zhiwei chong
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Pendulums: the Simple and the Physical
2023, arXiv (Cornell University)
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Abstract
This paper aims to show how to guide students with a familiar example to extract as much physics as possible before jumping into mathematical calculation. The period for a physical pendulum made up of a uniform rod is changed by attaching a piece of putty on it. The period for the combined system depends on the location of the putty. Simple reasoning without calculation shows that there are two locations for the putty that do not change the period of the physical pendulum: the axis and the center of percussion. Moreover, without calculation, we reason that there is at least one minimal period when the putty lies somewhere between these two locations. The underlying physics is that the period depends on the ratio between two quantities: the torque and the momentum of inertia, which depends on the location of the putty. They conflict with each other; a smaller (larger) moment of inertia is accompanied by a smaller (larger) torque. The uniqueness of the minimal period and the corresponding location for the putty can only be obtained by calculation. Zee who also emphasizes thinking by quoting John Wheeler in the preface to his book [1]: Never never calculate unless you already know the answer! Another input is the qualitative analysis of simple harmonic motion, where the sinusoidal shape of the velocity versus time curve is not obtained by calculation but based purely on physical reasoning instead . The content of this article was tested in my Year-10 class. They enjoyed this style of teaching, so did I. The journey starts from the following two basic facts. The period for a simple pendulum with length ℓ and the period for a physical pendulum made up of a uniform rod with length 𝐿 are
References (7)
- Anthony Zee, Fly by Night Physics: How Physicists Use the Backs of Envelopes, (Princeton University Press, Princeton, NJ, 2020).
- Zhiwei Chong, "A qualitative analysis to simple harmonic motion," arXiv preprint arXiv:2209.12662 (2022). ( Accepted by The Physics Teacher)
- The subscript in 𝑇 𝑝 stands for physical, while that in 𝑥 𝑝 actually stands for the center of percussion.
- Hugh G. Gauch, Scientific Method in Practice, (Cambridge University Press, Cambridge, 2003) p. 269.
- P. N. Raychowdhury, and J. N. Boyd, "Center of percussion," Am. J. Phys. 47(12):1088- 1089 (1979).
- J. Hass, C. Heil, and M. D. Weir, Thomas' Calculus, 14th ed. (Pearson Education, Boston, 2018), p. 191.
- S. Shan, J. A. Shore, and K. Spekken, "The falling rod race," Phys. Teach. 58(8):596-598 (2020).