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Unformatted text preview: Pre-Lab Required Homework Problem Preparation for Experiment 2: Part 1 X72 X6B X6D X6C The figure above represents a conical pendulum which is constructed with a spring rather than a solid shaft. We will consider the case where is constant and the mass is rotating in a horizontal circle of radius r . In the experiment the angle will be close to 90 o , but in this problem we will consider all possible values of . The spring is assumed to be massless and to exert a force given by k ( l- l ) where l and l are its actual and relaxed lengths respectively. The mathematics necessary to describe this situation is not very difficult, but the results have two unexpected subtleties. Before we tackle the math, note that there are two natural frequencies of oscillation in this mass-spring system. Consider the non-rotating system with the mass at rest hanging vertically. One can excite a small vertical oscillation where the restoring force is just due to the spring. Its frequency would be spring = p k/m . One can also excite a small pendulum oscillation, with no change in length of the spring, at a frequency pendulum = p g/l . [Note that l would be somewhat greater than l due to gravity pulling downward on the mass.] In order to make the physics as clear as possible, we will examine the system in two limits, first with no extension of the spring, then with spring extension but = 90 o . As we will see, there is a qualitative change in the dynamics of the system at each of these two characteristic frequencies. No spring extension This is in fact the common version of a conical pendulum treated in many textbooks. See example 5.21 in Young and Freedman, 12th ed. Here we treat l as a fixed quantity as would be the case if the spring were replaced by a massless rod. a) Use a free body diagram for the mass to get two equations, each involving the tension T in the spring. Divide one equation by the other to eliminate T and use r = l sin to eliminate r . Label the resulting equation as (1)....
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