This problem concerns a collection of N identical harmonic oscillators (perhaps an Einstein solid or the internal vibrations of gas molecules) at temperature T. The allowed energies of each oscillator are 0, hf, 2hf, and so on. We will use the infinite series 1 + x + x 2 + x 3 + · · · = 1 1 − x , 0 < x < 1 (a) Evaluate the partition function for a single harmonic oscillator. (b) Find an expression for the average energy of a single oscillator at temperature T. (c) What is the total energy of the system of N oscillators at temperature T? (d) Compute the heat capacity C = dU dT , and find the limit of C at T → 0: prove that it satisfies the third law of thermodynamics

Astone dropped into a well is heard to strike the water after 8seconds. find the depth of the well if the velocity of sound is 350m/s

Any color picture tube and electric field exerts a net force of magnitude 1.68x 10^-13 n on an electron the rest mass of an electron is 9.11 x 10^-13 what is the electron acceleration?

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.)