier, if it was moving sideways so fast?” The answers are the same: because you were pulling inward on the rock, changing its direction all the time to keep it close to you.

When the rock was zooming upward, you pulled sideways to change its direction, and soon it was zooming upward-and-a-little-bit-to-the-side. You kept pulling inward, and a few moments later it was at the top of its arc, zooming sideways. You pulled downward, changing its direction, and soon it was zooming sideways-and-a-little-bit-downward.

You kept pulling inward, and whatever direction the rock was moving, a moment later it was moving a different direction, always the same distance from your hand, namely the length of the string. Substitute rock and string for water and bucket, and I hope this made sense.

[King Author]

Spoiler:

Ah, okay, let's ignore the water in the bucket for a moment and just look at how objects spin, that seems to be the sticking point. The easiest thought is swinging around a yo-yo, and imagine it's tracing an anti-clockwise circle, which means at the top its velocity is to the left.
Now, you know from Newton's First Law that "an object at rest or in a state of rectilinear motion will continue in that state unless acted upon by an external force" i.e. the yo-yo wants to keep going left. What's stopping it though? The string — the string pulls on the yo-yo, and the only direction it can pull it is towards the centre.

Do you know how to use vectors? Let's say y velocity was {-,0} (i.e. I have some speed to the left (shown as a negative sign), and none down), and I accelerate towards the centre by {0,-} (i.e. I'm adding no new leftwards velocity, but some downwards). My new velocity is {-,-} i.e. at an angle to the centre.
A circle is where I keep changing this velocity so that my inertial tendency to travel in a straight line keeps getting pulled off-course by some force pulling me towards the centre i.e. the centripetal force.

But it seems to me that the string is pulling the yoyo counter-clockwise, not toward the center. I'm spinning the yoyo counter-clockwise, I'm not spinning it inward towards myself. Indeed, my spinning it so fast that it's flinging away from me is what's keeping it taut and in a circle, rather than slack and just dangling around.

I realize my intuition means nothing here, I'm just saying, it's keeping me from understanding this concept of an inward force.

The walls of the bucket are the inward force, acting against inertia.

And there's nothing strange about forces looking different for different people, because force is proportional to acceleration, which like velocity can depend on your frame of reference.

That makes more sense. Because me swinging the bucket, I feel like I'm definitely providing outward force, not inward. Although I don't see how you wouldn't see that just because you're not the one swinging the bucket.

(There's no "strikethrough" available, so this is me striking through the above. See my response to Qaanol at the end of this post.)

Look at the situation where the bucket is directly over your head.

If there were no force on the water, it would move in a straight line at constant velocity forever. But instead, it's moving in a circle. Why? Because there is a force pushing it towards the center of the circle.

"No," you protest, "there can't be a force pushing it towards the center of the circle. There must be a force pushing it away from the center, otherwise it would end up at the center." That is incorrect. Look at the difference between the trajectories with and without force:

centripetal.jpg

Now, notice that a force is causing the trajectory to bend. Which way is this force pointing? Do you still think the force is outward? If so, how do you explain the fact that the trajectory of the water bends inward, if the force is outward?

Nice graphic, that helps. Except that, in my mind, that image is showing the bucket, not the water in the bucket. I can see now how you could say the bucket had inward velocity, but I still don't see how the water is doing anything other than being flung outward, where it's stopped by the bucket.

In fact, we can calculate the amount of the inward force, using Newton's Second Law, to be mv²/r, where m is the mass of the water, v is the velocity, and r is the radius of the circle. Just for fun, let's make up some numbers. Let's say the bucket is going around a circle with a 1 meter radius, taking 1.5 seconds to do so, and there are 2 kilograms of water in the bucket. Now, if it takes 1.5 seconds to go around a circle with a 1 meter radius, the water is traveling at a velocity of 4.2 meters per second. Plugging everything into the formula we get F = (2 kg)*(4.2 m/s)²/(1 m) = 35.3 N. That is how much force we need just to keep the water moving in a circle. Remember, if you had less force than that, it would instead move in a way more like the straight line depicted in the diagram.

Now, the force due to gravity on the water is 19.6 N. This force is directed towards the center of the circle.

Whoa whoa whoa, what? Gravity is directed toward the center of the circle? Surely it's constantly directed toward the Earth?

Great -- that supplies 19.6 N out of the 35.3 N that we need to keep the water moving in a circle. But we still need an additional 15.7 N of downward force just to keep the water moving in a circle (otherwise it will fly out in more of a straight-line direction).

So, as you can see, the question is not "why doesn't the water fall?" The water is falling! In fact it is not falling enough! Gravity is helping to bend the path of the water from the straight line it would follow without force, to the curved line that it is actually following. But gravity isn't completely sufficient; as the calculation showed, we need another 15.7 N of force to do this. This extra force is provided by the bottom of the bucket.

Hmm...I feel like I'm this close to understanding that (well, as I say, "understanding"), but there's still some things I'm not grasping. Namely, as I said above, the force of gravity seems to me to be pulling in one constant direction ("down"), not towards the center of the circle. Thinking of the water as always falling is helpful, though.

A force in the direction you’re already moving will cause you to speed up or slow down. A force perpendicular to the direction you’re moving will cause you to turn. Similarly, a force at an angle from the direction you’re already moving will cause you to turn somewhat and also speed up or slow down somewhat. That should agree with your intuition.

For the case of uniform circular motion at constant speed, you never speed up or slow down, so the force must be entirely perpendicular to the direction you’re moving at any moment. Since you’re moving along the circumference of a circle, the direction you are currently moving points along a tangent line to the circle. That means the perpendicular direction must point along a radial line, namely toward the center of the circle.

Intuitively, imagine tying a string to a rock, and setting the rock on the table. To move the rock, you can pull on the string, and the rock will start moving toward your hand. You can only pull on a string, you can’t push on a string. The only direction you can make the rock move is toward your hand by pulling on the string.

If you swing the rock-on-a-string around in a circle, it will move just like the bucket of water, and it will feel just the same to you. But now it’s obvious that the force on the rock is pointed toward the center of the circle, because the only direction you can pull on the string is “inward”.

Now imagine you let go of the string at some angle, like David aiming at Goliath. The rock is moving really fast, and it will fly through the air. It’ll go up for a while, then fall back down, moving sideways the whole time. Indeed, the rock ends up going much higher than the top of the circle you were swinging it around.

So the real questions are “Why didn’t the rock fly that high earlier, if it was moving upward so fast?” and “Why didn’t the rock hit Goliath earlier, if it was moving sideways so fast?” The answers are the same: because you were pulling inward on the rock, changing its direction all the time to keep it close to you.

When the rock was zooming upward, you pulled sideways to change its direction, and soon it was zooming upward-and-a-little-bit-to-the-side. You kept pulling inward, and a few moments later it was at the top of its arc, zooming sideways. You pulled downward, changing its direction, and soon it was zooming sideways-and-a-little-bit-downward.

You kept pulling inward, and whatever direction the rock was moving, a moment later it was moving a different direction, always the same distance from your hand, namely the length of the string. Substitute rock and string for water and bucket, and I hope this made sense.

Okay, yeah. I see that now. That was a really good explanation, thanks.

But now I've moved onto a new problem -- it seems to me that it's the bucket which is being pulled towards the center of the circle. The water clearly seems to be trying to fling outward, but is stopped by the bucket. Because imagine if suddenly the bottom of the bucket disappeared -- the water wouldn't go inward, towards me, it wouldn't go along the circumference of the circle, it would clearly fling outward, away from the center.

[eSOANEM]

As JBJ correctly points out, there is an equivalent formulation that can be constructed for someone looking at the system from an outside point of view. In this case, there is a net force towards the centre. This is called the centripetal force. Which formulation that you use depends on what (or perhaps more accurately, where) you are trying to measure from. Depending on what you actually care about, you may choose to look at the problem from one point of view or the other.

This is where you lose me. Maybe I just don't understand frames of reference. Correction; I definitely don't "understand" frames of reference, being a layperson, but I thought I had a nice grip on them considering.

When we say "in x's reference frame" we mean "x will observe that".

So, in the accelerating car example, from someone standing on the road's reference frame, the car starts moving and the fluffy dice try to stay where they are but get pulled along by the string. From the driver's reference frame, the road starts moving backwards and the fluffy dice seem to feel some invisible force pulling them back.

When we talk about x's frame being inertial, we, roughly speaking, mean that x isn't accelerating. A non-inertial frame is one which is accelerating and so includes rotating frames.

There is no such thing as centrifugal force in an inertial frame of reference. There is however a centripetal force (e.g. the force from your arm acting on the bucket) acting inwards. Because the net force is thus not zero the bucket is accelerated (rotating).
You can however transform the whole system into the reference frame of the bucket (a rotating, non-inertial reference frame). In this reference frame the bucket isn't moving, thus the net force is zero and so there must be an additional force acting on the bucket. This force is called centrifugal force and it's acting outwards "compensating" the centripetal force.
Now one can argue that there is no such thing as centrifugal force because you only have it in non-inertial frames. But by using the same argument you can also deduce that there is no such thing as gravity because it also only appears because we have no global inertial frames of reference.

See, this is what gets me. Regardless of how we construct our explanations of things, some thing is objectively happening. Whether we look at my arm or the bucket or the water doesn't change what's physically happening in the world, it only changes the names and mathematics we use. That's what trips me up. You and I and a seven-year-old all know perfectly well why a swung bucket of water doesn't spill, we know this intuitively from our experiences. The confusion comes from the nomenclature. Whether we say centrifugal force is acting on the water or centripetal on, uh, my arm or whatever, the same thing is happening in reality.

It just always seemed to me that when someone sanctimonously says "there's no such thing as centrifugal force, you plebian" they're trying to say that reality is something different than what it obviously is. I feel like they're saying the physical force itself doesn't exist, rather than the more-acceptable but still-obnoxious "you're looking at things from the wrong perspective and using the wrong names for things."

A lot of people who say there's no such thing as centrifugal force will probably mean it that way.

They are wrong. The water feels a centrifugal force, this is undeniable and can be easily demonstrated by noting that, as you drive around a corner, you feel as if you're being pulled away from the centre.

The thing is that people who aren't rotating with the water won't see this force at all and will instead only see the water trying to keep going in a straight line.

The, "you're looking at things from the wrong perspective" is sometimes quite helpful. Moving into a non-inertial frame can be tricky because you have to make sure you include all the relevant inertial forces. If you miss one out, which in more complicated examples (such as a roller coaster with the carriage free to spin around on the end of an arm the other end of which is attached the wheels and the track itself) would be quite easy (I count up to four in this example), you're probably going to get the wrong answer.

So working in inertial frames is, in complicated situations, often much nicer. The problem is that it's hard for teachers to teach people when it is and isn't a good idea to use a non-inertial frame so it's easier to say they're wrong and therefore centrifugal force shouldn't be used.

This is where Newtons second law comes in:
Forces cause acceleration. In other words, Forces change the velocities (either their magnitude or their direction) of objects.
At the top of the circle the velocity is to the left, but at the left part of the circle the velocity is down. This means that there was a change in velocity! What caused that change in velocity? It has to have been some force. It turns out that this force is generated by the combination of gravity and the contact force between the bucket and the water. If you add up these two forces it turns out that the net result is a centripetal force (center seeking) which is directed exactly towards the center of the circle. This force acts to continuously tilt the velocity vector of the water to keep the water moving in a perfect circle.

...aaand lost again. How could a force towards the center stop the water from spilling?

I guess that's my big hangup with centriwhatever force -- intuitively, I think of a force pushing the water away from the center of the rotation. But apparently that's wrong, and I can't understand how.

It's not wrong at all. The water definitely feels a force pushing it away from the centre of rotation, but it only feels this because the bucket feels a force pulling it towards the centre.

Like the fluffy dice in the car, when you're in the accelerating reference frame, you feel a force in the opposite direction to the acceleration.

In fact, we can calculate the amount of the inward force, using Newton's Second Law, to be mv²/r, where m is the mass of the water, v is the velocity, and r is the radius of the circle. Just for fun, let's make up some numbers. Let's say the bucket is going around a circle with a 1 meter radius, taking 1.5 seconds to do so, and there are 2 kilograms of water in the bucket. Now, if it takes 1.5 seconds to go around a circle with a 1 meter radius, the water is traveling at a velocity of 4.2 meters per second. Plugging everything into the formula we get F = (2 kg)*(4.2 m/s)²/(1 m) = 35.3 N. That is how much force we need just to keep the water moving in a circle. Remember, if you had less force than that, it would instead move in a way more like the straight line depicted in the diagram.

Now, the force due to gravity on the water is 19.6 N. This force is directed towards the center of the circle.

Whoa whoa whoa, what? Gravity is directed toward the center of the circle? Surely it's constantly directed toward the Earth?

Yeah, that's a mistake.

Horizontal situations are a little nicer because then the speed is constant and the only non-inertial force is the tension in the string/your arm. When gravity gets involved, the force is not entirely towards the centre of the circle and so the speed varies (because some of the force is acting to speed up/slow down the bucket), this also means the tension varies because at the top it works with gravity (it is possible to spin things in circles such that at the top the string just becomes slack before continuing) but at the bottom it has to work against gravity which is trying to pull the bucket away from the centre.

The important thing is that for circular motion at a constant speed, the resultant force in an inertial frame (such as the centre of rotation's) must always be entirely towards the centre and of constant magnitude.

If the motion isn't quite circular, or the speed isn't constant then either the force must have a tangential component or its magnitude must vary or both.

But now I've moved onto a new problem -- it seems to me that it's the bucket which is being pulled towards the center of the circle. The water clearly seems to be trying to fling outward, but is stopped by the bucket. Because imagine if suddenly the bottom of the bucket disappeared -- the water wouldn't go inward, towards me, it wouldn't go along the circumference of the circle, it would clearly fling outward, away from the center.

You're quite right.

In the bucket over the head trick, the bucket is the one which has all the forces exerted on it.

The thing is, the bucket pushes on the water, in the same way the floor pushes you up and stops you falling through it under gravity. This is the force which keeps the water moving in a circle and so the tension in the string has to be larger by this same force than it would be for the empty bucket.

[firechicago]

But now I've moved onto a new problem -- it seems to me that it's the bucket which is being pulled towards the center of the circle. The water clearly seems to be trying to fling outward, but is stopped by the bucket. Because imagine if suddenly the bottom of the bucket disappeared -- the water wouldn't go inward, towards me, it wouldn't go along the circumference of the circle, it would clearly fling outward, away from the center.

This is where your intuition is misleading you and needs retraining. It's a little harder to observe with water, because water is a liquid, and is going to tend to go everywhere.

Let's try a simpler example, like swinging a weight around your head and then letting go of the rope. Did you ever play with one of these as a kid?

Or build your own. Tie a couple of washers to the end of a string. Take it somewhere where you've got room to swing it and won't damage anything and try swinging it around your head and then releasing it at a target. You'll very quickly learn that releasing it when the string is pointing at your target doesn't work. You have to release it when the string is perpendicular to your target.

Or think about a baseball pitcher. If you look closely, you can see that a baseball pitcher swings the ball through a more or less circular arc, and releases it near the top of that arc. By your logic that ought to result in the ball flying upward, rather than directly toward the batter.

[gmalivuk]

(There's no "strikethrough" available, so this is me striking through the above. See my response to Qaanol at the end of this post.)

Yes, there most definitely is "strikethrough" available...

But now I've moved onto a new problem -- it seems to me that it's the bucket which is being pulled towards the center of the circle. The water clearly seems to be trying to fling outward, but is stopped by the bucket. Because imagine if suddenly the bottom of the bucket disappeared -- the water wouldn't go inward, towards me, it wouldn't go along the circumference of the circle, it would clearly fling outward, away from the center.

Well, yeah. How is this a new problem? You're imparting inward force on the bucket, the bottom of which is thereby imparting inward force on the water. If it wasn't there, the water would fly outward. Though it would fly outward in a straight line tangent to the direction it was moving when the bottom disappeared, and not directly outward from the center. (This is ridiculously easy to check: just swing something around on a string once or twice and then let go. If you let go when it's directly in front of you, it won't fly forward, but instead will fly in whatever direction it's already traveling at that point.)

[++$_]

Whoa whoa whoa, what? Gravity is directed toward the center of the circle? Surely it's constantly directed toward the Earth?

I was talking only about the situation at the top of the circle, where the gravitational force does happen to be pointing towards the center of the circle.

[Qaanol]

But now I've moved onto a new problem -- it seems to me that it's the bucket which is being pulled towards the center of the circle. The water clearly seems to be trying to fling outward, but is stopped by the bucket. Because imagine if suddenly the bottom of the bucket disappeared -- the water wouldn't go inward, towards me, it wouldn't go along the circumference of the circle, it would clearly fling outward, away from the center.

An object in motion will remain in motion, moving in a straight line at constant speed, unless acted upon by an outside force. That’s Newton’s first law.

As you swing the bucket, at the top of the circle the bucket and all its contents are moving sideways. If you let go, then there will be no force on them (well there’s gravity, but meh) so they will move in a straight line at constant speed in the direction they are already moving. Namely, sideways.

At the top of the circle, the water and bucket are moving sideways, and you are pulling down to change the direction of the bucket. The bucket is then pushing down to change the direction of the water. Both bucket and water do change direction, keeping them in a circular path rather than the straight line that they would have followed had you let go, namely tangential to the circle.

Spoiler’d for somewhat off-topic

Spoiler:

Another way to phrase Newton’s first law is, “The center of mass of a closed system cannot change velocity”. A more accurate way is, “The center of mass of a closed system follows a geodesic path through space-time.”

Also, gravity as a force is just as fictitious as centrifugal force and the Coriolis force. By doing a coordinate-transform to the locally-inertial reference frame, meaning the one in free-fall along a space-time geodesic through the point in question, all three of these “forces” disappear. Of course, tidal effects from locally-curved space-time make for more complicated dynamics when considering multiple objects at once.

[Qaanol]

Sorry for double posting, but I wanted to add something and figured after this many days an edit would probably go unnoticed. When you swing a bucket around in a circle, there really is a real centrifugal force, and it is the reason people intuitively think “The water would fly straight outward radially if the bucket disappeared.”

So you’re holding the bucket and swinging it in a circle. At any moment the bucket (and water) is moving tangential to the circle, and you are pulling inward, applying a centripetal force to turn the path of the bucket away from a straight line, keeping it on the circle. The bucket is pushing inward, centripetally, on the water, making it turn the same way. Great.

Newton’s third laws says, “Every action has an equal and opposite reaction.” That means, when the bucket pushes on the water, the water pushes right back at the bucket, in the opposite direction. Namely, radially outward. Similarly, when you pull inward on the bucket, the bucket pulls right back at you, straight outward.

So you feel a real, radially-outward, centrifugal force on your hand. That force is really there, and I think it’s the source of much confusion. A person’s natural first reaction to “A force is pulling outward on the my hand” is, “There must be something trying to move outward, that’s pulling against my hand.”

The misunderstanding can be allayed by recognizing that the direction of force and the direction of motion do not need to be parallel. In this case, they are perpendicular. This might be more clear with the rock-on-string setup.

The rock is already moving around, trying to go off in a straight line tangential to the circle. You pull inward on the string, applying a centripetal force. The string pulls inward on the rock, applying a centripetal force. The rock applies an equal and opposite reaction force, pulling outward on the string with centrifugal force. The string pulls outward on your hand, applying a centrifugal force.

Notice that the original “action” force is caused by your hand, applies to the rock, and points inward. You are providing the force that turns the rock away from the straight tangential line it “wants” to follow. The reaction force is caused by the inertia of the rock saying “I don’t wanna turn”, applies to your hand, and points outward.

In general, action and reaction forces apply to different objects. Here one is centripetal and the other centrifugal. There is a centripetal force caused by your hand that acts on the rock, and there is a centrifugal force caused by the rock that acts on your hand. The string is in between, acting as an intermediary, and it feels both forces. Since the string is pulled in opposite directions, it gets stretched tight under the tension.

So the upshot is, your hand does in fact feel a real force pulling straight outward. There is an actual centrifugal force on your hand. But if you let go of the string, or if the bucket disappears, then your centripetal force would cease, and immediately the centrifugal reaction force would also cease. You would feel nothing pulling your hand outward. And the rock, or water, would feel nothing pulling it inward, so it would move in the straight line it’s already moving in, namely tangential to the circle.

The rock (and water) is not trying to move radially outward, it is trying to move straight forward, which is tangential to the circle. But you are preventing it from doing so by applying a force radially inward to turn it. The equal and opposite reaction force felt by your hand is directed radially outward, but nothing is actually trying to move radially outward, it’s just fighting back against your efforts to pull it off the straight tangential line.

Spoiler’d for astrophysics

Spoiler:

In the case of orbital dynamics, with say the moon orbiting the earth, both objects are pulled radially inward toward their mutual center of mass. In this case, both action and reaction are centripetal, even though they still point in opposite directions.