国际物理林匹克竞赛试题iphoSolutions.docVIP

Solutions to the problems of the 5-th International Physics Olympiad, 1971, Sofia, Bulgaria The problems and the solutions are adapted by Victor Ivanov Sofia State University, Faculty of Physics, 5 James Bourcier Blvd., 1164 Sofia, Bulgaria Reference: O. F. Kabardin, V. A. Orlov, in “International Physics Olympiads for High School Students”, eds. V. G. Razumovski, Moscow, Nauka, 1985. (In Russian). Theoretical problems Question 1. The blocks slide relative to the prism with accelerations a1 and a2, which are parallel to its sides and have the same magnitude a (see Fig. 1.1). The blocks move relative to the earth with accelerations: w1 = a1 + a0; (1.2) w2 = a2 + a0. Now we project w1 and w2 along the x- and y-axes: (1.3) ; (1.4) ; (1.5) ; (1.6) . Fig. 1.1 The equations of motion for the blocks and for the prism have the following vector forms (see Fig. 1.2): (1.7) ; (1.8) ; (1.9) . Fig. 1.2 The forces of tension T1 and T2 at the ends of the thread are of the same magnitude T since the masses of the thread and that of the pulley are negligible. Note that in equation (1.9) we account for the net force –(T1 + T2), which the bended thread exerts on the prism through the pulley. The equations of motion result in a system of six scalar equations when projected along x and y: (1.10) ; (1.11) ; (1.12) ; (1.13) ; (1.14) ; (1.15) . By adding up equations (1.10), (1.12), and (1.14) all forces internal to the system cancel each other. In this way we obtain the required relation between accelerations a and a0: (1.16) . The straightforward elimination of the unknown forces gives the final answer for a0: (1.17) . It follows from equation (1.17) that the prism will be in equilibrium (a0 = 0) if: (1.18) . Question 2. We will denote by H (H = const) the height of the tub