August 2025
Volume 93, Issue No. 8
Measuring optical force with a torsion pendulum: A platform for independent student experimentation
In this work, the force due to radiation pressure is measured with sub-10 pN sensitivity, corresponding to less than 2 mW of optical power. The apparatus adds homemade reflectors to a commercial Cavendish balance, which consists of a torsion pendulum with a built-in capacitance position sensor. When driven by four 5 mW laser diodes, with square wave modulation at the pendulum's natural frequency, the response is strong enough to easily discern in a short time series. The discrete Fourier transform of a longer dataset provides a more in-depth analysis, clearly showing the multiple frequency components from the square wave driving force. The driving power was controlled by adjusting the square wave duty cycle, allowing easy automation and avoiding additional optics or filters. For a 9-h dataset, white noise corresponding to about 2 pN was observed, enabling our most sensitive measurements. The pendulum operates in air. To minimize convective forces from differential heating and the resulting differential pressure, we use symmetrical reflectors encased in low-thermal conductivity material, namely, two glass-fronted mirrors attached back-to-back. This experiment could be used in a single lab session, allowing the optical force exerted by a laser pointer to be quickly and intuitively observed. It also demonstrates the power of Fourier analysis, builds student intuition about oscillator systems, and provides a compelling platform for student-driven projects.
EDITORIAL
In this issue: August 2025 by Harvey Gould; Raina Olsen; Beth Parks; B. Cameron Reed; Donald Salisbury; Jan Tobochnik; Keith Zengel. DOI: 10.1119/5.0287013
LETTERS TO THE EDITOR
A new Hookean center of mass theorem by J. West. DOI: 10.1119/5.0278672
Statistical interpretation of entropy by Kirk T. McDonald. DOI: 10.1119/5.0283836
Extending the discussion of entropy by Bruce D. Popp. DOI: 10.1119/5.0287588
PAPERS
Synchronized chaos in a system of two classical kicked rotors by Niklas Endler; Jesse Mullis; R. Benjamin Davis. DOI: 10.1119/5.0259010
Abstract: Synchronized chaos is demonstrated in a system of two classical kicked rotors where one rotor is driven by the momentum of the other. Despite sharing a common momentum, a synchronized response of the rotors is not inevitable. The prevalence of synchronized chaos is studied in changes in initial conditions and kick strength. While both synchronized quasi-periodic and chaotic motions are possible, both types of motion may also be asynchronous. In some rare cases, the driving rotor can exhibit quasi-periodic motion while the driven rotor exhibits chaos. System equations are solved using an iterative map, and chaotic motion is distinguished from quasi-periodic motion using a heuristic interpretation of the frequency response. The relative simplicity of the analysis enables an accessible introduction to the intriguing phenomenon of chaotic synchronization.
The upward-driven disk, a steadily forced chaotic pendulum Video Abstracts by Leo Maas. DOI: 10.1119/5.0147893
Editor's Note: A simple plastic disk, wedged between two wheels and driven upward by a small electric motor, demonstrates remarkably complex behavior. By varying the initial conditions, one can observe steady rotation, periodic motion, or chaotic motion. This paper introduces a simple model for the behavior that also points out how it can mimic the solutions of the Lorenz equations, which are frequently used to describe fluid convection. There are still plenty of open questions, especially regarding the best model for frictional damping, and the easy and inexpensive construction could make this the starting point for many student projects.
Visual relativistic mechanics by Karol Urbański. DOI: 10.1119/5.0233466
Editor's Note: The introduction to hyperbolic geometric depictions of energy-momentum vectors in special relativity offers amazing shortcuts to transformation relations that can be employed in introductory courses. One result that this approach makes simple to obtain is the acceleration of a rocket due to ejection of exhaust. The second, even more profound, is the analysis in different frames of reference of the relativistic Doppler effect, under the condition that there is an initial frame in which light is emitted with equal intensity in opposite directions.
Shortest derivation of time-independent perturbation theory by B. Blankleider; A. N. Kvinikhidze. DOI: 10.1119/5.0152132
Editor's Note: Perturbation theory is a mainstay of quantum mechanics. Developing the theory usually involves beginning with a complete set of eigenfunctions and eigenvalues for an unperturbed system and then applying a series-expansion procedure to establish expressions for perturbed eigenfunctions and energies to some desired order. This paper presents an alternate and more general approach by writing the bound-state equation for the perturbed wavefunctions directly in terms of the Green's function of the unperturbed system. This allows for a shorter derivation of the theory that does not rely on the orthonormality or completeness properties of the unperturbed eigenstates that can also be applied to energy-dependent potentials. Appropriate for upper-level students of mathematical physics and quantum mechanics.
INSTRUCTIONAL LABORATORIES AND DEMONSTRATIONS
Measuring optical force with a torsion pendulum: A platform for independent student experimentation ScilightFeatured by Leland Russell; Ezekiel A. Rein; Anatalya Piatigorsky; Jennifer T. Heath. DOI: 10.1119/5.0268585
Editor's Note: Radiation pressure is usually so slight as to be imperceptible. This elegant experiment shows how the force generated by low-power laser pointers acting on a modified commercial Cavendish balance can be measured to a level of tens of pico-Newtons. Appropriate for upper-level optics, dynamics, and laboratory courses.
COMPUTATIONAL PHYSICS
Agent-based Monte Carlo simulations for reaction–diffusion models, population dynamics, and epidemic spreading by Mohamed Swailem; Ulrich Dobramysl; Ruslan I. Mukhamadiarov; Uwe C. Täuber. DOI: 10.1119/5.0282284
Editor's Note: Agent-based models of reaction–diffusion, population dynamics, and epidemic spreading are of wide interest and provide excellent examples of nonequilibrium phenomena in a context accessible to students without a strong background in statistical mechanics. The authors provide a pedagogical introduction to how to simulate these models and discuss some fundamental ideas of nonequilibrium phenomena while providing results and helpful tips.
NOTES AND DISCUSSIONS
Hydrostatic-based proofs in geometry by Jaehyeon Kim. DOI: 10.1119/5.0267631
Editor's Note: Although mathematicians might not consider the proofs valid, physical laws can be used to “prove” mathematical identities. This paper shows how the forces associated with hydrostatic pressure can be used to verify two identities regarding polygons: that the polygon which encloses the maximum area for given side lengths must have all of its vertices on a common circle, and that the area enclosed by a polygon is given by a cross-product-like expression known as the “`Shoelace” formula. Appropriate for introductory-level students familiar with fluid pressure.
Additional Resources
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