Ps05sol - MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 Problem Set 5 Practice Problems Exam 1 Problem 1 A coin of mass

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01 Problem Set 5: Practice Problems Exam 1 Problem 1: A coin of mass m (which you may treat as a point object) lies on a turntable, exactly at the rim, a distance R from the center. The turntable turns at constant angular speed ! and the coin rides without slipping. Suppose the coefficient of static friction between the turntable and the coin is given by μ . Let g be the gravitational constant. What is the maximum angular speed max such that the coin does not slip? Solution: The coin undergoes circular motion at constant speed so it is accelerating inward. The force inward is static friction and at the just slipping point it has reached its maximum value. We can use Newton’s Second Law to find the maximum angular speed max . We choose a polar coordinate system and the free body force diagram is shown in the figure below. The contact force is given by ! C = ! N + ! f s = N ˆ k ! f s ˆ r . (0.1) The gravitational force is given by ! F grav = ! mg ˆ k . (0.2) Newton’s Second Law in the radial direction is given by ! f s = ! m R " 2 . (0.3) Newton’s Second Law, F z = ma z , in the z-direction, noting that the disc is static hence a z = 0 , is given by 0 N mg ! = . (0.4)
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2 Thus the normal force is N mg = . (0.5) As ! increases, the static friction increases in magnitude until at max = and static friction reaches its maximum value (noting Eq. (0.5)). s max ( ) f N mg μ = = . (0.6) At this value the disc slips. Thus substituting this value for the maximum static friction into Eq. (0.3) yields 2 max mg mR = . (0.7) We can now solve Eq. (0.7) for maximum angular speed max such that the coin does not slip max g R = . (0.8)
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3 Problem 2 A small ball of mass m is suspended by a string of length l . The string makes an angle ! with the vertical. The ball revolves in a circle with an unknown constant angular speed . The orbital plane of the ball is a height h above the ground. Let g be the gravitational constant. You may ignore air resistance and the size of the ball. Express your answers to the questions below in terms of the given quantities m , l , , h , and g as needed. a) Draw a free body diagram of the forces acting on the ball when it is attached to the string at the instant it passes above the x - axis. b) Find an expression for the angular speed . Later, the ball detaches from the string just as it passes the x - axis. It flies through the air and hits the ground at an unknown horizontal distance d from the point the at which it detached from the string. c) What is the horizontal displacement d of the ball during the time interval that the ball is freely falling?
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