Download PHYS4330 Theoretical Mechanics HW #3 Due 20 Sept 2011

PHYS4330 Theoretical Mechanics
HW #3
Due 20 Sept 2011
(1) The solution to the equation of motion for the one dimensional simple harmonic oscillator
can be written in complex notation as x(t) = Beiωt + Ce−iωt . Explain why, for a physical
mass-and-spring oscillator with k = mω 2 , we need to enforce that C = B ∗ , and show that
this gives the appropriate number of free parameters for a second order differential equation.
Write B = Ae−iδ for real parameters A and δ, and then derive x(t), ẋ(t), the kinetic energy
mẋ2 /2, the potential energy kx2 /2, and the total energy in terms of A, δ, m, and ω. Also
find the average kinetic and potential energies.
(2) In class, we separately derived the two solutions (each) for the under damped (β < ω0 )
and critically damped (β = ω0 ) oscillator. Here, derive the two critically damped solutions
by taking the limit as β → ω0 of the under damped solutions.
(3) This problem is not about “mechanics”, but uses the same mathematics we’ve addressed
in class. A resistor R, capacitor C, and inductor L are connected in series. An AC voltage
source V (t) = V0 eiωt is put across these elements. Find the amplitude of the voltage drop
VR0 (ω) across the resistor, after transients have died away, and find the resonant frequency.
(Assume the system is weakly damped.) For R = 400Ω, C = 10pF=10µµF, and L = 2mH,
make a plot of VR0 (ω)/V0 and also the phase δ(ω) versus ω, covering the resonance. [Recall
˙
that for a current I = q̇ through a circuit element, VR = IR, VC = q/C, and VL = LI.]
(4) A damped oscillator starts from rest at the origin at t = 0, at which time an oscillating
driving force F (t) = F0 cos ωt is applied. The oscillator has a “natural” frequency ω0 (that
is, the frequency it would have if there were no damping) and a damping parameter β. Find
the complete solution, including the initial conditions. In the following, set F0 /m = 1 and
ω0 = 2π and plot the position x(t) as a function of time, starting at t = 0 up to a large
enough time so that the transients have died away. For ω = ω0 , make plots for (a) β = 0.1ω0
and (b) β = 0.9ω0 . Repeat for ω = 2ω0 . Comment on your results.
(5) (See Taylor Problem 5.53.) An oscillator is driven by the following periodic force:
(a) FInd the long-term motion x(t) assuming a natural period τ0 = 2, damping coefficient
β = 0.1, and drive strength fmax = 1. Find the Fourier coefficients in the Fourier series for
x(t) and plot the sum of the first four terms in the series for 0 ≤ t ≤ 6. (b) Repeat, except
with natural period τ0 = 3. Why is the result so different? What about τ0 = 4?
You can work the integrals and use whatever computer program you’d like to make the plots.
However, you are also welcome to use a symbolic manipulation program like Mathematica
or Maple to do the work. If you use Mathematica, you would probably want to make
use of the built-in functions TriangleWave and FourierCosSeries.