Question

Three balls are constrained to move on the circumference of a circle of radius R. The balls are connected to each other along the circumference of the circle by three identical springs with spring constant k so that in the equilibrium configuration, the springs all have their common rest length, and the balls are spaced evenly around the circle. The mass of the first ball is M; the other two balls have mass m, m<M.a)  Choose as coordinates the angles a1, a2, and a3, the angular displacements of the three balls from the equilibrium configuration. Derive the Euler-Lagrange equations for the system in terms of these angles.b)  Using the result of part a, find the characteristic frequencies of the system.c)  What happens if m=M?

Answer 1

Answer circumference of the circle by three identical springs with spring constant k so that in the equilibrium configuration, the springs all have their common rest length, and the balls are spaced evenly around the circle. The mass of the first ball is M; the other two balls have mass m, m<M.a)  Choose as coordinates the angles a1, a2, and a3, the angular displacements of the three balls from the equilibrium configuration

Answer 2

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A double pendulum is undoubtedly an actual miracle of nature. The jump in complexity, which is observed at the transition from a simple pendulum to a double pendulum is amazing. The oscillations of a simple pendulum are regular. For small deviations from equilibrium, these oscillations are harmonic and can be described by sine or cosine function. In the case of nonlinear oscillations, the period depends on the amplitude, but the regularity of the motion holds. In other words, in the case of a simple pendulum, the approximation of small oscillations fully reflects the essential properties of the system.