Abstract
A method for evaluating finite trigonometric summations is applied to a system of N coupled oscillators under acceleration. Initial motion of the nth particle is shown to be of the order \(T^{2{n}+2}\) for small time T, and the end particle in the continuum limit is shown to initially remain stationary for the time it takes a wavefront to reach it. The average velocities of particles at the ends of the system are shown to take discrete values in a step-like manner.
Similar content being viewed by others
References
Calkin, M.G.: Motion of a falling spring. Am. J. Phys. 63, 261 (1993)
Unruh, W.G.: The falling slinky. arXiv:1110.4368v1 (2011)
Cross, R.C., Wheatland, M.S.: Modelling a falling slinky. Am. J. Phys. 80, 1051 (2012)
Sakaguchi, H.: Shockwaves in falling coupled harmonic oscillators. J. Phys. Soc. Jpn. 82(7), 073401 (2013)
da Fonseca, C.M., Kowalenko, V.: On a finite sum with powers of cosines. Appl. Anal. Discret. Math. 7, 354–377 (2013)
da Fonseca, C.M., Glasser, M.L., Kowalenko, V.: Basic trigonometric power sums with applications. Ramanujan J 42(2), 401–428 (2017)
Merca, M.: On some power sums of sine or cosine. Am. Math. Mon. 121(3), 244–248 (2014)
Olver, F.W.J., Daalhuis A.B. Olde, Lozier, D.W., Schneider, B.I. Boisvert, R.F., Clark, C.W., Miller, B. R., Saunders, B.V. (eds)., Digital Library of Mathematical Functions. http://dlmf.nist.gov/, Release 1.0.13 of (2016)