Undergraduate Mechanics Course Research Articles

A modern interpretation of Newton's theorem of revolving orbits

Newton's Theorem of Revolving Orbits derives the force that is necessary to explain a particular precession that leaves the shape of an orbit unchanged. Newton showed that for an orbiting body that is already subject to any central force, the inclusion of an additional inverse-cube central force will change that body's angular speed without affecting its radial motion; this leads to the orbit's precession. After reviewing the relevant concepts of Kepler orbits, we present a full description and explanation of Newton's theorem using modern physical and mathematical approaches aimed at a general audience. We specifically highlight the use of the transformation between a rotating reference frame and an inertial reference frame, making this a suitable topic for an upper-level undergraduate mechanics course.

Energy equations for projectiles with linear and quadratic drag

Although linear and quadratic drag are treated early in upper-level undergraduate mechanics courses, energy dissipation due to drag is not. In this paper, energy equations with exact analytic formulas for dissipation due to air resistance are developed for projectiles using linear and quadratic drag models, based on the work–energy theorem. Examples with analytic results for specific cases are presented for quadratic drag in one dimension, and linear drag in one and two dimensions suggested as student assignments. Similarly, three numerical examples of energy dissipation showing kinetic, potential, and/or dissipated energy versus either time or distance are presented, suggested as assignments that students could code and graph in a spreadsheet application.

Implementation of smartphone-based experimental exercises for physics courses at universities

A concept for undergraduate mechanics courses at universities is introduced where traditional pencil-paper based exercises are partially replaced by experimental exercises, in which smartphones are used as measurement devices. A detailed guidance for practical realization and implementation of these tasks formats into the course is presented. Three smartphone-based experimental exercises ‘The tilting smartphone’, ‘The oscillation balance’ and ‘Using the Smartphone in a Torsion Pendulum’ are presented. First empirical results with respect to the learning achievement indicate a mid size effect on the understanding of the physical concepts. Compared to the traditional pencil-paper based exercises, the students performance in the experimental exercises is slightly lower, although the motivation to solve these tasks is higher.

Failure analysis of engineering structures in undergraduate courses using optimization

ABSTRACTIn the design process for engineering structures, it is often of interest to determine the largest magnitude of a given form of load that can be applied on a structure before causing failure. This gives the factor of safety against failure. Determining the factor of safety entails analysis of a structure on the verge of failure, and is known as limit analysis. It is well known that limit analysis can be formulated as a constrained optimization problem. With robust computational tools for optimization having become readily available, it is possible to introduce the optimization‐based approach to limit analysis at various stages in the undergraduate civil engineering curriculum. In this article, we present three instances of limit analysis using an optimization‐based approach from (i) a sophomore level Statics course, (ii) a junior level Structural Analysis course, and (iii) a senior level Advanced Solid Mechanics course. These are based on materials presented by the author in undergraduate mechanics courses at the University of Colorado at Boulder. In each case, we develop a mathematical formulation and implement the computations in MATLAB. We use two optimization tools, one from the MATLAB Optimization Toolbox and the other called SeDuMi. Classroom experience has shown that the optimization‐based approach helps students recognize the underlying thread in seemingly disparate limit analysis problems. © 2011 Wiley Periodicals, Inc. Comput Appl Eng Educ 22:297–308, 2014; View this article online at wileyonlinelibrary.com/journal/cae; DOI 10.1002/cae.20555

Kepler's other 'law' of planetary motion

A calculation that can be used in an undergraduate mechanics course shows why, in the seventeenth century, there was another 'law' of planetary motion.

Precession of the perihelion of Mercury

The equation of motion for a test mass in the gravitational field of a spherical mass is written in a form that can be used in an undergraduate mechanics course. A simple calculation of the general relativistic precession of the perihelion of Mercury is presented.

An analysis of the rotational stability of the layout back somersault

A numerical solution of Euler’s equations for a freely rotating asymmetrical top is applied to an analysis of the relative stability of rotation of a gymnast performing a layout back somersault with and without a full twist. The results suggest that successful performance of the stunt without a full twist has less tolerance for error in the initial rotational velocity than does the stunt with a full twist. This analysis provides an interesting illustration that can be used in advanced undergraduate mechanics courses. The analysis also demonstrates the potential for increased utilization of computers and numerical techniques in the undergraduate curriculum.

Computer Graphics Exercises in an Undergraduate Mechanics Course

Summary Following the experience gained in the use of a Hewlett‐Packard Calculator and Plotter as well as a computer in the teaching of quantum mechanics,1,2 we are devising a number of exercises to enrich a first‐year mechanics course, taken jointly by 72 students in Electrical Engineering and Physical Science. Three ‘terminals’ were available: the HP Calculator and Plotter and two Computek graphics terminals attached to a Data General Nova Computer. The course was run on the Keller plan principle.3 The following will be presented: alfa A description of the exercises and of the aims they are designed to achieve. alfa An account of the way students reacted to the work. alfa An attempt to assess the relative advantages of the calculator and the computer. † Work supported by the National Development Programme in Computer Assisted Learning.