p0314 - 3.14. CHAPTER 3, PROBLEM 14 219 3.14 Chapter 3,...

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Unformatted text preview: 3.14. CHAPTER 3, PROBLEM 14 219 3.14 Chapter 3, Problem 14 Problem: A steel object of mass m rests on a steel turntable that starts from rest and has a constant angular acceleration, α, so that its angular velocity, Ω, increases linearly with time, t. The object is a distance R from the center of the turntable, and is constrained so that it can slide only in a straight radial line. (a) Find the time τ at which the object will begin moving radially. Express your answer in terms of R, α, static-friction coefficient, µs , and gravitational acceleration, g . (b) Assuming R = 1 cm and α = 108 sec−2 , compute τ for a clean and dry turntable and for a lubricated turntable. . . . .. . . . . . . . . . . ... . .... .... . ........ .. .... . ...... .. ... .... ..... . ..... . . ............... ......... . . . . . . ... ........ .. .............. .... . . . .. . ............................................................. ... ....................................................................... ... .............. ... . ... ............ ..... . ........................................................................ .. .. .. . . ........ . . .......... ................................................................................. . ................. ............................................................................................................ .. ... . .............................................................................................................. . . . .. . . .. ...... .. ................ .... ... ........... 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........ .. .. z Ω . . . . . . . . . . . . . . . . . . . . . ... ... . .. .. .. . . g = −g k • m R Solution: (a) Since the object is constrained to radial motion, in order to remain fixed relative to the turntable the static-friction force between the object and the turntable must exceed mar , where ar is the radial acceleration. Denoting the static-friction coefficient by µs , we have But, we know that ar = v /R, where v = ΩR = αtR. Consequently, 2 µs mg ≥ mar Thus, the object will begin to move at t = τ when τ= µs g ≥ α2 t2 R 1 α µs g R (b) The object and the turntable are both made of steel. Also, R = 1 cm = 0.01 m and α = 108 sec−2 . Thus, we have the following. Clean and Dry Turntable. From Table 3.1, the coefficient of static friction is µs = 0.78. Therefore, the time at which the object will begin moving is 1 τ= 108 sec−2 0.78 9.807 m/sec2 0.01 m = 0.256 sec = 256 msec Lubricated Turntable. From Table 3.1, the coefficient of static friction is µs = 0.11. Therefore, the maximum angular velocity is 1 τ= 108 sec−2 0.11 9.807 m/sec2 0.01 m = 0.096 sec = 96 msec ...
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This note was uploaded on 05/11/2011 for the course AME 301 taught by Professor Shiflett during the Spring '06 term at USC.

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