Assignment #3 Problems
For Exam 2, you will be expected to do problems of difficulty similar to those below without reference to the text or to any notes. Make sure that you can do this before you move on to the next assignment!
(1) Do Questions 1, 8, 13, 21, 24 and 25 on pp. 100 & 101 of Serway & Jewett.
(2) Do Problems 1, 2, 7, 10, 12, 16, 24, 28, 31, 41, and 71 on pp. 101-107 of Serway & Jewett.
(3) An aluminum ball is attached to the end of a string and whirled in a circular path in the horizontal plane (xy-plane). The length of the string from the center of the circle to the center of the aluminum ball is 1.5 meters, and the ball travels at a constant speed of 12 m/s in the counterclockwise direction. The origin of the xy-plane is taken to be the center of the circle.
At the instant that the ball is at the position x = 1.5 meters, y = 0 meters….
(a) Draw the r-hat and theta-hat vectors at this position.
|
(b) Does the ball have a non-zero tangential velocity at this point in its motion? If so, what is it? |
|
(c) Does the ball have a non-zero tangential acceleration at this point in its motion? If so, what is it? |
|
(d) Does the ball have a non-zero radial velocity at this point in its motion? If so, what is it? |
|
(e) Does the ball have a non-zero radial acceleration at this point in its motion? If so, what is it? |
|
(f) Does the ball have a non-zero centripetal acceleration at this point in its motion? If so, what is it? |
|
(g) Is the ball being subjected to a non-zero centripetal force at this point in its motion? Explain. |
(4) Repeat the questions (a)-(g) above for the instant that the ball is at the position x = 0 meters, y = 1.5 meters.
(5) No Question to answer here, but read carefully:
Suppose that we went to the frame of reference of the aluminum ball described in Problem (3) above. In other words, imagine riding along with the aluminum ball. In this frame of reference, the ball is not traveling on a circular path. In fact, in this frame, it is not moving at all. Nevertheless, there is still that string pulling on it. Now, if there is a string pulling on something and it doesn't move, you would be tempted to conclude that there is some other force working in the opposite direction to cancel it out. This is the origin of the idea of a "centrifugal force", a force that works opposite to the string and tries to pull the ball away from the center of the circle. Of course, this is just nuts. There is no force trying to fling the ball outwards. If there were, what would be causing it? There is nothing attached to the ball that could possibly pull it outwards, and gravity doesn't act horizontally. No, there is no real force acting outwards on the ball. Our impression that there is one comes from the fact that we are trying to analyze the situation from a point of view that is, itself, kind of nuts. If we are riding along with the ball, then our reference frame itself is being accelerated. And, as we shall see, whenever one attempts to analyze things from the point of view of an accelerated reference frame, fictitious forces appear. The supposed "centrifugal force" previously mentioned is but one example of such a fictitious force.
|