Coupled oscillations, other systems

Problem:

Two pendula, each of which consists of a weightless rigid rod length of L and a mass m, are connected at their midpoints by a spring with spring constant k.  Consider only small displacements from equilibrium.

image (a)  What are the frequencies of the normal modes of this system?  Briefly describe these modes.
(b)  At t = 0 the right pendulum is displaced by an angle θ to the right while the left pendulum remains vertical.  Both pendula are released from rest.  Describe the subsequent motion. 

Solution:

Problem:

A rigid uniform bar of mass M and length L is supported in equilibrium in a horizontal position by two massless springs attached at each end.  The identical springs have spring constant k.  The motion of the center of mass is constrained to move parallel to the vertical x-axis.  Furthermore, the motion of the bar is constrained to lie in the xz plane.  Let x1 and x2 be the departures of the two ends from their equilibrium positions, as shown.

 image

(a)  Show that the moment of inertia for a bar about the y-axis through its center of mass is ML2/12.
(b)  Construct the Lagrangian for the bar-spring arrangement assuming only small deviations from equilibrium.
(c)  Calculate the vibrational frequencies of the normal modes for small amplitude oscillations.
(d)  Describe the normal modes of oscillations.

Solution:

Problem:

A uniform horizontal rectangular plate (mass M, length L, width W) rests with its corners on four similar vertical springs with spring constant k.  Find the normal modes of vibration and prove that their frequencies are in the ratio 1 : √3 : √3.
Solution:

Problem:

Phonons are quantized lattice vibrations, and many aspects of these excitations can be understood in terms of simple mode counting.
(a)  Estimate the number of phonon modes in 1 cm3 of a crystalline material with an inter-atomic spacing of 2 Angstrom.
(b)  Assuming that in thermal equilibrium each phonon mode has kBT of energy, give a numerical estimate of the heat capacity ΔE/ΔT of this 1 cm3 of material, in [J/K].

Solution:

Problem:

A naïve model of a solid is that of a bunch of balls (atoms) connected by springs (bound by inter-atomic potentials which can be approximated by harmonic potentials near equilibrium).  If each inter-atomic spring has spring constant k, you can relate this microscopic value to the macroscopically measurable value of Young's modulus, Y.  Young's modulus is the ratio of stress (F/A, or applied force, F, perpendicular to the cross-sectional area, A, of a bar of material per unit cross-sectional area), to strain (ΔL/L, or the fractional change in length of the bar of material); thus, Y = (F/A)/(ΔL/L).  Evaluate k for the inter-atomic springs of aluminum, which has a Young's modulus of 70 GPa.  Assume that the aluminum atoms are arranged in a simple cubic lattice (they are really face-centered cubic); you can determine the inter-atomic spacing by knowing that the density of aluminum is 2.70 g/cm3 and that a mole of aluminum has a mass of 27 g.  Express your result for k in SI units.

 Solution:

Problem:

A pendulum consisting of a mass m and a weightless string of length l is mounted on a mass M, which in turn slides on a support without friction and is attached to a horizontal spring with force constant k, as seen in the diagram.  There is a slot in the support in order that the pendulum may swing freely.
(a)  Set up Lagrange's equations.
(b)  Find the normal mode frequencies for small oscillations. What are those frequencies to zeroth order in m/M, when  m << M?

coupled oscillator

Solution: