Using a Vortx to Teach Kepler’s Laws and Potential Energy

Abstract

For more than 10 years, I had taught Kepler’s laws and the law of universal gravity without a corresponding experiment. I sometimes remember how these topics were difficult for me in my high school days to understand. At that time, I thought that one of the reasons for my difficulty was that these phenomena cannot be visualized using physical objects. In this paper, I report on a method I developed by which students can actually watch some of the phenomena of Kepler’s laws and the law of universal gravity. In an earlier paper, theoretical and experimental results for a ball that rolls without slipping on a surface of revolution in cones and other funnels have been reported. Here I would like to introduce some experiments using a coin bank named Vortx®. The Vortx is a familiar and inexpensive apparatus (less than $50), and many Japanese physics teachers already have one. The Vortx, when used with the following experiments, can help students visualize a gravitational potential well, where the radius represents the distance to a massive body and the height from the funnel tip represents the potential energy per kg (for a description of a gravitational potential well, please refer to, for example, Ogborn). However, when we use the Vortx, we need to take care of some points as described in the following. One goal of this paper is to inform TPT readers how the Vortx can be used in the classroom.

Weighing the Sun with five photographs

“…For example, for photograph 1 on 26 March (day 83 and angle θ≈1.427 8), the error is | | q ~=  2 cos 0.48 E %. Therefore, our assumption (10) for revolution at a constant rate is justified, and in agreement with Kepler's second law for percentage errors of the order of 1%. This is the reason why our results are so accurate, even when, strictly speaking, we are violating Kepler's second law.…”

“…Therefore, we have proved the Kepler's first law. This law establishes that elliptical orbits are direct consequence of the Newton central gravitational force (that is inverse to the square of the distance) owe to the central star [7,8].…”

“…For Earth, during 2019, the perihelion occurred approximately during the night of January 2nd. The relation (10) between the angular position θ of the Earth's orbit and the date along the year is not correct in principle, as we will show in section 4. The Earth does not revolve at a constant rate during the complete orbit (that is Kepler's second law).…”

“…Compare this result with equation (10) for an orbital motion at constant rate. Taking the derivative of equation (10), we find…”