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General Physics 2, Spring '11

Homework 11

First: here's how you make "Gaussian (i.e., bell-curve) distributed random numbers" in Excel. (I had parentheses in the wrong places in class, which involved sometimes asking Excel to take the logarithm of a negative number, which is undefined.) The correct formula is:

z = sqrt(-2*ln(a))*cos(2*pi*b)

where a and b are "basic" random numbers (i.e., uniformly distributed between zero and one) and z is the thing that's supposed to be Gaussian distributed.

1. Make a bunch of these Gaussian distributed random numbers, and make an appropriate histogram to show that, indeed, they are distributed in something like a nice bell-shaped curve.

2. Now use these to re-do the thing you did last week: assign (now Gaussian) random numbers to each of the three velocity components of a molecule; compute its speed on the basis of those numbers; now do this for a bunch of molecules. (a) What is the average speed? (b) What is the RMS (root mean square) speed (i.e., the square root of the average of the square of the speeds)? (c) Make a histogram of the speeds. (d) What is the most probable speed? (e) Do a curve-fit to show that Equation 10.23 (the "maxwell speed distribution" formula) matches the distribution of speeds in your sample. (Note here that you'll have to choose values for the constants that appear in the formula appropriately; looking at Equation 20.21 and thinking about how it relates to your values for v_x might help.)

3. Now reflect on what you did in 2. How in the world can I be so sure that (if you do things right!) your speed histogram will match the formula in the book? After all, you created this histogram by assigning velocities at random! So shouldn't the speed histogram that emerges also be in some sense random? What's going on? How can something so random be so un-random, so predictable and well-defined? How does this relate to the ideas we talked about in class (with the poker hands example and the air molecules in the room example)?

4. QTD (not Project) 5: "Consider the gas, initially confined..."

Homework 11

First: here's how you make "Gaussian (i.e., bell-curve) distributed random numbers" in Excel. (I had parentheses in the wrong places in class, which involved sometimes asking Excel to take the logarithm of a negative number, which is undefined.) The correct formula is:

z = sqrt(-2*ln(a))*cos(2*pi*b)

where a and b are "basic" random numbers (i.e., uniformly distributed between zero and one) and z is the thing that's supposed to be Gaussian distributed.

1. Make a bunch of these Gaussian distributed random numbers, and make an appropriate histogram to show that, indeed, they are distributed in something like a nice bell-shaped curve.

2. Now use these to re-do the thing you did last week: assign (now Gaussian) random numbers to each of the three velocity components of a molecule; compute its speed on the basis of those numbers; now do this for a bunch of molecules. (a) What is the average speed? (b) What is the RMS (root mean square) speed (i.e., the square root of the average of the square of the speeds)? (c) Make a histogram of the speeds. (d) What is the most probable speed? (e) Do a curve-fit to show that Equation 10.23 (the "maxwell speed distribution" formula) matches the distribution of speeds in your sample. (Note here that you'll have to choose values for the constants that appear in the formula appropriately; looking at Equation 20.21 and thinking about how it relates to your values for v_x might help.)

3. Now reflect on what you did in 2. How in the world can I be so sure that (if you do things right!) your speed histogram will match the formula in the book? After all, you created this histogram by assigning velocities at random! So shouldn't the speed histogram that emerges also be in some sense random? What's going on? How can something so random be so un-random, so predictable and well-defined? How does this relate to the ideas we talked about in class (with the poker hands example and the air molecules in the room example)?

4. QTD (not Project) 5: "Consider the gas, initially confined..."

Last modified: Monday, December 19, 2011, 9:18 AM