Units, Physical Quantities, Vectors - Introduction to Physics

1. Chapter 1 Units, Physical Quantities, and Vectors

2. Goals for Chapter 1 • To prepare presentation of physical quantities using accepted standards for units • To understand how to list and calculate data with the correct number of significant figures • To manipulate vector components and add vectors • To prepare vectors using unit vector notation • To use and understand scalar products • To use and understand vector products

3. Introduction • The study of physics is important because physics is one of the most fundamental sciences, and one of the first applications of the pure study, mathematics, to practical situations. • Physics is ubiquitous, appearing throughout our “day-to-day” experiences.

4. Solving problems in physics • Identify, set up, execute, evaluate

5. Standards and units • Base units are set for length, time, and mass. • Unit prefixes size the unit to fit the situation.

6. Unit consistency and conversions • An equation must be dimensionally consistent (be sure you’re “adding apples to apples”). • “Have no naked numbers” (always use units in calculations). • Refer to Example 1.1 (page 7) and Problem 1.2 (page 8).

7. Uncertainty and significant figures—Figure 1.7 • Operations on data must preserve the data’s accuracy. • For multiplication and division, round to the smallest number of significant figures. • For addition and subtraction, round to the least accurate data. • Refer to Table 1.1, Figure 1.8, and Example 1.3. • Errors can result in your rails ending in the wrong place.

8. Estimates and orders of magnitude • Estimation of an answer is often done by rounding any data used in a calculation. • Comparison of an estimate to an actual calculation can “head off” errors in final results. • Refer to Example 1.4.

9. Vectors—Figures 1.9–1.10 • Vectors show magnitude and displacement, drawn as a ray.

10. Vector addition—Figures 1.11–1.12 • Vectors may be added graphically, “head to tail.”

11. Vector additional II—Figure 1.13

12. Vector addition III—Figure 1.16 • Refer to Example 1.5.

13. Components of vectors—Figure 1.17 • Manipulating vectors graphically is insightful but difficult when striving for numeric accuracy. Vector components provide a numeric method of representation. • Any vector is built from an x component and a y component. • Any vector may be “decomposed” into its x component using V*cos θ and its y component using V*sin θ (where θ is the angle the vector V sweeps out from 0°).

14. Components of vectors II—Figure 1.18

15. Finding components—Figure 1.19 • Refer to worked Example 1.6.

16. Calculations using components—Figures 1.20–1.21 • To find the components, follow the steps on pages 17 and 18. • Refer to Problem-Solving Strategy 1.3.

17. Calculations using components II—Figure 1.22 • See worked examples 1.7 and 1.8.

18. Unit vectors—Figures 1.23–1.24 • Assume vectors of magnitude 1 with no units exist in each of the three standard dimensions. • The x direction is termed I, the y direction is termed j, and the z direction, k. • A vector is subsequently described by a scalar times each component. A = Axi + Ayj + Azk • Refer to Example 1.9.

19. The scalar product—Figures 1.25–1.26 • Termed the “dot product.” • Figures 1.25 and 1.26 illustrate the scalar product.

20. The scalar product II—Figures 1.27–1.28 • Refer to Examples 1.10 and 1.11.

21. The vector product—Figures 1.29–1.30 • Termed the “cross product.” • Figures 1.29 and 1.30 illustrate the vector cross product.

22. The vector product II—Figure 1.32 • Refer to Example 1.12.