Homework 3 - Homework Set 3 Due Wednesday 1 Let x1(t and...

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Homework Set 3 Due: Wednesday, September 26, 2012 1. Let x 1 ( t ) and x 2 ( t ) be solutions to ¨ x 2 = bx . Show that x 1 ( t ) + x 2 ( t ) is not a solution to this equation. 2. A mass on the end of a spring (with natural frequency ω ) is released from rest at position x 0 . The experiment is repeated, but now with the system immersed in a fluid that causes the motion to be overdamped (with damping coefficient β ). Find the ratio of the maximum speed in the former case to that in the latter. What is the ratio in the limit of strong damping ( β ω 0 )? In the limit of critical damping? 3. For small oscillations, the period of a pendulum is approximately T 2 π radicalbig ℓ/g , independent of the amplitude, θ 0 . For finite oscillations, show that the exact expression for T is T = radicalBigg 8 g integraldisplay θ 0 0 cos θ cos θ 0 . (1) 4. Let the initial position and speed of an overdamped, non-driven oscillator be x 0 and v 0 , respectively. Show that if we can write this solution as x ( t ) = e - βt bracketleftbig A 1 e ω 2 t + A 2 e - ω
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