Harmonic oscillations

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Exercises on harmonic oscillations

The torsion pendulum

The torsion pendulum in the following figure has a cylinder with the moment of inertia $J = 3,2 kg \cdot m^{2}$ on a steel wire of length $l = 5 m$ hung up. The benchmark is ${D.}_{r} = 0,81 kg m^{2} s^{-2}$ . For small twists, there is the dependency for the returning torque $M.$ :

M. = - {D.}_{r} \cdot ϕ

Initial deflection $ϕ (0)$ of the torsion pendulum is $1 wheel$ . Let it be assumed that it vibrates without friction.

Establish the equation of motion for the torsion pendulum and solve the equation of motion.
How long is the period of the oscillation?
What are the formulas for potential and kinetic energy?

Lennard Jones Potential

Goal: Approach of potentials through a harmonic potential for small disturbances / deflections.

The Lennard-Jones potential is made up of a repulsive potential component of the form $\frac{B.}{{R.}^{12}}$ , which stems from the Pauli ban, and an attractive potential share of the form $- \frac{B.}{{R.}^{6}}$ , which comes from the van der Waals interaction (also London interaction or dipole-dipole interaction).

U (R.) = 4 ε [{(\frac{σ}{R.})}^{12} - {(\frac{σ}{R.})}^{6}]

Approach the potential $U (R.)$ with $ε = 14 \cdot 10^{- 23} J$ and $σ = 2,56 \cdot 10^{-10} m$ around the minimum by a harmonic potential (these values correspond to those for liquid helium at $0 K$ and $0 Pa$ ).

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