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Unformatted text preview: Problem Set 4: Problem 2.
Problem: A collar of mass m is constrained to travel in the horizontal plane along a frictionless ring of radius R. A nonlinear spring is attached to the collar as shown. The spring force is Fs = -k(x - xo ) - (x - xo )3 where k and are spring constants and xo is the spring's unstretched length. We wish to conduct an experiment to infer the value of . (a) What is the work done as the collar moves from Point A to Point B? (b) Now, assuming xo = 1 R, compute the speed of the collar, V , when it reaches Point B, assuming 2 the collar is at rest at Point A. Express your answer in terms of R, k, m and R2 /k. (c) If V is twice the value it would have with a linear spring with spring constant k, what is ? Express your answer in terms of k and R2 . Solution: To solve, we first determine the work done by the nonlinear spring force for general displacement from a point x1 to x2 . Then, we use the Principle of Work and Energy for the collar's motion along the horizontal ring as it moves from Point A to Point B. This will establish its speed when it reaches Point B. Then, we use the result to determine the nonlinear spring constant, . (a) The work done by the nonlinear spring force as the collar moves from arbitrary Point x1 to another arbitrary Point x2 is
x2 U1-2 = - = - x1 k(x - xo ) + (x - xo )3 dx
x=x2 x=x1 1 1 k(x - xo )2 + (x - xo )4 2 4 1 1 1 1 = - k(x2 - xo )2 - k(x1 - xo )2 + (x2 - xo )4 - (x2 - xo )4 2 2 4 4 (b) For the problem at hand, we have x2 = 1 R and x1 = 5 R. Thus, with xo = 1 R, 2 2 2 U1-2 1 = - k 2 1 1 R- R 2 2
2 2 4 4 - 1 5 R- R 2 2 1 - 4 1 1 R- R 2 2 - 1 5 R- R 2 2 1 1 = - k 0 - 4R2 - 0 - 16R4 = 2kR2 + 4R4 2 4 ...
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- Spring '06