Coupled oscillations, point masses and spring

Problem:

Find the eigenfrequencies and describe the normal modes for a system of three equal masses m and four springs, all with spring constant k, with the system fixed at the ends as shown in the figure below.  The motion can only take place in one dimension, along the axes of the springs.

 

 

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Solution:

Problem:

Two particles of mass m and one particle of mass M are constrained to move on a line as shown.  They are connected by massless springs with spring constant k.

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Find the normal modes and eigenfrequencies of the system, keeping M/m finite.

Solution:

Problem:

Two masses a and b are on a horizontal surface.  Mass b has a spring connected to it and is at rest.  Mass a has an initial velocity v0 along the x-axis and strikes the spring of constant k, compressing it and thus starting mass b in motion along the x-axis.
(a)  Find the maximum force exerted.
(b)  Find the resulting motion of mass a while in contact with the spring.

Solution:

A different approach:

Solution:

Problem:

Use Lagrange's equations to find the normal modes and normal frequencies for linear vibrations of the CO2 molecule shown below.

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Solution:

Problem:

A particle of mass m is attached to a rigid support by a spring with a force constant k.  At equilibrium, the spring hangs vertically downward.  To this mass-spring combination is attached an identical oscillator, the spring of the latter being connected to the mass of the former.

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(a) Show that by appropriate choice of coordinates and their zero-points the equations of motion can be expressed  as
md2x1/dt2 + 2kx1 – kx2 = 0, 
md2x2/dt2 + kx2 – kx1 = 0.
(b)  Calculate the characteristic frequencies for one-dimensional vertical oscillations. 
(c)  Qualitatively describe the normal modes of the system with a short discussion plus drawings. 
(d)  Quantitatively compare the characteristic frequencies with the frequencies when either of the particles is held fixed while the other oscillates. 

Solution:

Problem:

Four mass points of mass m move on a circle of radius R.  Each mass point is coupled to its two neighboring points by a spring constant k.  
(a)  Find the Lagrangian of the system, and derive the equations of motion of the system.  
(b)  Calculate the eigen-frequencies of the system, and discuss the related eigen-vibrations.

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Solution:

Problem:

Consider six equal masses constrained to move on a circle of fixed radius and connected by identical springs of spring constant k.
(a)  Find the normal mode frequencies of the system for small displacement of the masses.
(b)  Find the (time dependent) displacement of the masses for each normal mode.  Give a physical description of the motion of the masses for normal modes with the highest and lowest frequencies.

Solution:

Problem:

Three point masses of mass m move on a circle of radius R.  The equilibrium positions are shown in the figure.  Each point mass is coupled to its two neighboring points
by a spring with spring constant k.
(a)  Write down the Lagrangian of the system.
(b)  Find the normal modes of the system.
(c)  If at t = 0 mass 1 is displaced from its equilibrium position
clockwise by 9o, what is the subsequent motion of all the masses?

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Solution:

Problem:

Three point masses, one of mass 2m and two of mass m are constrained to move on a circle of radius R.  Each mass point is coupled to its two neighboring points by a spring.  The springs coupling mass 1 and 3 and mass 1 and 2 have spring constant k, and the spring coupling mass 2 and mass 3 has spring constant 2k.  Find the normal modes of the system.


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Solution:

Problem:

A large number N (N = even) of point masses m are connected by identical springs of equilibrium length a and spring constant k.  Let qi (i = 0 to N - 1) denote the displacement of the ith mass from its equilibrium position.  Assume periodic boundary conditions, qi = qi+N.  (You can, for example imagine the masses arranged on a large circle of circumference Na.)
(a)  Write down the Lagrangian for the system of N point masses.
(b)  Find the equation of motion for the jth point mass.
(c)  Assume solutions of the form qj(t)  = |A|exp(iφj) exp(-iωt)  = |A|exp(i(φj - ωt)) exist, where ω is a normal mode frequency. 
Assume the phase of the amplitude depends on the position of the mass and write φj = p*ja.
What are the restrictions on p due to the boundary conditions?
(d)  Find the N normal mode frequencies ωn.  Make a sketch of ωn as a function of mode number n.

Solution: