Interweaving the Principle of Least Potential Energy in School and Introductory University Physics Courses

Abstract

Abstract:Understanding advanced physical phenomena such as vertically hanging elastic column, soap bubbles, crystals and cracks demands expressing and manipulating a system's potential energy under equilibrium conditions. However, students at schools and universities are usually required to consider the forces acting on a system under equilibrium conditions, rather than taking into account its potential energy. As a result, they find it difficult to express the system's potential energy and use it for calculations when they do need to do so. The principle of least potential energy is a powerful idea for solving static equilibrium physics problems in various fields such as hydrostatics, mechanics, and electrostatics. In the current essay, the authors describe this principle and provide examples where students can apply it. For each problem, the authors provide both the force consideration solution approach and the energy consideration solution approach.

Interweaving the Numerical Kinematic Symmetry Principles in School and Introductory University Physics Courses

“…Accordingly, if we go for the simplest of atmospheric profiles, we get a fixed temperature and an exponential decay of air density as we go up. However, experimental evidence indicates that the density profile is better described by the second equation [12,13]. We notice the differences between the naive atmospheric equation and the more detailed barometric equation diverge around 40 km above the air surface, for the initial surface temperature of 288.15 K.…”

“…where ρ is the air density, → v is the projectile velocity, A the cross-section of the projectile along its movement direction, and C is some scale-less shape dependent coefficient empirically measured, which is aptly called the "drag coefficient" and it may depend on the projectile velocity [11,12]. This drag force, is independent of location, as it is sourced by velocity, which, in phase space, is independent of the coordinate.…”

“…In general, the air resistance for a body in motion depends on several factors: the direction of the body's velocity, its size, body's rotation, air characteristics, and the force of the wind. For small bodies moving at low speeds, the friction force (drag force) is proportional to the speed, and for high velocities, the friction force → F d is proportional to the speed square and can be written as above [11][12][13]. For this case then, the ballistic equations in the x and the y directions are written:…”

“…Accordingly, if we go for the simplest of atmospheric profiles, we get a fixed temperature and an exponential decay of air density as we go up. However, experimental evidence indicates that the density profile is better described by the second equation [12,13].…”

Energy, Christiaan Huygens, and the Wonderful Cycloid—Theory versus Experiment

“…This loss emphasized the need to solve the longitude problem. In 1714, the British Parliament announced the Longitude Act, which offered a prize of 20,000 pounds (a huge amount of money at the time) to anyone who could solve the longitude problem [1][2][3]. Already in the second half of the 17th century, it was clear that the solution to the problem lay in building a precise clock.…”