Note! There are lecture notes online now for my lecture on resonance!

1) If z = x + iy, then speed is .

2) Remember, parallel "resistors" satisfy

(Don't forget that this formula is giving you the inverse of the effective resistance!)

a) If "the angle of Z is 45 degrees", that means in polar form that z must be , in other words the real and imaginary parts of z must be equal!! (So, in standard Cartesian form, where z=a+ib, we have a=b!). Remember also

c) When R=C=L=1, and omega =2, I get z = 4/13 - 6i/13.

4) The definition of average is given in Boas 7.4, equation 4.3

Also, you should always remember .

5a) There are three forces in total, external, friction( =-cv=-c dx/dt), and spring force ( = -kx.) Newton says the sum of these forces = ma = .

If we write , etc, and then substitute these into Newton's equation above, and then just "forget" about the "Re" in front of everybody, then there should be an in front of each term, and hence cancels out. What is left behind now should be an algebra equation which can be solved for .

c) The answer for is given in the statement of #6, so you can check your result.

When , you can replace for wherever it appears, except when you have terms that look like , which you must leave alone (otherwise you get 0!) So, e.g.

Once you've done this, what must be for to drop from its maximum by a factor of 2? This tells you (half) the full width at half maximum. You expect it to be (maybe times some simple constant?) Can you see this from your equations?

(I have worked much of this problem out in my notes, so that may help too!)