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homework11 - UNIVERSITY OF COLORADO AT BOULDER PHYS 2210...

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PHYS 2210 UNIVERSITY OF COLORADO AT BOULDER CLASSICAL MECHANICS AND MATH METHODS, SPRING, 2011 Homework 11 (Due Date: Start of class on Thurs. March 31 ) 1. Consider a simple pendulum of length L = 10 m . (a) In an ideal world (assuming no damping, and making the small angle approximation) determine the period of oscillation. Now imagine we take into account air friction and determine that it causes the period to change by 0 . 1%. Using Taylor’s notation introduced in Eq. 5.28, what is the damping factor β ? By what factor will the amplitude of oscillation decrease after 10 cycles? (b) Which e ff ect of damping would be more noticeable - the change of the period or the decrease of the amplitude? Explain. 2. Imagine two concentric cylinders, centered on the vertical z axis, with radii R ± , where is very small. A small frictionless object of radius 2 is inserted between the two cylinders, so that it can be considered a point mass that can move freely at a fixed distance from the vertical axis. At time t = 0 the puck is released at height h with a purely angular initial velocity ω 0 . Figure 1: (a) Write down Newton’s second law first in terms of cartesian coordinates and then in terms of cylindrical polar coordinates. Which form would be easiest to use to solve for the motion of the ball described above? Why? ( If you need a reminder about what cylindrical polar coordinates are, problem 1.47 on p.
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