Stars form when vast clouds of gas and dust begin to swirl and collapse with a concentration of matter at the centre. Around the newly born star a proto-planetary disk rotates, out of which planets can later evolve by clumps arising and accumulating ever greater amounts of material. The smaller planets which arise tend to be largely composed of rock, like the first four in our Solar System, whereas the larger ones tend to be gas or ice giants, like the outer four in our system.

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.

Having developed the necessary mathematics in chapters 4 to 6, chapter 7 returns to physics Evidence for homogeneity and isotropy of the Universe at the largest cosmological scales is presented and Robertson-Walker metrics are introduced. Einstein’s equations are then used to derive the Friedmann equations, relating the cosmic scale factor to the pressure and density of matter in the Universe. The Hubble constant is discussed and an analytic form of the red-shift distance relation is derived, in terms of the matter density, the cosmological constant and the spatial curvature, and observational values of these three parameters are given. Some analytic solutions of the Friedmann equation are presented. The cosmic microwave background dominates the energy density in the early Universe and this leads to a description of the thermal history of the early Universe: the transition from matter dominated to radiation dominated dynamics and nucleosynthesis in the first 3 minutes. Finally the horizon problem and the inflationary Universe are described and the limits of applicability of Einstein's equations, when they might be expected to break down due to quantum effects, are discussed.