So I’m a teacher that usually overprepares. I have my lesson set up beforehand. Very little is set up for “free form.” This is even true for my Multivariable Calculus class of 5 students.
To be fair, though, in that class we do generally take a 20 minute tangent here or there. Like today we were resolving the acceleration vector of a vector valued function into normal and tangental components — and we spent 15 minutes deriving them because I just decided we should. Spur of the moment thing. Or a few weeks ago, I gave my student 50 minutes to come up with how to convert between rectangular, cylindrical, and spherical coordinates, with no help. But generally, the lessons are carefully planned out. Here’s an example of my introduction to triple integrals (which we do way later in the year) so you can get a sense:
slideshare id=1597835&doc=mvcalculustripleintegrals-090617102556-phpapp01
A few days ago, we had gone over the homework and somehow got on the topic of us being on the earth. I honestly can’t remember what prompted it. But we started talking about the force of gravity, which we feel because the earth is so massive. Then I had an insight — a direction we could take the conversation.
We are also spinning: “Does that change anything?”
I stopped class. I paused for 15 seconds, told the students to hush while I considered whether to go down this route. I felt this pang of deviating from my preplanned lesson. We were going to be behind. Do I really want to possibly come to a dead end?
I almost pushed it off. I was going to “leave it as an exercise to the reader” — tell my students they could think about it independently. But just as I was about to brush it off, I thought: WTFrak. Tangents are more interesting and more memorable, when the kids are interested in them.
My kids seemed interested.
So I threw away the lesson I had planned completely, and we went off the cuff, without a known destination in sight.
So back to the spinning earth. I didn’t know. I hadn’t thought about that kinda obvious fact before — we’re spinning, so that should have some consequences
We learned in our previous class that if something is spinning at a constant velocity in a circular motion, it must have an acceleration pointing inward to the center of the circle. So since we are spinning, once around our latitude every day, we must also feel a force pulling us to the center of the circle.
If we model the earth as a sphere, not tilted, and put us at an angle 45 degrees from the equator… we feel a force pulling us to the center of the earth (from gravity), and also a force pulling us directly inwards (centripetal – from rotating) :
But I don’t feel that centripetal force. I jump up, I come down. I don’t feel like I’m being pulled in any other direction.
So we decided to calculate the magnitude of the two forces, and figure out what’s going on.
Awesome.
I left giddy. We figured that the centripetal force was about 1/400 the force of gravity. Afterwards, I did a few more calculations, and realized that actually some of this centripetal force will be in the direction of gravity, so it will feel even less powerful.
I’m leaving for a wedding tomorrow, so I’m having my kids do a formal writeup of what they found. I can’t wait to see it. I am going to show it to their AP Physics teacher.
(As an aside, I think I’ve found the physics term for what we discovered: the Coriolis force. If anyone knows anything about it, or any good resources on it, let me know!)
Continuous Everywhere but Differentiable Nowhere 
Sam,
I think you’ve found the centrifugal force, not the corolios force which is the force that arises from having motion on a rotating body (it’s also a great application of durl). The centrifugal force is an outward force that any object on a rotating body will feel. Both of these are ficticious forces, meaning they aren’t exerted by real objects, instead, they are forces we have to “invent’ because we are on a non inertial reference frame. But this is an awesome lesson, and I might use this with my physics classes.
Sad! I just stumbled across Coriolis and thought “hey, that looks about right!” I was planning on reading up on it more later.
Can you help me with this:
(a) do I feel the centripetal/centrifugal force, but it’s too weak for me to perceive?
(b) does this force have any practical consequences in terms of calculations? Do scientists ever have to make corrections/adjustments?
(c) if you draw a circle (the earth) and put a person on it (just not on the equator or poles, so we can keep things general), there will be 3 forces acting on the person: gravity towards the center of the earth, the normal force, and the centrifugal/centripetal force inwards.
If you resolve the centrifugal/centripetal force into two forces, one in the direction of gravity, and the other perpendicular, we can see that some of the centrifugal/centripetal force will “feel” like gravity. However, there is a bit of it which will not point in the direction of gravity, but rather perpendicular to it. Well, anytime there is a force, there must be an acceleration in that direction. Why don’t I move in this direction (perpendicular to gravity), just when standing still? (Friction?)
Is this making any sense? It’s late, so probably not.
I can clarify, by scanning in some stuff later.
Sam
Here’s what I’d say
You’re clearly accelerating b/c you’re moving around a circle, so you need to have an inward acceleration at the equator of about 0.033 m/s^2 (this is a centripetal–direction changing acceleration) This means the gravitational force must be slightly larger than the normal force of the ground pushing on you. You feel the normal force on your feet—it’s part of what makes your feet ache. the centripetal force is just the part of the net force that causes your centripetal acceleration.
Where things get weird is when we try to ignore the fact that we are in a non inertial reference frame. In this case, we’d measure our mass, and calculate the gravitational force using universal law of gravity, and find that it is slightly bigger than the number we see on the scale. In order to account for this, we’d say there is a centrifugal (the ‘f’ is for fictitious, I say) force pushing radially outward on us. This force is indeed tiny, and I don’t think it’s perceptible. But if you speed the rotation of the earth up, or your reduce the radius, you could literally feel lighter standing on scale (b/c the normal force isn’t having to push on you as hard).
The centrifugal force is also how you might create artificial gravity, a la 2001. If you’re inside a big rotating ring and the parameters are just right, you could get this fictitious centrifugal force to be equal to mg, and you would feel like you are being pushed into the ring with a force equal to your weight (of course, in reality, you need the ring to push with this force to keep you moving in a circle).
b. I think the centrifugal force may need to be taken into account in long range ballistics. I know the corolios force, which is a force that acts perpendicular to the radial direction must be taken into account in these situations.
c. Not quote. See my explanation above. I tell my kids, forces have to be exerted by objects, and every real force can be described as The [type of force] of the [object exerting the force] on the [object experiencing the force], but this convention, gravity is a real force exerted by the earth, but the coriolis and centrifugal forces are fictitious—no object exerts these forces, and the only reason we perceive them is we don’t fully recognize we are in a reference frame that is accelerating. It’s very similar to the idea that if we were in a rocket in deep space that was accelerating at g, we’d find everything “falls” to the bottom of the rocket, but not because of gravity. Things would fall because the floor of the rocket is accelerating up to meet the ball. (In fact, this rocket could feel exactly like earth—this is one of the bigger ideas of general relativity). We’d say there’s a downward force on the ball, only because inside the rocket, we’d think the ball is accelerating downward (in reality it has constant velocity, and the floor is accelerating), and so we’d invent a downward force to explain this.
Your last question is a bit harder for me to understand. If you are standing still, I don’t think there has to be any force that is parallel to the surface of the earth. However, if you start to move along the surface, then it does turn out that there is a fictitious corolios force , and this force acts at right angles to angular velocity of the earth and the velocity of the object (its the cross product of the angular velocity of the earth and the velocity).
You can most easily see the coriolis force in action if you imagine yourself on a merry go round, trying to toss a ball into a basket in the center of the merry go round. If you aim straight for the center, the ball will appear to be deflected from the straight line path toward the center (and you would say a coriolis force is responsible for this). In reality, an observer watching you would say that the ball had some velocity tangential to the circle when you released it, and so it kept that velocity which made it not go solely in the radial direction.
Wikipedia has a pretty nice entry on coriolis and centrifugal forces.
Reminded immediately of this!
The spinning of earth causes you to be lighter, not heavier. It will make your weight on a scale less at the equator than at the poles (unless you use a mass balance instead of a force scale).
Okay, I’ve been thinking about this for awhile. You definitely didn’t calculate the coriolis effect.
What you calculated is what portion of the gravitational force is keeping you going around in a circle. Every portion of gravity beyond that calculated force needs to be counteracted by the normal force on you by the ground. So if you could decrease the magnitude of gravity by the right factor, you would be weightless at Lat 45 degrees. At the equator, you would need to grab on to the ground to keep from being flung off into space. At the poles, you’d be cold, but gravity would keep you on the ground.
Another way of saying this: you’ve found how much of the gravitational force is counteracting the fact that your body would shoot off the earth tangentially.
It’s also important to note that the Coriolis is an effect, and there is no basic force involved. It’s more like a piece of several other forces, that we can treat independently, like we’ve defined a new function. Check out the wikipedia page on Coriolis, it’s pretty good.
The Coriolis force is generally denoted as the fake force you would feel in a rotating frame if you are changing your radius of rotation. Compare this to the centrifugal force which is the fake force you feel in a rotating frame because you are moving in a circle.
Really, they are just two parts of the same force. The corilios part is essentially the cross product of the velocity and the angular velocity.
Here is a post about these two forces – http://scienceblogs.com/dotphysics/2010/01/rp_11_when_the_centrifugal_for.php