Assignment #6 Reading, Objectives, & Problems
A. Availability and Due Dates
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Assignment #6 is available Tuesday, October 26, 2004. The target date for its successful completion is Thursday, November 4, 2004. In any event, all of the Type A problems from Assignment #6 must be successfully completed and turned in by 5:00 pm on Friday, November 5, 2004. If this requirement is not met, the student will not be allowed to take Quiz 3, to be given in class on Tuesday, November 9, 2004. |
B. Reading
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As preparation for completing the problems in Assignment #6, read Chapters 12 & 13 in Cohen's The Fundamentals of College Physics, Volume IB. |
C. Objectives
In addition to the Objectives listed on Assignments 1- 5, after completing Assignment #6, the student should
[1] Know what the term period means when applied to circular motion.
[2] Understand the meaning of the terms radial and tangential. As special cases, the student should understand what these terms mean when applied to quantities such as velocity, acceleration, and displacement.
[3] Understand the definition of polar coordinates. In particular, the student should know what is meant by the unit vectors
and 
.[4] Know what is meant by the phrase Uniform Circular Motion. Realize that an object undergoing uniform circular motion is accelerating, and realize why. Know also what are meant by the terms centripetal and centrifugal.
[5] Understand the details of Uniform Circular Motion. Specifically, the student should be able to explain the relationship between the velocity and acceleration vectors for an object undergoing uniform circular motion (qualitatively and quantitatively, magnitude and direction).
[6] Realize that the centripetal acceleration of an object undergoing uniform circular motion implies the existence of a corresponding net centripetal force acting on the object, and be able to calculate the magnitude of this force.
[7] Know that the often mentioned "centrifugal force" is actually a fictitious force, and be able to explain its (apparent) origin.
[8] Be able to analyze arbitrary dynamical problems in which objects are undergoing circular motion.
[9] Realize that an object experiencing a restoring force proportional to its distance from equilibrium will execute simple harmonic motion. Be able to express the position of such an object as a function of time.
[10] Be able to explain the link between uniform circular motion and simple harmonic motion.
[11] Understand what the terms amplitude, period, frequency, phase, and angular frequency mean when applied to simple harmonic motion, and be able to calculate these quantities given sufficient information.
[12] Be able to explain why simple harmonic motion is arguably the most relevant type of motion as far as the natural world is concerned.
D. Type A Problems
[1] Problem 12.1 on p. 286 of Cohen.
[2] Problem 12.3 on p. 286 of Cohen.
[3] Problem 12.4 on p. 287 of Cohen.
[4] Problem 13.1 on p. 298 of Cohen.
[5] 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.
(i) At the instant that the ball is at the position x = 1.5 meters, y = 0 meters….
(a) Draw the r-hat and t-hat vectors at this position.
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(b) Does the ball have a non-zero tangential velocity at this point in its motion? If so, what is it? |
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(c) Does the ball have a non-zero tangential acceleration at this point in its motion? If so, what is it? |
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(d) Does the ball have a non-zero radial velocity at this point in its motion? If so, what is it? |
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(e) Does the ball have a non-zero radial acceleration at this point in its motion? If so, what is it? |
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(f) Does the ball have a non-zero centripetal acceleration at this point in its motion? If so, what is it? |
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(g) Is the ball being subjected to a non-zero centripetal force at this point in its motion? Explain. |
(ii) Repeat the questions (a)-(g) above for the instant that the ball is at the position x = 0 meters, y = 1.5 meters.
E. Type B Problems
[6] Problem 12.2 on p. 286 of Cohen.
[7] No Question to answer here, but read carefully:
Suppose that we went to the frame of reference of the aluminum ball described in Problem [5] 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 have discussed, 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.
[8] If you watch the stars of the night sky long enough, you will notice that they appear to travel along circular paths. What is the period of their apparent circular motion?
[9] A ball of mass 40 grams is attached to the end of a string of length 60 cm. The ball is then swung on a horizontal circular path at constant speed v. If the string snaps when the tension in it exceeds 350 N, what it the largest speed v that the ball can experience?
[10] A block of mass 5 kg is attached to the end of a spring having spring constant
K = 20 N/m. The spring and mass are oriented horizontally, and the mass is free to slide upon a frictionless plane. The mass is pulled a distance 5 cm to the right of its equilibrium position, and is then released from rest.
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Determine the amplitude of the resulting motion. |
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Determine the maximum speed experienced by the mass. |
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Determine the maximum acceleration experience by the mass. |
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Determine the total energy of the oscillating system. |
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How long does it take the mass to complete one full cycle of motion? |
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