In the preceding chapter we described the kinematics of a rigid body. In this chapter we study the dynamics of a rigid body that is rotating in an arbitrary manner about a fixed point. To make the analysis easier I will introduce tensors and dyads, physical quantities that may be new to you. (You may be using tensors for the rest of your life, so this is a good chance to learn them!) Once the necessary mathematical tools have been developed we will use them to obtain expressions for the energy and angular momentum of rotating bodies. Next we develop the equations of motion for rotating bodies; these are called Euler’s equations. Finally, we use these concepts to study two problems, a freely rotating body and a rotating body acted upon by a torque. As examples we consider the Earth and a spinning top. The Earth is approximately a torque-free spinning body and a top (or gyroscope) is an example of a spinning body subjected to an external torque.

This chapter presents the basic ideas of the special theory of relativity. Relativity theory involves serious modifications to classical mechanics as well as to many of our basic notions about time, space, and causality.

This chapter (and much of the rest of this book) deals with dynamics, that is, the relation between the forces acting on a body and its motion. A force is an interaction between a body (or particle) and its environment, often described as a push or a pull in a specified direction. In this book you will encounter a number of familiar forces, such as the gravitational force and forces exerted by springs, as well as a few less-familiar ones.

By now you have heard me say many times that Newton’s laws of motion are applicable only in inertial (nonaccelerated) frames of reference. The question arises: How do we deal with motion in an accelerated reference frame? After all, most real reference frames are accelerating and, for many of them, the acceleration cannot be neglected. Since we live on the surface of a large rotating sphere, it is important for us to be able to solve physics problems in noninertial systems.

This chapter analyzes the conservation of energy in a much more rigorous manner than the review in Section 1.3. You will be exposed to a number of useful concepts such as energy diagrams, metrics, and a review of the del operator. The techniques presented here are used quite frequently, not only in this course but also in other advanced physics courses.

Recall that kinematics is the study of motion and, specifically, it is the study of the relationships between position, velocity, acceleration, and time.

As an appendix, we can look briefly at the central ideas of General Relativity (though we are limited, since much of the maths is beyond our scope). We prepare the ground with a number of thought experiments, and then discuss, in outline, the geometrical ideas we have to use. We can get a sense of what Einstein's equation is doing, and we look at some solutions of Einstein's equation (including the Schwarzschild metric), describing possible spacetimes.

We introduce the two axioms on which SR is based, namely the Principle of Relativity and the constancy of the speed of light.

Introducing the Minkowski diagram and Minkowski space; how do we represent motion? And how can we represent the phenomena of length contraction and time dilation graphically?

In this chapter we will examine some immediate consequences of the axioms of , and develop some qualitative understanding of these before we mathematise things in the .