Homework #7

Due at the end of Week 8 (5pm Friday).

1. On Day 17 we calculated the integral that describes x(t) when a harmonic oscillator is excited with a pulse of constant force. We found solutions for t < tau. Finish this calculation for the time period t > tau.

Once you have a final answer, check your answer to see if it is physically reasonable. Use simple concepts such as Hooke's Law and Newton's Second Law to make quantitative comparisons. Discuss any aspects of the solution that you checked.

Note for next year: I should rewrite the paragraph about checking the final answer - “Once you have a final answer, check to see if it is physically reasonable. Did you find harmonic motion at the resonant frequency? Does the time averaged position make sense? Do the solutions for t < tau and t > tau match each other for an arbitrary value of tau (check at least two easy cases)?

2. On Day 18 we used separation of variables to find a steady state temperature distribution as a function of x and y. In this problem we will investigate a different geometry and find a transient temperature distribution which is a function of x and t.

You live in a house with single-pane windows of thickness 4 mm. It is cold outside (zero Celsius) and warm inside (20 deg Celsius). Assume that the outside and inside surfaces are clamped at 0 and 20 deg Celsius respectively. There is a temperature gradient inside the glass. Let's call the direction of the gradient the x-direction.

(a) Just out of curiosity, how much heat energy (Watts/m^2) is being transmitted through the window?

(b) At t = 0 you open the window to get some fresh air. The window swings outward. Both surfaces of the window pane are now clamped at zero Celsius. The core of the window pane will remain hot for some time. The function Temp(x,t) can be expressed as a converging series. Find this converging series. Sketch the temperature distribution to show the time evolution.

3. On Day 20 we set up a calculation for the Green's function of the operator (∇² + κ²). Complete this calculation to show that the Green's function has the form of a spherical wave.