Homework #2, Due date Feb 11th
Problem 1: In the lecture we solved the problem of adding vibrations of N oscillators as below
If we represent the sum of the series by the complex exponential form, we have
Show that
,which is the amplitude of the sum of the series.
Problem 2: Let’s consider a LCR series circuit as drawn below.
(1) An alternating voltage, amplitude V0 is applied across an LCR series circuit. Show that the
voltage at current resonance across either the inductance or the condenser is QV0.
(2) Show that the maximum potential across the capacitor occurs at a frequency
where
and
(3) show that the maximum potential across the inductance occurs at a frequency
Problem 3: Suppose light of wavelength 0.6 m (or 600 nm) is emitted by an electron in an atom
behaving as a lightly damped simple harmonic oscillator with a Q-value of 5*107.
(1) Find out the width of the spectral resonance line from such an atom.
(2) Find out the width of the power absorption resonance from such an atom
Problem 4: When the masses of the coupled pendulums m1 and m2 below are not equal, write
down the equations of motion in terms of coordinate x and y marked. Show that we may choose
the normal coordinates as
. Find out two normal mode frequencies. Also find out the effective mass in the Y mode.
m1
m2
x
y