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<jats:p>An alternative approach to the n-dimensional small oscillations problem is presented. This method is based on the finding of n new independent constants of motion to get the n eigenfrequencies and the n normal coordinates of the problem. These constants of motion exist and may be explicitly constructed for any small oscillations problem. Three examples are presented. One of them involves solving a five-dimensional small oscillations problem whose solution is usually obtained by finding the roots of a quintic algebraic equation. The approach constructed here is especially suited to deal with high-dimensional problems. Applications to small oscillations as well as to high-degree algebraic equation solutions are discussed.</jats:p>

<jats:p>We have experimentally demonstrated dissipative coupling in a double pendulum system through observation, which shows three distinctly different patterns of motion over the accessible parameter space. The described dissipative coupling apparatus is easy to manufacture and budget-friendly. The theoretical calculations are also suitable for the undergraduate level. Our experiment can serve as a novel demonstration for ubiquitous dynamic coupling effects encountered in many disparate physical systems. Unlike the well-known spring-coupled pendulums, our experiment employs Lenz's effect to couple the pendulums through electromagnetic damping, which, to the best of our knowledge, has not been demonstrated in the classroom. Our pendulums exhibit level attraction behaviour between two modes, induced by the dissipative coupling. This stands in contrast to the traditionally taught concept of level repulsion (avoided crossing) with spring-coupled pendulums. This experiment showcases distinctly different time domain dynamics of the dissipatively coupled pendulums over the parameter space, characterized by different oscillation patterns, damping rates, and relative phase between the two pendulums, which is a valuable lesson elucidating the dynamics of synchronization in linear systems for undergraduate students.</jats:p>

<jats:p>Orbital interception scenarios typically involve a chaser that is actively maneuvered to encounter an inertial target and may be undertaken for a variety of purposes, including docking spacecraft or colliding with an asteroid for planetary defense studies. Viable intercept trajectories are constrained by the free-fall path of the target and by auxiliary conditions such as the available time or fuel budget. Whereas a constraint on the time to intercept is central to the (extensively studied) Lambert problem, a less common but more visually compelling constraint is that of the available fuel for intercept. This was the basis of a recent study [E. M. Edlund, Am. J. Phys. 89, 559–566 (2021)], which analyzed one of the two families of possible intercept solutions that were identified. The second family, studied in more detail here, describes intercepts at all points in the orbit and has the interesting property that it admits fast-intercept solutions. This work concludes the analysis of this problem; it develops a general condition that describes both families of intercepts, presents representative solutions, and considers the sensitivity of these solutions to errors in the control parameters.</jats:p>

<jats:p>We show that a two-dimensional square lattice of magnets can be studied by placing small cylindrical neodymium magnets inside plastic spherical shells and floating them on water, leaving their magnetic moments free to re-orient within the plane. Experimentally, anti-correlated dipole orientations between nearest neighbors appear to be favored energetically. This motivates the construction of a simplified single-variable energy function for a 2D square lattice of magnetic dipoles. For odd numbers of spheres, this ansatz yields a continuum of dipole configurations with the same energies, matching the observed behavior that the orientation of the dipoles in these lattices can be rotated freely. The behavior of square lattices with even numbers of spheres is strikingly different, showing strongly preferred orientations. While the energy calculated in this simplified model is larger than that of the actual ground state for finite size clusters, its asymptotic value in the limit where the number of spheres goes to infinity is in good agreement with the literature value. Additionally, rectangular arrangements of magnetic spheres with and without a defect are analyzed within the class of the single variable energy function. Simple experimental demonstrations qualitatively reproduce several interesting results obtained from all these analyses.</jats:p>

<jats:p>We describe student difficulties in applying the superposition principle in combination with Gauss's law. We addressed these difficulties by developing a tutorial that uses guided inquiry. Students who used this tutorial following lecture-based instruction performed significantly better on these topics than those who did not. Instructors can assign the tutorial as classwork or homework.</jats:p>

<jats:p>We drop a circular disk magnet through a thin coil of wire, record the induced voltage, and compare the results to an analytic model based on the dipole approximation and Faraday's law, which predicts that the difference between the voltage peak magnitudes corresponding to the entry and exit of the magnet should be in proportion to z0−1/2, where z0 is the initial height of the magnet above the center of the coil. Agreement between the model and experimental data is excellent. This easily reproduced experiment provides an opportunity for students at a range of levels to quantitatively explore the effects of magnetic induction.</jats:p>

<jats:p>The canonical equation for self-inductance involving magnetic flux is examined, and a more general form is presented that can be applied to continuous current distributions. We attempt to clarify and extend the use of the standard equation by recasting it in its more versatile form.</jats:p>