MATHCAD DIDACTIC TOOL FOR STUDYING THE ELLIPTICAL OSCILLATOR
MATHCAD DIDACTIC TOOL FOR STUDYING THE ELLIPTICAL OSCILLATOR

Summary/Abstract: This paper describes an interactive teaching tool developed with the aid of the Mathcad program for studying the elliptical oscillator. For the tool creation we explored the equations of motion of a point mass in a central force field with a spring potential energy. The motion of a point body attached to a spring can be mathematically modelled when to the body is given a velocity in a certain direction relative to the spring. We considered as input data the characteristics of the oscillator, the mass and the spring constant, the initial conditions and an arbitrary moment of time. The initial conditions were specified by the position vector represented in polar coordinates and the velocity vector characterized by the magnitude and orientation towards the direction of the vector radius. The position, velocity and acceleration of the oscillator were calculated at the time specified in the input data. The calculations were performed using both the Cartesian and the polar coordinates. The graph showing the elliptical trajectory of the point mass was placed next to the numerical results. For the characterization of the trajectory, the amplitudes and initial angles phases corresponding to the motion on the two coordinate axes were determined. By modifying the time moment in the input data we can track the motion of the point mass on the trajectory compared to the initial position. At the same time, the modification of the initial conditions allows for the observation of the feedback on the oscillator trajectory. Thus, the particular situations of orientation of the ellipse can be obtained with respect to the reference frame and the cases when the ellipse change into a straight line. The use of this tool in the classroom helps the students to extend the study of the oscillatory motion from the one-dimensional case to the two-dimensional one. Students can interactively observe the influence of initial conditions on the trajectory and analyze a number of particular cases. Furthermore, the students have the opportunity to understand more easily the role of the coordinate transformations, from Cartesian coordinates to polar coordinates, when tackling a problem related to the motion in a central force field. The students can be also encouraged to develop their own Mathcad tools for solving various physics problems.

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