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In this study, the numerical solution of the ordinary kind of differential equation for a simple pendulum with large-angle of oscillation was introduced to obtain the time period. The analytical solution is obtained in terms of elliptic functions, and numerical solution of the problem was achieved by using two numerical quadrature methods, namely, Simpson?s 3/8 and Boole?s method. The period of a simple pendulum with large angle is presented; a comparison has been carried out between the analytical solution and the numerical integration results. In the case of error analysis, absolute and relative errors of the problem have been presented. A numerical algorithm has been developed by Matlab software 2013R and used for analyzing the result. It is established that the results of the comparison guaranty the ability and the accuracy of the present method.

The well-known effects of the spin-orbit interaction of light are manifestations of pair mutual influence of the three types of the angular momentum of light, namely, the spin angular momentum, the extrinsic orbital angular momentum and the intrinsic orbital angular momentum. Here we propose the convenient classification of the effects of the spin-orbit interaction of light and we observe one of the new effects in the frame of this classification, which is determined by the joint influence of two types of the angular momentum on the third type of the angular momentum, namely, the influence of the spin angular momentum and the extrinsic orbital angular momentum on the intrinsic orbital angular momentum. We experimentally studied the propagation of circularly polarized light through an optical fiber coiled into a helix. We have found that the spin angular momentum and the helix parameters affect the spatial structure of the radiation transmitted through the optical fiber. We found out that the structure of the light field rotates when changing the sign of circular polarization. The angle of rotation depends on the parameters of the helix. The results can be used to develop the general theory of spinning particles and can find application in metrology methods and nanooptics devices. Structured light beams carry three types of angular momentum [1][2][3][4]. The spin angular momentum is associated with polarization, the extrinsic orbital angular momentum is determined by the propagation path of the light beam, and the intrinsic orbital angular momentum is determined by the structure of the light field of the beam [3]. The effect of one of the angular momenta on another angular momentum leads to the spin-orbit interaction of light (a photon) [5,6]. There are six variants of such effects.The spin angular momentum affects the extrinsic orbital angular momentum, the effect can be observed as the longitudinal shift of the centroid of a linearly polarized light beam and the transverse shift of the centroid of a circularly polarized light beam at reflection and refraction and in focused light beams. These shifts are known as the Goos-Hanchen shift [7], the Imbert-Fedorov shift [8][9][10], the Hall effect for light [11], the optical Magnus effect [12] and the shift of the beam waist [13][14][15].The extrinsic orbital angular momentum affects the spin angular momentum, the effect manifests itself as the rotation of the linear polarization of light when changing * kundikovand@susu.ru the light propagation path [16][17][18][19]. The effect is known as the Rytov-Vladimirski-Berry-Chao-Wu-Tomita geometric polarization rotation. It can be observed in a single mode fiber, coiled into a helix [20], or in a multimode optical fiber [21].The intrinsic orbital angular momentum affects the extrinsic orbital angular momentum, the effect manifests itself as the shift of the centroid of a vortex light beam under reflection and refraction [3,[22][23][24].The extrinsic orbital angular momentum affects the intrinsic orbital angular momentum, t...