Physics 321 -- Spring 2003
Homework #6, due Wednesday
Feb. 26
1. (5) Marion & Thornton, problem 3-20. Do this problem by hand (using algebra), not
using the computer. You need to find
the two angular frequencies on either side of the resonance (call them w1 and w2) where the velocity
amplitude is equal to the maximum velocity (on resonance) divided by Ö2.
The “full width” of the resonance is then defined as w2 - w1.
2. (5) Marion & Thornton, problem 3-24. Rather than put three plots on a single graph,
I suggest that you plot xp(t) and xc(t) together, but
then plot their sum x(t) separately.
(Otherwise it will be too hard to read.) Recall from the previous problem set that to put two plots on the
same graph using Mathematica, put the two functions inside curly brackets
separated by a comma, like this:
Plot[{x^2, Sin[x]},{x,0,4}]
One more note about using Mathematica, or any other
computer program you use to solve problems.
On the computer there are no units and all variables are dimensionless. Make sure you know what units you are using
for the time axis (usually seconds).
Set the time range on your plots so you can see at least a couple of
complete oscillations of the slower of the two frequencies, w and w1.
3. (5) Marion & Thornton, problem 3-28. Do this problem from scratch. Start by deriving the differential equation
for the charge on the capacitor using Kirchhoff’s laws, with the driving
voltage of the form V(t) = V0cos(wt). Now try a solution of the form q = q0cos(wt-d), and calculate both q0
and d.
After you have done all this, you will see that the amplitude of the
current qdot(t) is proportional to the driving frequency w when wRC << 1. (The product RC is the time constant of the
circuit.) Ignore the answer given in
the back of the book – we are interested in the amplitude of the
oscillating current, not the instantaneous current.
4. (5) Marion & Thornton, problem 3-29. When they say that the “oscillation
frequency” is 1 kHz, that means that w1=2p*103 s-1, where w1 is given in Eqs. (3.38) and
(3.40). From that you must calculate
the value of the capacitance, so you can convert the initial voltage on the
capacitor to its initial charge.
5. (5) Marion & Thornton, problem 3-32. You may use Mathematica to make the plots.