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CHAPTER 2 Some Tools of the Trade: Numbers, Quantities, and Units The language of physics: symbols and formulas Positive and negative numbers Zero Numbers, huge and tiny: powers of 10 Precision: Significant figures Quantities and units Ratios and proportional reasoning Tables, graphs, equations, and functions Right-angled triangles Once more the four forces, this time quantitatively The gravitational force The electric force The other forces So far we have used words, almost exclusively.We have talked about size, force, and energy, but mostly without using numbers, although a few times they almost forced themselves on us. But most of the time we have to know how big a distance is, or a force, or any other quantity, and that requires numbers and units. We also want to describe relations between different quantities, and do that with symbols, such as E for energy, and M for mass. All of that is mathematics. Mathematics is the language in which the ideas, facts, and relationships of physics are best expressed. Sometimes it’s just shorthand . It is much easier to write v = 32 m/s than “the velocity is thirty-two meters per second,” or v̄ = Δx Δt , rather than “the average velocity of an object is equal to the displacement along the x-axis divided by the time it took to make that displacement.” Sometimes the relationships are more complex, like F = Ma: “the force, or, if there is more than one, the sum of all the forces acting on an object, is equal to the mass of the object multiplied by its acceleration.” You can see that just as a way of writing things down in shorthand notation with symbols like v and x, math is very helpful. But it does more than that. It lets us write down relationships between different quantities, and change them so that they lead to other relationships. To do the same with words would be cumbersome even for the simplest ones, and close to impossible for others. 2.1 The language of physics: symbols and formulas / 17 Mathematicians can make up their own rules. All that is required is that they are not in conflict with one another, i.e., that they are internally consistent. Physics, and more generally science, does something quite different. It describes the world as it exists.We can make up rules and laws, but more than consistency is required of them.The more severe test is whether they help us in the primary task, the description of the dynamic, interacting, ever-changing, endlessly new world that we observe. 2.1 The language of physics: symbols and formulas Look at the equation y = 3x2. It could be seen as a “formula,” or “recipe” that expresses the quantity y in terms of the quantity x: when x = 1, y = 3, when x = 2, y = 12, and so on. Given x we can follow the rules and find y. A physicist is more likely to see it as a relationship, for which the mathematical equation provides the description . The equation shows how y depends on x. In other words, it shows y as a function of x. It shows that y is proportional to x2, and brings to mind a picture of the graph of y against x that shows how the two quantities are related. Some symbols stand for quantities, like distance , L, and their units, like meters, m. Others describe a procedure or operation, like plus (+) or times (×), or a relation like equals (=) or “is proportional to” (∝). A page with unfamiliar symbols does not give us understanding or comprehension. It is like a page of musical notes that comes alive to the musician as he or she looks at it, but remains hidden to those who do not know musical notation . Moreover, to know what each written note means is far from knowing what an orchestral passage sounds like. It is the same with a page of physics filled with mathematical notation. Our intention is to help you see, follow, and “hear” what is being described. When we speak or write we use the “parts of speech,” the nouns, verbs, adjectives, and so on. Their proper use leads to understanding and communication, while their improper use can lead to confusion. Similarly, to “speak physics” we need mathematical components, such as numbers (positive, negative, and zero), graphs, proportionalities , and equations. In the following sections we review some of the mathematical procedures and ideas that we will use to communicate . Positive...

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