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Rotational motion is of fundamental importance in physics and engineering, and an essential topic for undergraduates to master. This accessible yet rigorous Student's Guide focuses on the underlying principles of rotational dynamics, providing the reader with an intuitive understanding of the physical concepts, and a firm grasp of the mathematics. Key concepts covered include torque, moment of inertia, angular momentum, work and energy, and the combination of translational and rotational motion. Each chapter presents one important aspect of the topic, with derivations and analysis of the fundamental equations supported by step-by-step examples and exercises demonstrating important applications. Much of the book is focused on scenarios in which point masses and rigid bodies rotate around fixed axes, while more advanced examples of rotational motion, including gyroscopic motion, are introduced in a final chapter.
In this chapter, we extend perhaps the most famous law in mechanics, Newton’s Second Law, to study objects and systems of objects executing rotational motion. Emphasis is placed on developing an intuition for the effects of torques on the rotational dynamics of systems by comparing and contrasting them to the effects that forces have on the linear motion of such systems.
In this chapter, we begin by defining the concept of the angular momentum for a point mass, systems of discrete masses, and continuous rigid bodies. We then use the most general form of Newton’s Second Law for rotational motion to study the impulse due to a torque, the angular momentum impulse theorem, and finally the conservation of angular momentum. To develop these theorems, we draw from our understanding of the analogous theorems in linear motion.
Just as force is a ubiquitous concept in linear mechanics, torque is ubiquitous in rotational mechanics. We, therefore, begin this chapter with the definition and detailed description of torque, which we then use to study static equilibrium. Our discussion includes descriptions of common forces and their points of application, as well as subtleties associated with studying systems of objects in static equilibrium. The chapter ends with some useful theorems commonly found in the literature.
The most general motion of a rigid body can be described by the combination of the translational motion of its center of mass and the rotational motion of all points of the body about an axis through the center of mass. In this chapter, we apply kinematics, dynamics, and conservation laws to investigate rolling motion, which is a special case of this most general motion. This chapter represents the culmination of all the topics we cover in the first six chapters of this book.
In this chapter, we begin by examining the work due to a torque. We then define the concept of the rotational kinetic energy for a point mass, systems of discrete masses, and continuous rigid bodies. We develop the angular work-kinetic energy theorem and use it to study the conservation of energy and the conservation of mechanical energy in systems involving rotational motion. To develop these theorems, we draw from our understanding of the analogous theorems in linear motion.
We begin our study of rotational motion with the definitions and detailed examination of the fundamental quantities which we will use throughout this book. We then proceed with a description of kinematics in rotational motion by drawing analogies from our knowledge of one-dimensional kinematics in linear motion.
The journey through rotational motion is not quite done. In fact, we are just beginning. This chapter introduces a few topics which would be covered in an intermediate-level mechanics course. The topics include more advanced physical phenomena, such as gyroscopic precession, and the mathematical formalism of parameterizing rotations using matrices.
The concept of mass in linear motion was quite simple. However, the rotational analog, the moment of inertia, is comparatively complicated. In this chapter, we present a thorough introduction to the moment of inertia, and we develop the tools needed to compute this quantity for point masses, systems of discrete masses, and continuous rigid bodies about different axes of rotation. The chapter ends with some useful theorems that allow us to extend the application of these fundamental tools.
The Australian SKA Pathfinder (ASKAP) is being used to undertake a campaign to rapidly survey the sky in three frequency bands across its operational spectral range. The first pass of the Rapid ASKAP Continuum Survey (RACS) at 887.5 MHz in the low band has already been completed, with images, visibility datasets, and catalogues made available to the wider astronomical community through the CSIRO ASKAP Science Data Archive (CASDA). This work presents details of the second observing pass in the mid band at 1367.5 MHz, RACS-mid, and associated data release comprising images and visibility datasets covering the whole sky south of $\delta_{\text{J2000}}=+49^\circ$. This data release incorporates selective peeling to reduce artefacts around bright sources, as well as accurately modelled primary beam responses. The Stokes I images reach a median noise of 198 $\mu$Jy PSF$^{-1}$ with a declination-dependent angular resolution of 8.1–47.5 arcsec that fills a niche in the existing ecosystem of large-area astronomical surveys. We also supply Stokes V images after application of a widefield leakage correction, with a median noise of 165 $\mu$Jy PSF$^{-1}$. We find the residual leakage of Stokes I into V to be $\lesssim 0.9$–$2.4$% over the survey. This initial RACS-mid data release will be complemented by a future release comprising catalogues of the survey region. As with other RACS data releases, data products from this release will be made available through CASDA.
A model of dynamical evolution of meteoroid swarm is applied to study the problem of difference in mass spectra of meteoric bodies during meteor showers and for sporadic meteors. It is demonstrated that mass spectra forms within meteoroid stream. Qualitative behavior of mass index in model is consistent with observational data.
Deep-learning algorithms have gained much popularity in the past decade. However, their supervised nature makes them fully dependent on the quality and completeness of the data samples used in the training processes. And when it comes to galaxy spectra for instance, there is simply no suited training sample available yet. This is where unsupervised classification (clustering) can come to the rescue.
Using the discriminant latent mixture-model based algorithm Fisher-EM, we 1) demonstrate the discriminative capacity of the method and its robustness with respect to noise on a sample of galaxy spectra simulated with the code CIGALE, 2) manage to classify ∼700 000 spectra of galaxies from the SDSS, and 3) extend this classification to higher redshifts thanks to the VIPERS data (∼80 000 spectra up to a redshift of 1.2). Finally, it appears that FisherEM is efficient for images of galaxies as well.
We present a machine learning method to estimate the physical parameters of classical pulsating stars such as RR Lyrae and Cepheid variables based on an automated comparison of their theoretical and observed light curve parameters at multiple wavelengths. We train artificial neural networks (ANNs) on theoretical pulsation models to predict the fundamental parameters (mass, radius, luminosity, and effective temperature) of Cepheid and RR Lyrae stars based on their period and light-curve parameters. The fundamental parameters of these stars can be estimated up to 60 percent more accurately when the light-curve parameters are taken into consideration. This method was applied to the observations of hundreds of Cepheids and thousands of RR Lyrae in the Magellanic Clouds to produce catalogs of estimated masses, radii, luminosities, and other parameters of these stars.
In this work we reviewed the Gauss method to infer the orbits of minor bodies of the solar system, as the identification of optimization parameters to infer the orbital elements of two asteroids near to the Earth (NEAs): 5587 (1990 SB) and 4953 (1990 MU)), already cataloged and named by the Minor Planet Center - MPC. We used the database of JPL - Horizons and also included an analysis using data of Gaia Data Release 3 (DR3). We did an statistic analysis between distribution of points correspondent to orbital elements that we obtained and the inferred by the Jet Propulsion Laboratory-JPL. When data is crossed with the orbital parameters of Small-Body Database (JPL), we analyzed the deficiency grades of the method by statistic comparation with state vector method between orbital elements inferred with the implemented method and the accepted by the community, obtaining a relative error for the orbits calculated of 0.424149% and 0.416237% for the body 5587 (1990SB), and 0.257968% and 0.223521% for the body 4953 (1990MU). The first error value mentioned corresponds to an orbit calculated with the database JPL- Horizons, and the second value to an orbit calculated with the database Gaia (DR3), for each body in study. In the first phases of implementation of the code, it was found that the restrictions of the traditional method are overcome under the additional parameters that are proposed, resulting in orbits that are better approximated than those determined by NASA Team at the time of observation corresponding to the collection of data for the different bodies analyzed in the framework of this work.The best approximations are established with respect to the calculation of the orbital elements through the numerical solutions incorporated in the NASA SPICE kernel in python for a period of 100 years, with a step of 1 month. We found that our data are quite close to the curves that represent the variations of each of the 5 orbital elements involved in our analysis, namely: a, e, i, ω, Ω. Finally, it should be noted that our method could contribute to the estimation of orbits for minor bodies of the Solar System from observational data, which could easily be taken by using small telescopes. Thus, it would enrich processes that seek to expand the coverage of observatories focused on estimating the orbits of minor bodies in the solar system.