# a3 - Phys 263/Amath 261 Assignment 3 Due Wednesday June 4...

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Phys 263/Amath 261 Assignment 3 Due: Wednesday, June 4, 2008 (in class) 1. A simple harmonic oscillator with mass m = 0 . 5 and k = 2 is initially at the point x = 3 when it is projected towards the origin with speed 2. (a) Show that in the subsequent motion x = 3 cos(2 t ) - sin(2 t ) . (1) (b) Deduce the amplitude of the oscillations. (c) How long does it take for the particle to first reach the origin? 2. For the simple harmonic oscillator (no damping), the time average of a quantity f ( t ) is defined by f = 1 t 2 - t 1 t 2 t 1 f ( t ) dt (2) (a) Calculate the time average of the potential and kinetic energies, U time and K time , in terms of k and A , over one complete time period T 0 . (b) Defining the spatial average in a similar way, calculate U space and K space over one complete time period. (c) Prove that E space = E time 3. Recall the overdamped harmonic oscillator with equation of motion: x ( t ) = e - γt A 1 e ω 2 t + A 2 e - ω 2 t (3) with initial position and speed x 0 and v 0 , respectively.

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