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Unformatted text preview: Problem Set 3: Problem 4.
Problem: A small object of mass m is placed on the inner surface of a conical dish at a radial distance R from the axis of rotation. The dish is rotating at a constant angular rotation rate . The friction coefficient between the object and the dish surface is s , and the object is constrained so that it can slide only in a straight radial line, i.e., there can be no angular acceleration. (a) In what direction do you expect the object to move for very large values of and for very small values of ? (b) Determine the range of values of for which the object will remain at a fixed position on the dish without slipping. Express your answers as a function of s , R, and gravitational acceleration, g. HINT: Solve for the maximum rotation rate and then explain how changing the sign of terms proportional to just one parameter yields the minimum rotation rate. Solution: The first thing we must do is identify the forces acting on the object. There are three. As shown in the figure below, they are the weight of the object, mg, the reaction force from the dish wall, N , and the friction force, s N . .. ...
.... .... .... .... .... .... .... ..... .... .... . . .... .... . ... ........ .. . ... .. ..... . .. . .... ... . .... . . .... .. ...... ..... ......... . .. .. ... ..... ........ .... .. ............ . .... . . s . .. . .. .... . . .... . .. ... . . . .. .... . .... .. . . .... .... .. . .. . .... .... . . .. . .. .... . .. . .... . .... .. . .. . .. . .... .... . . .. .. .. . ... .... .... .. . .. .. .. . .. .. .. .... .... . . . . .. .... .... . .. . .... . .... . . .. . . .... . .... . ................................................................................................. ................. ............................................................................... .. . m N mg N (a) When the angular rotation rate is very high, we expect the centrifugal force to cause the object to move toward the top of the dish, overcoming the combined effects of gravity and the friction force. When the rotation rate is very small, we expect the gravitational force to dominate and cause the object to slide toward the center. The friction force would act in the opposite direction to that in the sketch as it would oppose the motion. (b) From Newton's Second Law the vertical force balance tells us that N cos = s N sin + mg while the radial force balance yields N sin + s N cos = mv2 ...
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This note was uploaded on 05/12/2010 for the course AME 301 taught by Professor Shiflett during the Spring '06 term at USC.
 Spring '06
 Shiflett

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