physics chapter 5 exercise problems and numericals pdf

Physics Chapter 5 Exercise Problems And Numericals Pdf

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In the following example, try to identify the number of times the velocity of the ball changes. Also identify the agent supplying the force in each case.

Not a MyNAP member yet? Register for a free account to start saving and receiving special member only perks. In addition to ascertaining that the internal vitality of the mathematical sciences is excellent, as illustrated in Chapter 2 , the current study found a striking expansion in the impact of the mathematical sciences on other fields, as well as an expansion in the number of mathematical sciences subfields that are being applied to challenges outside of the discipline. This expansion has been ongoing for decades, but it has accelerated greatly over the past years.

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Not a MyNAP member yet? Register for a free account to start saving and receiving special member only perks. In addition to ascertaining that the internal vitality of the mathematical sciences is excellent, as illustrated in Chapter 2 , the current study found a striking expansion in the impact of the mathematical sciences on other fields, as well as an expansion in the number of mathematical sciences subfields that are being applied to challenges outside of the discipline. This expansion has been ongoing for decades, but it has accelerated greatly over the past years.

Some of these links develop naturally, because so much of science and engineering now builds on computation and simulation for which the mathematical sciences are the natural language. In addition, data-collection capabilities have expanded enormously and continue to do so, and the mathematical sciences are innately involved in distilling knowledge from all those data. However, mechanisms to facilitate linkages between mathematical scientists and researchers in other disciplines must be improved.

The impacts of mathematical science research can spread very in some cases, because a new insight can quickly be embodied in software without the extensive translation steps that exist between, say, basic research in chemistry and the use of an approved pharmaceutical. When mathematical sciences research produces a new way to compress or analyze data, value financial products, process a signal from a medical device or military system, or solve the equations behind an engineering simulation, the benefit can be realized quickly.

For that reason, even government agencies or industrial sectors that seem disconnected from. And because that enterprise must be healthy in order to contribute to the supply of well-trained individuals in science, technology, engineering, and mathematical STEM fields, it is clear that everyone should care about the vitality of the mathematical sciences.

This chapter discusses how increasing interaction with other fields has broadened the definition of the mathematical sciences. It then documents the importance of the mathematical sciences to a multiplicity of fields. In many cases, it is possible to illustrate this importance by looking at major studies by the disciplines themselves, which often list problems with a large mathematical sciences component as being among their highest priorities.

Extensive examples of this are given in Appendix D. Over the past decade or more, there has been a rapid increase in the number of ways the mathematical sciences are used and the types of mathematical ideas being applied. Because many of these growth areas are fostered by the explosion in capabilities for simulation, computation, and data analysis itself driven by orders-of-magnitude increases in data collection , the related research and its practitioners are often assumed to fall within the umbrella of computer science.

But in fact people with varied backgrounds contribute to this work. The process of simulation-based science and engineering is inherently very mathematical, demanding advances in mathematical structures that enable modeling; in algorithm development; in fundamental questions of computing; and in model validation, uncertainty quantification, analysis, and optimization. Advances in these areas are essential as computational scientists and engineers tackle greater complexity and exploit advanced computing.

These mathematical science aspects demand considerable intellectual depth and are inherently interesting for the mathematical sciences. People with mathematical science backgrounds per se can bring different perspectives that complement those of computer scientists and others, and the combination of talents can be very powerful. The discipline known as the mathematical sciences encompasses core or pure and applied mathematics, plus statistics and operations research, and extends to highly mathematical areas of other fields such as theoretical computer science.

The theoretical branches of many other fields—for instance, biology, ecology, engineering, economics—merge seamlessly with the mathematical sciences. The Odom report implicitly used a similar definition, as embodied in Figure , adapted from that report.

Figure captures an important characteristic of the mathematical sciences—namely, that they overlap with many other disciplines of science, engineering, and medicine, and, increasingly, with areas of business such as finance and marketing. Where the small ellipses overlap with the main ellipse representing the mathematical sciences , one should envision a mutual entwining and meshing, where fields overlap and where research and people might straddle two or more disciplines.

Some people who are clearly affiliated with the mathematical sciences may have extensive interactions and deep familiarity with one or more of these overlapping disciplines. And some people in those other disciplines may be completely comfortable in mathematical or statistical settings, as will be discussed further. These interfaces are not clean lines but instead are regions where the disciplines blend. It is easy to point to work in theoretical physics or theoretical computer science that is indistinguishable from research done by mathematicians, and similar overlap occurs with theoretical ecology, mathematical biology, bioinformatics, and an increasing number of fields.

This is not a new phenomenon—for example, people with doctorates in mathematics, such as Herbert Hauptman, John Pople, John Nash, and Walter Gilbert, have won Nobel prizes in chemistry or economics—but it is becoming more widespread as more fields become amenable to mathematical representations. This explosion of opportunities means that much of twenty-first century research is going to be built on a mathematical science foundation, and that foundation must continue to evolve and expand.

Mathematics: A Plan for the s. National Academy Press, Washington, D. Note that the central ellipse in Figure is not subdivided. The committee members—like many others who have examined the mathematical sciences—believe that it is important to consider the mathematical sciences as a unified whole. It is true that some mathematical scientists primarily prove theorems, while others primarily create and solve models, and professional reward systems need to take that into account. But any given individual might move between these modes of research, and many areas of specialization can and do include both kinds of work.

Overall, the array of mathematical sciences share a commonality of experience and thought processes, and there is a long history of insights from one area becoming useful in another. Thus, the committee concurs with the following statement made in the International Review of Mathematical Sciences Section 3. A long-standing practice has been to divide the mathematical sciences into categories that are, by implication, close to disjoint.

Furthermore, such distinctions can create unnecessary barriers and tensions within the mathematical sciences community by absorbing energy that might be expended more productively. In fact, there are increasing overlaps and beneficial interactions between different areas of the mathematical sciences.

What is this commonality of experience that is shared across the mathematical sciences? The mathematical sciences aim to understand the world by performing formal symbolic reasoning and computation on abstract structures.

One aspect of the mathematical sciences involves unearthing and understanding deep relationships among these abstract structures. Another aspect involves capturing certain features of the world by abstract structures through the process of modeling, performing formal reasoning on these abstract structures or using them as a framework for computation, and then reconnecting back to make predictions about the world—often, this is an iterative process. A related aspect is to use abstract reasoning and structures to make inferences about the world from data.

This is linked to the quest to find ways to turn empirical observations into a means to classify, order, and understand reality—the basic promise of science. Through the mathematical sciences, researchers can construct a body of knowledge whose interrelations are understood and where whatever understanding one needs can be found and used.

The mathematical sciences also serve as a natural conduit through which concepts, tools, and best practices can migrate from field to field. A further aspect of the mathematical sciences is to investigate how to make the process of reasoning and computation as efficient as possible and to also characterize their limits. It is crucial to understand that these different aspects of the mathematical sciences do not proceed in isolation from one another.

On the contrary, each aspect of the effort enriches the others with new problems, new tools, new insights, and—ultimately—new paradigms. Put this way, there is no obvious reason that this approach to knowledge should have allowed us to understand the physical world. Yet the entire. The traditional areas of the mathematical sciences are certainly included. But many other areas of science and engineering are deeply concerned with building and evaluating mathematical models, exploring them computationally, and analyzing enormous amounts of observed and computed data.

These activities are all inherently mathematical in nature, and there is no clear line to separate research efforts into those that are part of the mathematical sciences and those that are part of computer science or the discipline for which the modeling and analysis are performed.

The number of interfaces has increased since the time of Figure , and the mathematical sciences themselves have broadened in response.

The academic science and engineering enterprise is suggested by the right half of the figure, while broader areas of human endeavor are indicated on the left. Within the academy, the mathematical sciences are playing a more integrative and foundational role, while within society more broadly their impacts affect all of us—although that is often unappreciated because it is behind the scenes.

It does not attempt to represent the many other linkages that exist between academic disciplines and between those disciplines and the broad endeavors on the left, only because the full interplay is too complex for a two-dimensional schematic. It is the collection of people who are advancing the mathematical sciences discipline.

Some members of this community may be aligned professionally with two or more disciplines, one of which is the mathematical sciences. This alignment is reflected, for example, in which conferences they attend, which journals they publish in, which academic degrees they hold, and which academic departments they belong to. The collection of people in the areas of overlap is large. It includes statisticians who work in the geosciences, social sciences, bioinformatics, and other areas that, for historical reasons, became specialized offshoots of statistics.

It includes some fraction of researchers in scientific computing and computational science and engineering. It includes number theorists who contribute to cryptography, and real analysts and statisticians who contribute to machine learning. It includes operations researchers, some computer scientists, and physicists, chemists, ecologists, biologists, and economists who rely on sophisticated mathematical science approaches. Some of the engineers who advance mathematical models and computational simulation are also included.

It is clear that the mathematical sciences now extend far beyond the definitions implied by the institutions—academic departments, funding sources, professional societies, and principal journals—that support the heart of the field. As just one illustration of the role that researchers in other fields play in the mathematical sciences, the committee examined public data 4 on National Science Foundation NSF grants to get a sense of how much of the research supported by units other than the NSF Division of Mathematical.

Sciences DMS has resulted in publications that appeared in journals readily recognized as mathematical science ones or that have a title strongly suggesting mathematical or statistical content. It also lent credence to the argument that the mathematical sciences research enterprise extends beyond the set of individuals who would traditionally be called mathematical scientists. This exercise revealed the following information:. These publication counts span different ranges of years because the number of publications with apparent mathematical sciences content varies over time, probably due to limited-duration funding initiatives.

For comparison, DMS grants that were active in led to 1, publications. Therefore, while DMS is clearly the dominant NSF supporter of mathematical science research, other divisions contribute in a nontrivial way. Analogously, membership figures from the Society for Industrial and Applied Mathematics SIAM demonstrate that a large number of individuals who are affiliated with academic or industrial departments other than mathematics or statistics nevertheless associate themselves with this mathematical science professional society.

A recent analysis tried to quantify the size of this community on the interfaces of the mathematical sciences. Over the same period, some 75, research papers indexed by Zentralblatt MATH were published by faculty members in other departments of those same 50 universities. The implication is that a good deal of mathematical sciences research—as much as half of the enterprise—takes place outside departments of mathematics.

Higher Education 61 6 : This figure shows the fraction of 6, nonstudent members identifying with a particular category.

That analysis also created a Venn diagram, reproduced here as Figure , that is helpful for envisioning how the range of mathematical science research areas map onto an intellectual space that is broader than that covered by most academic mathematics departments. The diagram also shows how the teaching foci of mathematics and nonmathematics departments differ from their research foci.

The tremendous growth in the ways in which the mathematical sciences are being used stretches the mathematical science enterprise—its people, teaching, and research breadth.

If our overall research enterprise is operating well, the researchers who traditionally call themselves mathematical scientists—the central ellipse in Figure —are in turn stimulated by the challenges from the frontiers, where new types of phenomena or data stimulate fresh thinking about mathematical and statistical modeling and new technical challenges stimulate deeper questions for the mathematical sciences.

But the cited paper notes that only about 17 percent of the research indexed by Zentralblatt MATH is classified as dealing with statistics, probability, or operations research. FIGURE Representation of the research and teaching span of top mathematics departments and of nonmathematics departments in the same academic institutions. Subjects most published are shown in italics; subjects most taught are underscored. Higher Education 61 6 , Figure 8. Many people with mathematical sciences training who now work at those frontiers—operations research, computer science, engineering, economics, and so on—have told the committee that they appreciate the grounding provided by their mathematical science backgrounds and that, to them, it is natural and healthy to consider the entire family tree as being a unified whole.

Many mathematical scientists and academic math departments have justifiably focused on core areas, and this is natural in the sense that no other community has a mandate to ensure that the core areas remain strong and robust. But it is essential that there be an easy flow of concepts, results, methods, and people across the entirety of the mathematical sciences. For that reason, it is essential that the mathematical sciences community actively embraces the broad community of researchers who contribute intellectually to the mathematical sciences, including people who are professionally associated with another discipline.

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Question 5. Hence net force on the cork is zero. A pebble of mass 0. Give the direction and magnitude of the net force on the pebble, a during its upward motion,. Give the magnitude and direction of the net force acting on a stone of mass 0. Neglect air resistance throughout. No force acts on the stone due to this motion.

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No, the accelerating elevator will affect the weight of both sides of the beam balance. So, the net effect of the accelerating elevator cancels out, and we get the actual mass. So, the coin will remain stationary w. If no force acts on the particle it cannot change its direction. So, it is not possible for a particle to describe a curved path if no force acts on it.

Question 1. The product of resistivity and conductivity of conductor depends upon : a the cross-sectional area b the temperature c the length d none of these Answer: d none of these. Question 2.

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