Efficient conditions guaranteeing the existence and multiplicity of T-periodic solutions to the second order differential equation u '' = h(t)g(u) are established. Here, g : (A, B) -> (0, +infinity) is a positive function with two singularities, and h is an element of L(R/TZ) is a general sign-changing function. The obtained results have a form of relation between multiplicities of zeros of the weight function h and orders of singularities of the nonlinear term. Our results have applications in a physical model, where from the equation u '' = h(t)/sin(2)u one can study the existence and multiplicity of periodic motions of a charged particle in an oscillating magnetic field on the sphere. The approach is based on the classical properties of the Leray-Schauder degree.