Question 1: Simple Harmonic Oscillator A 0.5kg mass hangs from a spring with a force constant of 500. The mass is lifted 8em above its equilibrium position and allowed to oscillate. Write equations that model the vertical position, vertical velocity, and vertical acceleration of the mass as a function of time. Assume the mass/spring system is a simple harmonic oscillator. (15 points) W = 1/ K 500 = 31.6 = 0.08 m = A m 0 . 5 100 gm = Acos(wt+ ()) - 10.08cos( 31.6t ) O = - Awsin (wt+Q) - 08x 31.6 sin ( 31.6t) =-2.5 sin ( 31.6+) = - Aw-cos( wit) - - 0.08x131.6) cos ( 31.60) $-79.9 cos (31. 6t) What is the magnitude of the maximum velocity and maximum acceleration that the mass experiences while oscillating? What is the total mechanical energy of the mass while oscillating? (9 points) Vmax = Aw = 0.08 x 31.6 = (2.5) Amax AW = 0.08 (31,6) = (79.9 ) = _ mv 2 Z - mu ? + { k x 2 = SKA ,. E = 2500(0.08) = 11.6], tal = KE+ PE Consider the same vertical mass/spring system as described above, but with a damping coefficient of b = 0.0249. If the damped oscillator was given the same initial displacement of 8cm upwards, as the undamped oscillator, how long (in seconds) would it take for the amplitude of the damped oscillator to be equal 08 = 0_04 to half that of the undamped oscillator? 2 0. 04= 0.08e 2x5 cos 500 (0.02) 2 0.5 (2.51 2 1000 -0.000004