In this lecture we will be discussing the topic of Uniform Circular Motion. The “circular motion” part should be clear enough — we will be talking about objects which are moving around in a circular path (whether they actually complete the circular motion or not is irrelevant — the only thing that concerns us is that the object is currently following a path which is at least a part of a circle of a given radius). The “uniform” in uniform circular motion means that the object is moving around in the circle with a constant speed. This actually brings us to the first problem associated with circular motion — that of constant speed versus constant velocity. Do you remember the difference between speed and velocity? (Hopefully so!)
Remember that speed is just the magnitude of the velocity vector. A vector is a quantity which has a magnitude and a direction. The speedometer in your car measured speed (it doesn’t care which direction you are moving in — it just tells you how fast you are going). You can build a velocitometer in your car by sticking a compass on your dashboard. The combination of the information from your speedometer (magnitude) and your compass (direction) give you the full velocity vector information. If either one of these bits of information change, then your velocity is changing.
Now let’s say that you drive around a circular race track. Is it possible to drive around this track with a constant speed? Of course! We just drive such that our speedometer stays at the same reading (maybe 50 mi/h). And what about the velocity — is it possible to drive around this track with a constant velocity? No way!! It is impossible to drive around a circular track (or any curve, for that matter) without changing your direction (as your compass will more than clearly show you!). Now see if you can correctly answer the following question: You are driving around the circular race track with a constant speed. Do you have an acceleration? Yes or no?
Hopefully you were careful about the difference between speed and velocity in answering the question above. Remember that velocity is a vector, and so is acceleration. As a matter of fact, acceleration is a vector because it is defined in terms of velocity: the acceleration is the change in velocity over a corresponding change in time. Thus, if you have a change in velocity, you do have an acceleration (and the direction of the change in velocity is the same as the direction of the acceleration!). Since, as we discussed above, if you are moving along a curved path your velocity must be changing (even if your speed remains a constant), it then follows that, if you are moving along a curved path, you must be accelerating! This acceleration is due not to a change in speed of the car (as we’ve had in the lectures up until this point), but rather to a change in direction of the car. This special acceleration is called the centripetal acceleration, and will be discussed further in the next section.
For now, we go on to give some basic definitions dealing with circular motion. You will have to have these definitions memorized (if you don’t know them already!).
The figure at left represents an object of mass m moving with a constant speed v in a circle of radius R. Note that the velocity vector, which points in the direction that the object is heading at any particular instant, is always tangent to the circular path at every instant of time.
DEFINITIONS:
PERIOD (T): the amount of time required for the object to complete one repetition of its motion. For circular motion, this is just the amount of time it takes the object to go around once. Units: seconds (s) or seconds per revolution (s/rev)
FREQUENCY (f): This is also sometimes called the linear frequency, and gives the number of times the motion is repeated per second, or, in the case of circular motion, the number of times the object completes the circular motion per second. Units: hertz = cycles per second = repetitions per second = revolutions per second (Hz)
ANGULAR FREQUENCY (w): (This is the Greek lower-case letter omega.) This gives the angle measured in radians that the object moves through (as viewed from the center of the circular motion) per second. Units: radians per second (rad/s)
Note that the meaning of frequency, revolutions per second, is just the inverse of the meaning of period, seconds per revolution. This implies that the two quantities f and T are just inverses of one another: f = 1/T. Also, recalling that there are 2p radians in one complete circle (360o) or one revolution, we can see that we can symbolically write
w = rad/s = (2p rad/rev)(rev/s) = 2pf.
Combining all of these relations then gives us that
In addition, since the speed of the object is constant, it follows that the following relations must also hold true (remember that the circumference of a circle is just given by 2pR):
Note that we have applied the equations above in order to obtain these relations. You should know these definitions and the resulting relations given above. It should not be a matter of memorizing these relations, but rather knowing them as a result of understanding their meanings. If you understand the meanings, then the relations follow, no memorization required!
This is the end of this rather wordy section. The next section discusses the concepts of centripetal acceleration and the related centripetal force. That will then be the end of the new ideas for this lecture. The remaining part of the lecture will then simply be examples to show how these ideas are applied.