In the opening page of this lecture, we were discussing a bus ride in which you and a bunch of friends were riding in the back of a big bus as it rounded a curve toward the right. We posed the question, “In which direction is a force acting which is a result of the bus moving around the curve?”, and we stated that the answer was to the right, into the curve. It’s now time to resume that discussion and to (hopefully) clear up any remaining confusion or misconceptions.
Fist of all, remember Newton’s 1st law: An object moving with constant velocity will continue to move with that constant velocity unless acted upon by a net, external force. Among other things, this means that an object moving in a straight line will continue to move in that straight line until a net, external force acts on it. Well, here are a bunch of bodies moving together in a bus along a straight line (before the curve). Just to help the discussion, let’s say that this is a special convertible bus, and that we have the top down. We also have some friends helping us out who are hovering above us in a helicopter, videotaping us as we go along the road and around the curve.
We travel along the straight road, and then round the curve to the right. As we do this, we “feel” our upper bodies “pulled” to the left. Is this really what happens? It certainly feels that way!
This force that we think we feel pulling us to the left is really a fictitious force; it is called the centrifugal force. To see what’s really happening, we have to view the videotape made by our friends in the helicopter.
In the videotape made from directly above the bus, we see a bunch of happy, handsome people (especially one person!) traveling merrily along a straight-line path. Then we see the bus starting to move to the right. As the bus starts curving to the right, we see something that makes very much sense as viewed from above: we see the bodies of the people in the bus trying to move in a straight line. That is, we see the people’s bodies trying to continue moving in the same straight-line motion that they had before the bus started making the turn. However, there is a new net, external force acting on them. What is it? It is basically the frictional force between their merry asses and the bus seats! As the bus moves toward the right, so of course do the bus seats. This means that the seats are trying to pull the people towards the right along with them. But since only the people’s derrieres are in contact with the seats, the people’s lower halves tend to move to the right with the bus, while the their top halves continue to move in the same straight-line motion that they had previously (since the net, external seat-friction force has not yet acted on them)! Eventually the top halves have to follow the bottom halves (or vice versa), and the people straighten up again or lean into the curve (to keep themselves from sliding if static friction isn’t enough to hold them in the circular motion).
The net result of this discussion is the following: As the bus turns towards the right, a net, external force (the friction force from the bus seats) acts on the people towards the right to pull them into the circular motion along with the bus. If this did not happen (or if the seats were too slippery so the frictional force was not large enough), the people would slide across the seats, trying to remain in the same straight-line motion that they had prior to entering the curve. (This would happen until they bumped into the side of the bus, thereby encountering a normal force from the side of the bus which would then exert a force towards the right to hold them in the circular motion).
Of course, all of this is possible only because there was sufficient friction between the road surface and the bus tires to pull the bus into the circular motion around the curve. If the frictional force were not large enough, the entire bus would tend to continue in its straight-line motion, and run right off the road (this is what tends to happen to people driving too quickly in icy weather).
There is a lot of important information in this bus discussion which has taken us through a good part of two sections of this lecture. The reason for all of the words is that the ideas presented here focus on the underlying confusion that people tend to have about circular motion, which is at the root of why students tend to have so much trouble with circular motion problems. Please take some time to sit back and think about this discussion about the bus. Draw some sketches as viewed from above in the helicopter, and make sure that you really understand what’s going on. Make sure that you see that, in order for an object to move around a curve, it needs a force pulling it inward towards the center of the curve. When you understand this, you are ready to go on in this lecture.
Remember that acceleration is the change in velocity divided by the corresponding change in time. For uniform circular motion, this change in velocity is a result of the changing direction of the velocity vector. In what direction does this vector change? Newton’s 2nd law tells us that, in whatever direction the acceleration vector (or change in velocity vector) points, this must be the same direction as that of the net force acting on the object undergoing the circular motion.
Imagine that you have a beetle that can only walk in a straight line. You draw a large circle on a piece of paper, and try to get the beetle to walk around the circle. You accomplish this feat with a straw: as the beetle walks around, you gently tap it to make it walk along the circle. QUESTION: In what direction must you tap the beetle in order to get it to follow the circular path? It should (hopefully) be clear that you must continually tap the beetle towards the center of the circle. If you don’t do this, the beetle will walk in a straight line and leave the circular path that you’ve drawn. This again says that, in order for an object (the beetle) to follow a circular path, it must experience a net, external force (the tapping force of the straw) towards the center of the circular motion. This means that the resulting acceleration must also point towards the center of the circle. This acceleration is called the centripetal acceleration, ac. (The word centripetal comes from the Greek, and means “towards the center”.)
CENTRIPETAL ACCELERATION (ac): the component of acceleration pointing towards the center of circular motion which results from the change in direction of the velocity vector. This special acceleration has a special equation giving its magnitude:
We can call the direction towards the center of circular motion the x direction if we wish. Or we could call it the y direction or the s direction (for strange!). But to make life easier and to remind us of what we’re talking about, we will call it the c-direction (to remind us that this is the direction towards the center of the circular motion, or the centripetal direction). It is very important to remember that this subscript is simply labeling a component of vectors: the component along an axis pointing towards the center of the circular motion. We can then call the direction perpendicular to this the y-direction (for example). In this case, Newton’s 2nd law in component form would be written:
This is just like the x- and y-components of Newton’s 2nd law that we had previously. The only difference is that we are using a c subscript instead of an x to remind us that we are talking about a special direction towards the center of circular motion, and that we therefore have a special expression for the acceleration in this direction. If you want to continue to call this direction the x-direction, and if you can remember the special things about this direction, then you may do so.
The unfortunate thing about Newton’s 2nd law as applied to circular motion is that the sum of the components of forces in the c-direction acting on the object undergoing circular motion is itself given a special name, and this name tends to confuse an already confusing subject! This sum of forces is called the centripetal force:
It is extremely important to keep in mind, however, that this is not a force in itself. That is, this is not some magical circular-motion force that suddenly appears whenever an object decides to go in circular motion. Rather, it is simply a sum of components of forces that are already acting on the object in question (such as the friction between the bus seats and the people’s lower extremities, or between the bus tires and the road surface). You should NEVER have a force in your free-body diagram called the “centripetal force”. (It is for this reason that we have not given this “force” a symbol, as many physics texts like to do!) There is nothing more magical about the centripetal force than there would be if we decided to call the sum of forces in the y-direction the “uppity force” because it is the direction that usually points up:
It doesn’t change the fact that we are still just drawing a FBD of the forces acting on the object in question, and summing the components of those forces in a specified direction!
OK — again, we’ve gotten carried away with lots of words, but, as before, these words were delivered trying to dispel misconceptions about some confusing points dealing with circular motion. We end this section with an outline of how to approach solving circular-motion problems. We will then simply go on to show how to apply these steps in some examples.
Solving Circular-Motion Force Problems
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