Bill Goodwine's Courses

Reading: Chapter 6, sections 6.1-6.4.

Exercises: 6.2 (A9 only), 6.4 (A3, A5 and A6 only).

Project Problem: This problem starts the theoretical work you need to do for the pendulum project. Determine the equation of motion for a pendulum of length l with a mass attached to the end with mass m as illustrated in the following figure where tau is an applied torque. Is the equation linear or nonlinear? If the angle is small so that you can assume sin(t) is approximately t, if you replace any sin(theta) by theta and any cos(theta) by 1, is the resulting equation linear or nonlinear? Using the same assumption (small angle) what is the equation if the angle is zero when the pendulum is hanging straight down instead of up?

For problem 6.5 it says there will be complex eigenvalues, by this do you mean there will be imaginary components or just they will be more difficult? for matrix A5 I found imaginary eigenvalues but for both A6(as well as in A5) I only find real eigenvalues however they are repeated without following theorem 6.1 or 6.2 making them harder to deal with. Thanks

Jessie wrote:For problem 6.5 it says there will be complex eigenvalues, by this do you mean there will be imaginary components or just they will be more difficult? for matrix A5 I found imaginary eigenvalues but for both A6(as well as in A5) I only find real eigenvalues however they are repeated without following theorem 6.1 or 6.2 making them harder to deal with. Thanks

I just sent an email to the whole class about this. A5 and A6 should have been from 6.4, which I have now edited the homework to say.

Save your work for those two matrices because you will be able to use it for next week.

goodwine wrote:

Jessie wrote:For problem 6.5 it says there will be complex eigenvalues, by this do you mean there will be imaginary components or just they will be more difficult? for matrix A5 I found imaginary eigenvalues but for both A6(as well as in A5) I only find real eigenvalues however they are repeated without following theorem 6.1 or 6.2 making them harder to deal with. Thanks

I just sent an email to the whole class about this. A5 and A6 should have been from 6.4, which I have now edited the homework to say.

Save your work for those two matrices because you will be able to use it for next week.

Thanks!