6.1 INTRODUCTION [LO1]

In earlier chapters, we have discussed axisymmetric problems in two-dimensional (2D) polar coordinate systems. Thick cylinders fall into this class of problems. Cylindrical pressure vessels, hydraulic cylinders, gun-barrels, and pipes carrying fluids at high pressure develop radial and tangential stresses (circumferential). Longitudinal stresses can also be developed if the ends are closed. Therefore, ideally, this is a triaxial stress system as shown in Figure 6.1.

• (a) Circumferential or hoop stress (σθ)

• (b) Longitudinal stress (σz)

• (c) Radial stress (σr)

If the wall thickness of a hollow cylinder is less than about 10% of its radius, it may be treated as a thin cylinder. Cylinders with higher wall thickness are considered to be thick cylinders. Before analyzing the stress in a thick cylinder, we should briefly consider the stress state in thin cylinders, where radial stress is small compared to the other stresses, and this can be neglected. Stress variation across the thin wall is also negligible. Analysis of thin-walled pressure vessels may therefore be carried out on the basis of biaxial stress system. Since the presence of shear stress at the cut section would lead to incompatible distortion, the longitudinal and circumferential stresses in this case are both principal stresses. We now take another section of the cut section as shown in Figure 6.2 (a) to consider the equilibrium of the section, and this is shown in Figure 6.2 (b).

The section is acted upon by internal pressure p and the circumferential stress developed at the cut section is σθ. Force on an infinitesimal small area subtended by angle dθ at θ inclination from the horizontal axis is pridθ.

Introduction

Communication through optical fiber is one of the remarkable discoveries of the twenty-first century that have brought a revolution in modern times. The transfer of information over long distances was earlier performed through copper wires and coaxial cables. The limitation of these devices, such as limited bandwidth, could not fulfill modern needs and hence were replaced by glass fiber. It was the effort of Alexander Graham Bell, who in 1880 used the light as a carrier of signal. Since the attenuation in the optical fibers was quite high, an attempt to minimize it was done to improve it, and today its features are so fantastic that optical communication through glass fiber with low loss has become a reality. A large number of advantages of optical fibers over the traditional wires and coaxial cables are not hidden now and have been accepted over the entire globe. Apart from their use in communication, optical fibers are widely used in other areas also. In a nutshell, we can therefore say that fiber optics is a backbone of communication infrastructure.

The optical fiber is a cylindrical waveguide system operating at optical frequency. It consists of a core at the center and a cladding outside the core. The core is generally a cylindrical dielectric glass, and cladding is the second dielectric cover usually of glass with a lower refractive index n2, as shown in Figure 13.1.

5.1 INTRODUCTION [LO1]

In any three-dimensional (3D) elasticity problem, there are 15 unknown parameters: 6 stress components, 6 strain components, and 3 displacements. There are 15 related equations: 3 equations of equilibrium, 6 compatibility equations, and 6 constitutive equations. Solutions to a particular elasticity problem require evaluation of these 15 unknown parameters using 15 equations, satisfying all the boundary conditions. As discussed in the earlier chapters, there may be displacement or stress, or mixed boundary conditions. In many cases, solutions to 3D problems are not easy analytically. Even numerical solutions may be difficult.

There are mainly three methods of simplification of solution techniques:

• (a) If the boundary conditions are in terms of stresses, stress function approach may be made as discussed in the earlier chapter. This makes the solution simpler.

• (b) Assumptions of plane stress and plane strain reduce 3D problems to two-dimensional (2D) ones and this also makes the solution simpler.

• (c) Use of St. Venant's principle and superposition principle also makes the solution of elasticity problems simpler.

An introduction to stress function approach has been discussed in Chapter 4. We therefore start our discussion on plane stress and plane strain approaches.

5.2 PLANE STRESS AND PLANE STRAIN PROBLEMS [LO2]

The idealizations of both plane stress and plane strain states are suitable for certain classes of problems that are made to reduce the complexity of solutions. We shall consider the plane stress state first.

A renewed focus on “the common” in contemporary political theory, as it gained momentum in the work of Michael Hardt and Antonio Negri, is a direct response to the failure of community in political theory. I argue that Hardt and Negri develop what we can understand as an “insurgent democracy of the common.” On the one hand, such an insurgent democracy of the common seeks to preserve the revolutionary potential of democracy. On the other hand, this democracy of the common also tends toward political romanticism: the legitimacy of “the common” depends on a political subjectivity tied to a paradoxical nostalgia for failed revolutions. As such, the democracy of the common entails an escape from the realities of the political world that undercuts its emancipatory potential. This becomes particularly obvious in the hope that Hardt and Negri place in social movements of resistance, but also in their critique of neoliberal capitalism.

A2.1 Mathematical operations

Try the following operations and commands on the command line in the command window of Matlab.

A2.1.1 Defining an array and arranging in ascending and descending order

In this chapter, I demonstrate that Hegel removes three Kantian obstacles that stand in the way of an elaboration of autonomy as a form of life. Hegel rearticulates the form of autonomy in such a way that we can recognize living beings as a basic case of autonomy. Secondly, Hegel shows that internal purposiveness is not a derivative concept, making positive knowledge of natural purposiveness intelligible. Thirdly, Hegel provides a positive account of the lived reality of freedom. Taken together, these shifts open up the possibility of understanding practical autonomy not just as analogous to living self-organization but as an actual form of living self-organization. The second half of the chapter shows how this account is underwritten by Hegel’s new understanding of the distinction between the realm of nature and of freedom. By reference to Hegel’s Philosophy of Nature, the chapter shows how he modifies Kant’s distinction in crucial ways. Firstly, he gives a new substantive account of the realm of nature, revealing how it includes a form of natural freedom. Secondly, Hegel clarifies that the realms of freedom and nature are not externally juxtaposed and argues that the differentiation of these two realms is internal to spirit. Thirdly, Hegel considers the ways in which spirit reproduces the forms of a realm of nature within itself in the shape of a second nature.

Introduction

Numerous microscopic events, including atomic stability, blackbody radiation, the photoelectric effect, and atomic spectroscopy, could not be explained by classical physics. When Max Planck presented the idea of the quantum of energy in 1900, it marked the first significant advancement. Only after positing that the energy exchange between radiation and its surroundings occurs in discrete, or quantized, amounts was he able to replicate the experimental findings in his attempts to understand the phenomenon of blackbody radiation. He claimed that an electromagnetic wave of frequency v and matter can only exchange energy in integer multiples of h, or what he termed a quantum's energy, where h is a fundamental constant known as Planck's constant. The concept of quantizing electromagnetic radiation proved to have far-reaching effects.

Blackbody radiation was correctly explained by Planck's hypothesis, which inspired fresh thinking and set off a wave of new findings that provided answers to the most pressing issues of the day.

Planck's quantum idea received a potent reinforcement from Einstein in 1905. Einstein realized that Planck's theory of the quantization of electromagnetic waves must also apply to light when attempting to comprehend the photoelectric effect. So, adopting Planck's methodology, he proposed that light itself is composed of discrete energy units (or minuscule particles) called photons, each of which has energy hv, which corresponds to the light's frequency.

This Introduction provides an overview of the main themes of the book and questions the democratic potential we often attach to “community” and to “the common.” Community might be what we desire from political life, and it is tempting to hope that a return to community can correct much of the current disillusionment with liberal constitutional democracy and the state of civil society. Instead, I argue that, as models for the normative organization of political life as a whole, neither community nor the common are compatible with the normative demands of democracy. The communitarian desire of much political thought often stands in sharp contrast to the pluralism of democracy.

Introduction

The word laser is an abbreviation for “light amplification by stimulated emission of radiation.” Historically, the laser is the outgrowth of maser, which means “microwave amplification by stimulated emission of radiation.” If the stimulated radiation lies in optical region, the device is called optical maser or laser. Laser beam is highly monochromatic, highly coherent, intense, and highly collimated with a small diversion.

Using ruby as the active material, T. H. Maiman invented the first laser system in 1960. Thus, it is known as Ruby Laser.

ABSORPTION, SPONTANEOUS EMISSION, AND STIMULATED EMISSION OF RADIATION

Absorption of Radiation:An atom has a number of quantized states. Initially, an atom is in ground state, that is, all its electrons possess lowest possible energy state. When energy is given to an atom, it goes to excited state, that is, its electron jumps to a higher energy state by absorbing a quantum of radiation or photon. This process is called the absorption of radiation shown in Figure 5.1.

If E1 and E2 are the energies of an electron in initial and final states and ν is the frequency of absorbed radiation, then

where h is Plank's constant. Thus, absorption is a stimulated process.

SPONTANEOUS EMISSION

An atom in an excited state remains for only about 10−8 sec. After staying 10−8 sec, the atom jumps automatically to a lower energy state, emitting a radiation. If initially an atom is in excited state, then it spontaneously jumps to ground state, emitting a photon of frequency ν, given by

In this chapter, we shall see how new topological spaces may be created from given topological space(s). We already saw one such technique Section 2.4 of Chapter 2, where, given a subset Y of a topological space X, we endow it with the subspace topology, also called the relative topology inherited from X. In the first section, we shall see a possible way to define a topology on a set by considering a function from the set to some topological space satisfying specific properties. On the other hand, the same can be done by means of a function from a topological space to our given set. Furthermore, given topological space, not necessarily distinct, we shall see how topologies can be defined on their Cartesian product and their disjoint union (coproduct). In Chapter 2, we saw many topologies on ℝ, such as the usual topology, the lower limit topology, and the K-topology, which were defined by means of a basis. Most of the topologies that we shall see in this chapter are defined by means of a subbasis. It is advisable to refer Section 2.3.2 once before proceeding with this chapter for a better grasp of these concepts.

As we discuss the topic of traversal within a graph, it becomes necessary to define the concept of distance in a graph. The definition of distance within a graph follows the same style of definition of distance between two cities or from one's location to one's destination. Just as it is logical to seek shortest paths in one's day-to-day life, we discuss the distance between vertices, in terms of the shortest path from vertex u to vertex v. In a connected graph G, the distance between two vertices u and v denoted as d(u,v) is the shortest path between two vertices in a graph. In a disconnected graph, there are no paths between two vertices which lie in different components and so it follows that if u and v lie in different components of a disconnected graph, then d(u,v) = ∞. In this chapter, we will also be introduced to shortest path algorithms and graph traversal algorithms, that help one to navigate a graph while also paying attention to distance and connectivity.

This chapter introduces the fundamental idea of The Life of Freedom in Kant and Hegel: the notion that we can only make sense of autonomy by returning to the concept of life. This return is needed to understand fully the genesis, the form, and the reality of human freedom. Such an account can be developed by means of a systematic reconstruction of Kant’s and Hegel’s philosophies of freedom. As we can learn from Kant’s account, the notion of autonomy is threatened by the paradox of self-legislation and an opposition of freedom and nature that makes the reality of freedom unintelligible. As Kant already indicates and Hegel goes on to develop, we can overcome these problems by reconceiving of autonomy as a form of life. The chapter outlines the reading of Kant and Hegel supporting this view, situates the resulting systematic position in current debates on the sources of normativity and the nature of human freedom, and defines its relation to other approaches norm and nature (ethical naturalism, forms of life, and biopolitics).

Object tracking dealswith the estimation of the trajectory of a movingobject in the image plane. The tracking takes placein a temporal sequence of images or frames in agiven input video. The task of object tracking isperformed in various contexts. Accordingly thecomputational problems are formulated with varyingset of objectives. For example, the tracking may berequired for either a single object or multipleobjects. It can either be a trajectory intwo-dimensional (2-D) space, as in the image space,or it can be a three-dimensional (3-D) trajectory,where the object is moving in a world coordinatesystem. It can either be a tracking of a set ofpoints, or it can be tracking of the entire body ofan object. The motion involved with the object caneither be a rigid body motion, or it can be adeformable body motion. The processing can either beperformed offline on recorded videos, or it may berequired to process the video in real-time or quasireal-time, while capturing the videos. On the otherhand, it could be an online tracking, which meanstracking results are obtained within a very shorttime interval of capturing of video frames. Thechoice and design of a tracking algorithm depends onthe context and application. A few examples ofapplications of tracking are shown in Fig. 8.1. TheFig. 8.1 (a) illustrates tracking of the vehiclesand pedestrians to check if they are on the correctlane. In (b), visitors of a monument in crowd arebeing tracked and in (c), cars are being tracked toensure the speed limit.

Climate change, global warming, and shifting to sustainable ways of growth are major concerns of present times. Although these issues were identified many years ago, for a long time the exact nature of these phenomena and their impact were under debate. However, recent years have seen clearly visible and regularly occurring phenomena that confirm the adverse effects and grave threat of global warming and climate change.

It is noteworthy that 17 November 2023 was an important day for climate change, as on this day the average temperature of the earth exceeded by more than 2°C compared to the pre-industrial age temperature for the first time. The Copernicus Climate Change Service of the European Union (EU) has confirmed that in 2023 the average temperature of the earth's surface was 1.48°C higher than the temperature during the pre-industrialization days. In absolute terms also, the global emissions of CO2 are rising; in 2022, for example, these emissions were at least a billion tonnes higher than in 2019.

The Conference of Parties (CoP) is the supreme decision, making body under the United Nations Framework Convention on Climate Change (UNFCC). CoP is an important annual event initiated about 30 years ago. But the increasing interest of public, over the years, in the event shows that we have come a long way in understanding the adverse impacts of climate change. From sceptics doubting the very idea of climate change to it becoming the most debated topic in the world is a major change.