Solving the Noneuclidean Uniform Circular Motion Problem by Newton's Impact Method

Students’ understanding of forces in circular motion is often incomplete. The problems are not limited to confusions about centripetal acceleration and centrifugal forces. This paper considers possible effects of different interventions by a teacher who has discovered the many types of free-body diagrams drawn by students for circular motion in a vertical plane. Students’ free-body diagrams often show a lack of attention to lengths of force arrows and to the direction and size of the sum of forces. The strategies used by students may differ depending on the situation and questioning by the teacher can remind students of critical aspects they have overlooked. The revisions of responses following interventions reveal information that is not available from just a single response. This paper discusses how a matrix can be used to express the probability for revisions of responses for different teacher interventions and also help teachers decide on future interventions.

American Institute of Mining, Metallurgical, and Petroleum Engineers, Inc. Discussion of this paper is invited. Three copies of any discussion should be sent to the Society of Petroleum Engineers office. Such discussion may be presented at the above meeting and, with the paper, may be considered for publication in one of the two SPE magazines. Introduction The use of centrifugal force to separate fluids is a proved process that has been applied by many industries. The force is explained in college physics textbooks under uniform circular motion where Newton's laws of motion are discussed. A classic textbook example describes an object of known weight that is held in a circular path by a string of known length, and the weight is moving at a constant speed. The inward force that is developed is known as the centripetal force. The equal and opposite reaction that is in the outward direction is known as the centrifugal force. This is the force that we use to separate gas well fluids in a centrifugal-type separator. The flowing pressure at the wellhead furnishes the energy for the process. Upon expansion to a lower pressure at the wellhead choke, the natural gas, condensate, and water flow to the centrifugal separator for rotation and subsequent separation. The fluids tend to separate in layers of uniform density. Water collects on the internal surface of the separator, condensate flows on the water, and the gas flows in a circular pattern in the center of the vessel. FUNDAMENTALS The following formula for centripetal - force summarizes the process. 2WVFc = ———,gR where Fc = centripetal force, lbW = weight of the fluids, lbV = velocity, ft/secg = gravity constant, 32 ft/sec/secR = radius of the circle, ft The centripetal force, Fc, is that force in the inward direction, expressed in pounds, that is developed when the fluids are directed in a circular path in the separator. The centrifugal force is an equal and opposite reaction to the centripetal force; therefore, calculated values for Fc are vend for the centrifugal forces in the outward direction. The weight of the fluids can be calculated from the fixed dimensions of the internals of the vessel, the operating pressure, and temperature in the separator, and the gas-condensate ratio and gas-water ratio of the flowing stream.

IntroductionThe simplest model of circular locomotion is a point mass moving uniformly on a circle. To produce circular motion, a centripetal force is generated. This study investigated contributions of individual limbs to the net centripetal force in horses trotting on a circle.MethodsSix horses trotted on a 6 m diameter circle at their preferred speed. Optical motion capture (Motion Analysis Corp) recorded kinematics, while six force plates (AMTI) recorded ground reaction forces (GRF). Whole body center of mass (CoM) was computed using a 23‐segment rigid body model. The GRF of each limb was expressed at the CoM in cylindrical coordinates centered at a best‐fit circle of the ground path of the CoM. Average vertical, tangential and radial force components were computed for each limb over the whole stride duration. Forces were analyzed using ANOVA with Tukey HSD post‐hoc tests.ResultsThe maximum ground‐path deviation of the CoM from the best‐fit circle among all trials was only 2.8% of the radius. The forelimbs contributed an average centripetal (radial) force of 0.49 ± 0.16 N/kg (mean ± SD, normalized to body mass), which was significantly greater (P<0.001) than the hind limb contribution of 0.36 ± 0.13 N/kg. There was no significant difference between inside and outside limbs.ConclusionsThe contribution of individual limbs to the centripetal force provided a direct indication of their role in circular locomotion. The forelimbs generated significantly more centripetal force, possibly in direct relation to their greater support of the body mass against gravity.Ethical Animal ResearchThe study was approved by the MSU Institutional Animal Care and Use Committee, protocol #06/11‐112‐00. Sources of funding: This study was funded by the McPhail Endowment. SD Starke's collaborative research visit for this project was funded by the British Society for Animal Science, Murray Black Award. Competing interests: none.

In this undergraduate laboratory experiment we measure the centripetal force acting on a bob in uniform circular motion. As the experiment was originally designed, it consistently yielded large errors due to a subtle twist of the bob as the mass was increased incrementally. This error is due to the fact that the center of mass changes relative position as the mass is incremented; therefore, the spring that provides the centripetal force for the apparatus causes an unintended torque on the bob. A solution to the problem consists of positioning the incremental masses so that the center of mass does not change position. This simple correction provides a useful lesson on redesigning an undergraduate laboratory experiment to obtain better agreement with theory.

A detailed analysis is presented for the problem of uniform circular motion in which the centripetal force is provided by a stretched spring whose mass is not negligible compared with that of the whirling body. It is shown that the force required to stretch the spring statically to the length it has during the rotation exceeds the centripetal force it exerts on the whirling body by an amount depending in a simple way on the spring mass; the excess can be erased by adding to the body mass approximately one-third the spring mass in computing the centripetal force. Some experimental results that tend to support the theory are exhibited and discussed. A computation of radial-oscillation frequencies is included.

We study the two-dimensional extension of the energy or charge transfer problem in the long-wave limit along a mechanical lattice of oscillators for the interplay between linear longitudinal and periodic on-site interactions in the Holstein approach [S. Reza-Mejía and L.A. Cisneros-Ake, Wave Motion 130, 103382 (2024)0165-212510.1016/j.wavemoti.2024.103382]. It is found that the coupling between the linear Schrödinger and sine-Gordon equations, governing the modeling, holds coupled localized and traveling solutions radially symmetric in shape, contrary to the well-known decoupled case where solutions collapse in finite time. We then use the variational description of the problem to find the parameters regime where localization and further propagation take place. For the steady-state scenario, we find a decreasing and oscillatory branch of solutions for the wave function's power, as depending on the frequency of the phase, and then obtain an extension of the Vakhitov-Kolokolov stability criterion that predicts the bifurcation points. We validate these findings by solving the reduced coupled system in the stationary state by means of the Newton method and the Townes soliton condition in the polar coordinates. Finally, we extend our variational approach to the moving case to find the appropriate model and wave parameters where radial motion and the existence of a critical ratio for the strength between longitudinal and on-site interactions take place. These findings are numerically confirmed by solving the full coupled system with the help of the pseudospectral method.

We derive the exact equations of motion for the circular restricted three-body problem in cylindrical curvilinear coordinates together with a number of useful analytical relations linking curvilinear coordinates and classical orbital elements. The equations of motion can be seen as a generalization of Hill’s problem after including all neglected nonlinear terms. As an application of the method, we obtain a new expression for the averaged third-body disturbing function including eccentricity and inclination terms. We employ the latter to study the dynamics of the guiding center for the problem of circular coorbital motion providing an extension of some results in the literature.

Parametric instabilities of the radial motions of non-linearly viscoelastic shells under pulsating pressures

Students experience a force every time the car they drive rounds a corner, and yet they have difficulty with the concept of centripetal force. Some common misconceptions that need to be addressed in the study of circular or centripetal motion are the following:No force is needed for circular motion.The force that throws you out from the center is the centrifugal force.An object that travels at a constant speed in a circle or around a curve cannot be accelerating.If the force causing the object to travel in a circular path is suddenly removed, the object will continue to travel in a circle or curved path.

The two-body problem is extensively studied in open systems and asymptotically flat spacetimes. However, there are many systems where radiation is trapped: they range from radiating charges in cavities to low-energy excitations of massive degrees of freedom, to anti-de Sitter spacetimes. Here, we study the problem of motion of a pointlike particle orbiting a massive compact object inside a cavity. We first show that - assuming circular motion - there are initial conditions for which the self-force vanishes and the binary is eternal. We then consider the evolution of the system under radiation reaction in a toy model which we argue captures the essentials of orbiting particles. We show that eternal circular binaries may exist. We also show that the presence of cavity modes leads to chaos in regimes of strong coupling or when the system is initialized close enough to a resonance. Our results have implications for physics in anti-de Sitter spacetimes and possibly for binaries evolving within dark matter haloes, if it consists on massive fundamental fields.