The difference between "centripetal" and "centrifugal" force

This one's in response to another high-school student's question: What's the difference between "centripetal" and "centrifugal" force? Why do we say this has something to do with relativity?

Well, defining centripetal force is easy. Centripetal force is what keeps pulling a rotating object towards the center of what it's rotating around so that it doesn't fly away. For example, if you swing a ball on a rope around horizontally over your head, the centripetal force is your pulling on the rope so the ball doesn't fly away. Or in the case of the Moon rotating about the Earth, the centripetal force is the force of gravity constantly pulling the Moon toward the center of the Earth, otherwise the Moon would fly away in a straight line too. Remember Newton's First Law - a body in motion wants to stay in motion in a straight line at a constant speed unless something else (gravity or friction or whatever) specifically acts to change that motion. Anyhow, so centripetal force points toward the center of the rotating system.

The definition of centrifugal force on the other hand, and its difference from centripetal force, is a little more involved; it's ironic that it is the term more familiar in popular usage. From popular usage we at least know that centrifugal force has something to do with rotation and with getting thrown off merry-go-rounds. But before I define centrifugal force and compare it to centripetal force, I have to take a second and talk about different "frames of reference" so the rest makes sense.

The difference between centripetal and centrifugal force has to do with different "frames of reference", that is, different viewpoints from which you measure something. When I'm measuring how far away a baseball that I throw lands, I measure it in comparison to something. For example, if I'm standing on the street and throw the ball down the street I might measure it having landed 50 feet away from where I'm standing on that street. We call that a measurement in the frame of reference of the street. However, if I were standing in the back of a pickup truck while it is being driven down that same street (which of course is unsafe to do but that's a separate issue!), and I throw my baseball just as hard as I did before, how far does it go? It depends how I choose to measure it - do I measure the distance from where I am in the moving truck when the ball lands, or do I measure from the point on the street that the truck was at when I threw the ball? If I measure it from where I am in the moving truck when the ball lands, I find that the ball traveled 50 feet again. But if I measure from that point on the street the truck was at when I threw the ball from it, I find that the ball went farther than the 50 feet, because the movement of the truck acted in addition to my own throwing. Both measurements are valid even though they disagree in the distance; you just have to be clear what you were measuring in comparison to. In other words, you have to be clear what your frame of reference was.

It turns out there are two types of these frames of reference. If we are in a frame that is either at rest or moving in a straight line at a constant speed in comparison to everything we consider around us, we call this an inertial frame of reference. In contrast, a non-inertial frame of reference is one that is accelerating in some way compared to everything we consider around us. I should mention, by the way, that to us physics people, "accelerating" can mean both changing the speed you are going as well as changing the direction you are going (notice that above for the inertial frame of reference I made a point of saying constant speed in a straight line). Anyhow, an accelerating frame, that is, a non-inertial frame, has extra forces in it that don't come from something directly obvious to someone who can't see the broader picture outside that frame of reference. For example, we are all familiar with being pressed back into our seats in the car when it accelerates from a stop, and if a ball is tossed upward in the front seat and then the car accelerates, the ball can land in the back seat(*). Some "mysterious" extra force - related of course to the acceleration of the car - makes the ball move toward the back of the car. I say "mysterious" because this force on the ball would be surprising to someone who didn't realize they were in an accelerating car at the time.

Now, here's the important part in that example: a person watching this same ball while standing at the side of the road instead sees the ball move merely straight up and back down, while the car accelerates underneath it - to this person on the side of the road there never was any force pushing the ball to the back of the car because the ball never moved sideways to them, it only moved up and down. So this mysterious force, which we call an inertial force, exists in some frames of reference but not others.

The same phenomenon happens in a rotating frame of reference like a merry-go-round. Here the acceleration is a change of direction rather than a change in speed, but it's still an acceleration, and a rider on the merry-go-round similarly feels these mysterious forces that to an observer on the ground are not forces at all but the result of something else. The observer on the ground remembers from his or her science class that Newton's First Law stated that an object in motion (the person on the merry-go-round) prefers to stay in motion in a straight line unless something specifically acts to change it (the merry-go-round doesn't go in a straight line). So the person on the ground sees that the person on the merry-go-round is moving around in a circle and that the force pulling them to the center of that circle is the friction from the merry-go-round floor - that's centripetal force. But the person on the merry-go-round doesn't see themselves moving at all (if they don't look up that is!). To their mind they're standing or sitting still on the merry-go-round. That is, according to their frame of reference they aren't moving. But they feel this mysterious force trying to pull them off the merry-go-round; this force is pointing away from the center of the merry-go-round and is the centrifugal force.

So we have non-inertial (accelerating) frames of reference which have the "mysterious" extra forces which don't show up in other frames of reference. In fact, for this reason physicists prefer to write basic equations of motion in inertial frames of reference, which are either at rest or moving at a constant velocity and so do not have any extra forces in them. Centrifugal force doesn't show up in that case then, and this is why sometimes centrifugal force doesn't show up at all in your science textbook and you only learn about centripetal force - that way you can stick with a consistent set of equations of motion and sidestep this whole frames of reference issue.

So to conclude, centripetal force and centrifugal force are really the exact same force, just in opposite directions because they're experienced from different frames of reference. And this is what the concept of relativity centers on - comparing what happens in different frames of reference like this. For example, one of the interesting points in "general relativity" is the principle of equivalence, which harkens back to the car example above. If the person in the car had earplugs and a blindfold on and so didn't know they were in an accelerating car, they might just as validly think that the car was tipped up a bit and that gravity was what was pulling the ball (and themselves) toward the back of the car. Without additional knowledge, like taking off the blindfold and earplugs, one can't tell the difference. Relativity is all about comparing frames of reference, and that's the connection with relativity here. Now with any luck this explanation about the difference between centripetal and centrifugal forces may help serve as an introduction to some concepts useful in learning about relativity as well.

*(Thank you to Fernando Del Rio for catching the original misleading wording in this sentence! I have edited the sentence in the text above from its original to be more accurate. Alas your email server bounces my email reply to you, so I will post my reply to you below, which others may find interesting too.)
[...back up to (*) above...]

On 4 Apr 2014, Fernando Del Rio wrote:
Andy; great write-up on the difference between centripetal an centrifugal forces. But you said if a ball is thrown up in a car, it will land in back seat. That actually does not happen regardless of the speed on the car. The accelerating force exerted on the ball by the moving vehicle makes the ball fall back down to the same spot it was thrown from. Try it. You certainly know a lot more about Physics than I do, but I think you might have slipped on this. Thanks for the explanation. Good job. Fernando Del Rio

Hi Fernando,
Ha ha, clearly you don't drive as recklessly as I do.  ;-)

But seriously, I just went back and looked up that post I wrote years ago - here's the relevant sentence:  "For example, we are all familiar with being pressed back into our seats in the car when it accelerates from a stop, and during this acceleration a ball tossed upward in the front seat can land in the back seat."

Aha, yes I see what you mean - I wrote that sentence poorly.  Pardon me and brilliant catch!  The meaning as you read the sentence was, "the ball is tossed upward during a constant acceleration."  The accurate version of the sentence as I'd originally meant it would be, "...during this horizontal acceleration, a ball that had been tossed upward before the horizontal acceleration began can land in the back seat."  So the sentence as you read it in the blog post implies the car begins a constant acceleration and THEN the ball is tossed upward, while I'd meant that the ball is tossed upward while the car is stopped and THEN the car begins the acceleration.

Note that the phrase "from a stop" above is not the only possibility here - it's the change in acceleration that matters, whether the acceleration goes from zero to value A, or from value A to value B.  So even in the meaning that you had read the sentence, ie ball toss while accelerating, if the acceleration increases (via increasing torque from engine) after the ball is tossed up, then the ball could still end up in the back seat.  Say perhaps you have the gas pedal partway down while tossing the ball and then you floor it, or you toss the ball while gradually pushing the gas pedal downward to the floor.

Here's a quick proposition for you - by your wording it sounds like you have enough background to do this:  What do you say we each go figure out mathematically whether this thing could happen in a real car?  That is, what constant acceleration would we need - AFTER tossing the ball up - to get the ball to the back seat 1m behind, given a toss where the ball goes up 0.5m?  (Ignore air resistance etc.)  In this simple case does the weight or size of the ball matter?  Let's express this acceleration in terms of how many seconds would it take to go from 0-100kph.  Is this acceleration achievable in a real car?  Typical sports-car abilities are 0-100kph in 3-5sec, while regular cars/trucks are more like 0-100kph in 10sec.  F1 racing cars can do 0-100kph in 1-2sec (I went to one of those races last year, very cool)!  Honestly I don't know the answer to this little problem yet - I'll go do it myself too.  And if you need assistance after all, I'm very happy to help you set up the problem...

Note I am NOT suggesting that you try it as a stunt in a real car, only mathematically; I do not wish for anyone to get hurt!

Best regards, -Andy

(at F1 Grand Prix du Montréal 2013: me! Vettel! Hamilton! )