hw4.pdf - Homework 4 PHY 5200 Fall 2016(due on 1 A particle...

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Unformatted text preview: Homework 4, PHY 5200, Fall 2016 (due on 10/05/2016) 1. A particle of mass m is moving on a frictionless horizontal table and is attached to a massless string, whose other end passes through a hole in the table, where I am holding it. Initially the particle is moving in a circle of radius r0 with angular velocity ω0 , but I now pull the string down through the hole until a length r remains between the hole and the particle. (a) What is the particle’s angular velocity now? (b) Assuming that I pull the string so slowly that we can approximate the particle’s path by a circle of slowly shrinking radius, calculate the work I did pulling the string. (c) Compare your answer to part (b) with the particle’s gain in kinetic energy. 2. (extra 2 pts) Consider a small frictionless puck perched at the top of a fixed sphere of radius R. If the puck is given a tiny nudge so that it begins to slide down, through what vertical height will it descend before it leaves the surface of the sphere? 3. Prove that the direction of ∇f (r) at any point r is the direction in which f increases fastest as we move away from r. 4. Which of the following forces is conservative? (a) F~ = k(x, 2y, 3z) where k is a constant. (b) F~ = k(y, x, 0). (c) F~ = k(y, x, 0). For those which are conservative, find the corresponding potential energy U . 156 5. A metal ball (mass m) with a hole through it is threaded on a frictionless vertical rod. A massless string (length l) attached to the ball runs over a massless, frictionless pulley (at a distance b from the rod) and supports a block of mass M , as shown in Figure below. The positions of the two masses can be specified by the one angle θ. (a) Write down the potential energy U (θ). (Assume that the pulley and ball have negligible size.) (b) Chapter 4 Energy By differentiating U (θ) find whether the system has an equilibrium position, and for what values of m and M equilibrium can occur. Discuss the stability of any equilibrium positions. Figure 4.27 Problem 4.36 (The subscript "o" is to emphasize that this is the period for small oscillations.) 4.35 ** Consider the Atwood machine of Figure 4.15, but suppose that the pulley has radius R and moment of inertia I. (a) Write down the total energy of the two masses and the pulley in terms of the coordinate x and I. (Remember that the kinetic energy of a spinning wheel is i ho2 .) (b) Show (what is true for any conservative one-dimensional system) that you can obtain the equation of motion for the coordinate x by differentiating the equation E = const. Check that the equation of motion is the same ...
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