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hwk4 - Phys 7268 Assignment 4 Due Problem 1 Identication of...

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Phys. 7268 Assignment 4 Due: 2/24/11 Problem 1 Identification of a Dimensionless Stress Parameter. To see how a dimensionless stress parameter like the Rayleigh number might be discovered from known dynamical equations, consider the evolution equation for a damped driven pendulum, m ¨ θ + α ˙ θ + C sin( θ ) = A sin( ω 0 t ) , (1) where m is the mass of the pendulum, θ ( t ) is the angle of the pendulum with respect to the vertical (so θ = 0 means the particle hangs down vertically underneath the fulcrum), α is the damping coefficient, A is the amplitude of the external driving, ω 0 is the frequency of the external driving, and overhead dots denotes differentiation with respect to time, e.g., ¨ θ = d 2 θ/dt 2 . One parameter in this equation can be eliminated immediately by dividing both sides of Eq. (1) by m , which redefines m = 1 and the other parameters to the values α/m , C/m , and A/m . A second parameter can be eliminated by defining a new dimensionless time coordinate ¯ t by the transformation: t = c t ¯ t, where c t is a new unit of time. (a) Write down Eq. (1) in the new variable ¯ t by substituting and using the chain rule of calculus. (b) Describe the three possible ways to choose the value of c t so as to eliminate a second coefficient in the new equation by setting the coefficient equal to one. Discuss also the physical meaning of the three different choices
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