8 Home exercise sheet 8.1 Small oscillations

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Home exercise sheet
8.1
Small oscillations
Exercise 8.1: Tri-atomic molecule
A molecule consists of three identical atoms located at the vertices of a 45◦ right triangle.
Each pair of atoms interacts by an effective spring potential, with all spring constants equal
to k. Consider only planar motion of this molecule.
1. Find three ’zero modes’ for this system (i.e normal modes whose associated eigenfrequencies vanish).
2. Find the remaining three normal modes.
Exercise 8.2: Masses and springs
For the given system of masses and spring which can oscillate in 3-Dimensions do the
following:
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1. Find the eigen-vibrations of the system. All particles and springs are identical. The
tension in the springs at equilibrium is f = κl, where l is the equilibrium distance
between the particles.
2. Find the eigen-bivrations of the system of four identical particles of the system for the
case where the mass of particles 5 is put equal to zero.
Exercise 8.3: Zero Modes
Three distinct masses m1, m2, m3 move around a friction less hop of radius R. The
masses are connected to their neighbors by identical springs of force constant k.
Find the eigenmodes of small oscillations around the equilibrium point. Is there a zero mode
oscillations - i.e. a mode with ω 2 = 0? what does is correspond to?
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