Homework 4, due October 3, 2007.

[goodwine]

Reading assignment: Chapters 4 and 16.

Problems: 4.1, 4.8, 4.9 and 16.1.

Also, return to problem 4.1 and plot the particular solution and forcing function on the same graph. Change the scale of one of them so they have approximately the same magnitude. Does the relationship between the applied force and particular solution depend on whether or not the forcing frequency is greater or less than the natural frequency of the system? If so, offer an explanation of your observation.

Also, return to problem 4.1 again and let m=1, k=4, F=1 and the initial conditions be zero. In one case let omega=1.98 and in the other let omega=2.0. Write a computer program to determine an approximate numerical solution for the differential equation. Plot x(t) versus t for each case. Explain the relationship between the two solutions.

[mbrundag]

In problem 4.1 is the problem supposed to contain just omega n or omega n ^2. I wasn't sure since we defined omega n as radical k/m and in all the examples in class we used omega n ^2.

[goodwine]

In problem 4.1 is the problem supposed to contain just omega n or omega n ^2. I wasn't sure since we defined omega n as radical k/m and in all the examples in class we used omega n ^2.

It should be omega_n^2. If you work it out with just omega_n, then you'll just get square roots throughout the sine and cosines.

[wdonalds]

Problem 4.8 in the book is different from problem 4.8 in the most recent online version of the book. Which problem would you like us to do?

[goodwine]

Problem 4.8 in the book is different from problem 4.8 in the most recent online version of the book. Which problem would you like us to do?

The numbers always refer to the printed version for the class.

[goodwine]

I have some questions about 4.8 on the homework for this week. I asked you yesterday if the equation of motion is a solution or just the differential equation and you stated it was the latter, so I wanted to double check that. Secondly, I don't want you to give me the answer, but my thinking is that in part one, the kx term goes away because at the instance being considered (x = 0), there is no spring force, but the problem with that is there would still be a force at other positions. For the second part, I am thinking that the kx and mg terms are gone because they cancel. Am I on the right track or completely off target here? If it would be easier to explain in person, could I come by your office some time today? I know that I missed office hours last night, but I hadn't seen this problem coming until after they were over.

The "answer" is just a differential equation. You don't have to solve it. The point of the problem is that if there is gravity and you measure from the unstretched position of the spring, then a gravity term appears in the equation. If you measure x from the equilibrium position, which is the amount it would be deflected by the weight of the mass, then gravity disappears from the differential equation, which is handy because you then don't have to deal with it.

[colinheye]

The question says to plot forcing function and particular solution and adjust one to fit the other. Since the only difference is a factor of 1/(w_n^2 - w^2), the two plots can be adjusted to have approximately the same magnitude, but we do not have any numeric values to plug in and plot. What exactly should we use for the unknown values?

[asaville]

Would you possibly mind clarifying what you would like us to do for plots relating to 4.1?

Is the forcing function the inhomogenous term of the given problem ((F/m)sin(wt)) and the particular solution that which is found using undetermined coefficients? If so, how are we supposed to plot these? Would you like us to pick arbitrary values for F and m and w (and w_n which appears in the particular solution)? Because if not we have four unknown variables and I'm not sure how to graph that. Or are we to assume to values below for these plots as well (F = 1, m =1, etc.)?

Then, for the other part, by "other omega" do you mean w_n so that F = 1, m = 1, w = 1.98, and w_n = 2.0 and then plug these values into the solution x(t) where w does not equal w_n? Or should we use the solution x(t) where w does equal w_n and then just plot two separate graphs, one for w = 1.98 and the other w = 2.0?

I apologize if this is confusing but any further clarification on plotting 4.1 would be very helpful. Thanks.

[goodwine]

Would you possibly mind clarifying what you would like us to do for plots relating to 4.1?

Is the forcing function the inhomogenous term of the given problem ((F/m)sin(wt)) and the particular solution that which is found using undetermined coefficients?

Yes.

If so, how are we supposed to plot these? Would you like us to pick arbitrary values for F and m and w (and w_n which appears in the particular solution)? Because if not we have four unknown variables and I'm not sure how to graph that. Or are we to assume to values below for these plots as well (F = 1, m =1, etc.)?

If you want to, you may pick values for the parameters. You should also be able to plot them qualitatively for arbitrary values for the parameters (as long as m, b, k and F are positive). The point to be observed is whether or not they are in or out of phase. The latter case should be somewhat counter-intuitive.

Then, for the other part, by "other omega" do you mean w_n so that F = 1, m = 1, w = 1.98, and w_n = 2.0 and then plug these values into the solution x(t) where w does not equal w_n? Or should we use the solution x(t) where w does equal w_n and then just plot two separate graphs, one for w = 1.98 and the other w = 2.0?

The two cases are for two different frequencies in the forcing function, i.e., case 1: sin(1.98 t) and case 2: sin(2 t). m, k and F don't change in either case.