Observation of Oscillation Superposition and Theoretical Description · IZhO 2024 #1
Physics · mechanics
Problem
Context: The oscillatory system is a torsion pendulum consisting of a metal screw-threaded rod suspended on two threads of equal length, tied to a horizontal rod fixed in a tripod leg. Two elongated screw nuts can be moved along the rod. The system exhibits longitudinal oscillations (rod and threads move in the same vertical plane) and torsional oscillations (rod rotates around a fixed vertical axis).
Place the threads at a distance of a=20 cm, their lengths should approximately be L=45 cm, and the screw nuts should be placed at the ends of the movable rod. Excite mixed oscillations of the rod and carefully observe the movement of one of its ends. The frequencies of longitudinal and torsional oscillations are close, so the shape of the trajectory of the rod end changes slowly and regularly, periodically returning to the initial one. Let the process of sequential restoration of the original shape of the trajectory be called a cycle, the corresponding time as the cycle time TC, and the number of complete longitudinal oscillations corresponding to it as NC.
Introduce a coordinate system on the horizontal plane: the x axis is directed along the rod in the position of its equilibrium, the y axis is perpendicular to it, and the origin is placed under the center of the rod in the position of its equilibrium. Denote the angle of deflection of the threads during longitudinal oscillations as α, the angle of rod rotation during torsional oscillations as β, the period of longitudinal oscillations as T0, the period of torsional oscillations as T1, the length of the threads as L, and the length of the rod as l.
1.1 Draw several successive characteristic shapes of the rod end trajectories in one cycle.
1.2 Write down the exact dependences of the coordinates of the rod end x(t) and y(t) on time t, and then obtain the corresponding approximate expressions for the conditions of this experiment.
1.3 Within the framework of the proposed model, obtain a formula for the cycle period TC, expressing it through the periods of longitudinal and torsional oscillations T0 and T1.
1.4 Write down the formula for the number of longitudinal oscillations in the cycle NC, expressing it in terms of periods T0 and T1.