(a) consider the classical mechanical description of a one-dimensional harmonic os- cillator of mass 0.4kg, with spring constant k damping with b 1.3nsm. 1nm, and subject to linear (i) write down the equation of motion of the oscillator. (ii) assume that the oscillator is initially at r(t 0) (t 0) ims. solve the equation of motion for the oscillator. is the motion over- or under-damped? 0 and has velocity (b) consider the quantum mechanical description of a 1d harmonic oscillator with mass m and angular frequency w. at t 0 the wave-function is given by: (r, t 0) av'o (2) - where o and are the normalized ground and first excited states respectively (i) determine the real positive constant a (ii) give the wave-function as a function of time.

Aseries circuit consists of a resistor with r = 20 ? , a capacitor with c = 0.01 f, and a decaying battery with e = 300e? 5t. if the initial charge is 0, find the charge as a function of time t

Suppose that recording station n is located 200 km north of the epicenter of an earthquake, and recording station s is located 100 km south of the epicenter of the same earthquake. which of the following statements would be true?

Suppose water is leaking from a tank through a circular hole of area ah at its bottom. when water leaks through a hole, friction and contraction of the stream near the hole reduce the volume of water leaving the tank per second to cah 2gh , where c (0 < c < 1) is an empirical constant. a tank in the form of a right-circular cone standing on end, vertex down, is leaking water through a circular hole in its bottom. (assume the removed apex of the cone is of negligible height and volume.) (a) suppose the tank is 20 feet high and has radius 8 feet and the circular hole has radius 2 inches. the differential equation governing the height h in feet of water leaking from a tank after t seconds is dh dt = − 5 6h3/2 . if the height of the water is initially 8 feet, how long will it take the tank to empty? (round your answer to two decimal places.)