Physics 222 Assignment 4

Due February 16

Directions

(Aside for those you who like numerological oddities. Try the following commands in a terminal window:

date +%s
date -d @1234567890

Note that those numbers are Unix times - the time in seconds since January 1 1970.)

Please turn in this assignment (and all future assignments for this course) by emailing them to me . Please use the subject of "Physics 222 HW4" for these assignments (if you use the link above, the subject will be added automatically). Please attach the source code to any programs that you write for this assignment. If you write more than one program, attach each separately. When answering the questions themselves feel free to answer directly in the body of your message (or attach your answers). I may take off points for assignments that are not sent to me in the correct form.

Number Checker

Write a program that will use a separate function or subroutine to calculate all of the following quantities for a given integer:
1. Whether the number is prime or not. For the purpose of this assignment consider numbers to be prime if their absolute value is prime.
2. The reverse of the digits of the number — 1234 goes to 4321.
3. What the square root of the number is.
Feel free to reuse your prime number code, but beware that you will have to take care of cases such as 0, 1, and negative numbers.

Show your results when using the following numbers as input: -121, 53, 0, -13459, 457, 77779, -1, and 17.

Heat Conduction Through a Window

In this problem you will write a program to find the steady-state temperature profile of a window (or wall). The physical concept being explored here is heat conduction. The relatively simple equation that we will be dealing with for heat flow is:

 P/A = -k dT/dx

where P/A is the power per area (or energy per time per area) flowing through a surface, k is the conductivity and dT/dx is the temperature gradient in the direction the heat is flowing.

To solve this equation numerical we will convert the differential equation to a difference equation. The idea behind a difference equation is that derivatives are approximated as small (but non-infinitesimal) Δ's. So here we have

 ΔT = - ((P/A)/k) Δx.

In order to apply this difference equation in a program we have to convert the Δ's into differences between variables. In this case I am going to give you the temperature on the outside of the window. To find the temperature a small distance from the outside of the window we would say:

 T₁ = T₀ + (((P/A)/k)) Δx.

(Note that by starting at the outside and working are way into the house, we have flipped the sign on the heat flux term - if you prefer you can think of P/A as negative).

We can generalize this scheme to find the temperature at any place along the window based on the temperature at the previous location:

 T_i+1 = T_i + ((P/A)/k_i+1) Δx.

The last equation is the one that you will want to translate into code for this program. Your program should contain a loop which iterates through to find the temperature based on the temperature at the previous location. Your program should save the temperature at all locations using an array. Your program should output the temperature at the end of each layer of material and create a plot of your results with PLPlot. Note that you may want to use lines [ pls->line(npts, x, y); ] in the plots instead of points. Listed below are the conditions that you should test your program with. Note that you can either read these conditions in as inputs, or initialize them in the code directly. Use a Δx of 1.0x10^-6 m and an outside temperature of -20.0 degrees C for all runs.

P/A	k₁	width₁	k₂	width₂	k₃	width₃
W/m²	W/m/deg. C	m	W/m/deg. C	m	W/m/deg. C	m
0.0	0.84	0.006	0.018	0.003	0.84	0.006
400.0	0.84	0.006	0.018	0.003	0.84	0.006
400.0	0.84	0.006	0.84	0.003	0.84	0.006
400.0	0.84	0.006	0.024	0.003	0.84	0.006
400.0	0.84	0.006	0.018	0.006	0.84	0.006

The cases above correspond to several physical situations. The material with the highest conductivity is glass, the material with the middle conductivity is air, and the material with the lowest conductivity is argon gas. The first situation is a window with no heat source. In the second case there is a heat source and a window that has two panes of glass with a smaller argon gap between the glass layers. In the fourth case the argon gas is replaced with air. In the third case the argon is replaced by glass. In the last case a larger argon gap is used.

For this program, turn in the output of your program running for each of the cases. Comment on the results of each run of the program. Do the results seem reasonable?