1. Fundamental Tools Chapter 1

2. Fundamental Tools Expectations After this chapter, students will:  understand the basis of the SI system of units  distinguish between units and dimensions  be able to perform dimensional analyses  distinguish between fundamental and derived units  be able to convert a quantity to different units  know standard powers-of-ten prefixes  be able to solve right triangles

3. Fundamental Tools Expectations (continued) After this chapter, students will:  distinguish between vector and scalar quantities  be able to resolve vectors into orthogonal components  be able to add and subtract vectors  know how vectors can be multiplied by scalars  know how vectors can be multiplied by vectors  know how many significant figures are in a given number

4. What is Physics? The mapping of mathematics onto the material world. A mathematical description of the interactions of space, time, matter, and energy. An experimental science: theory is judged by how well it predicts the results of experiments.

5. Numbers and Units “I often say that when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot express it in numbers, your knowledge is of a meager and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of Science, whatever the matter may be.” --- William Thomson (Lord Kelvin) 1824 - 1907

6. Numbers and Units Quantity: a characteristic of an object or material that can be expressed quantitatively (in numbers) examples: height, weight, volume, density Dimension: the name of the class or category of units that express a physical quantity examples: length, mass, time, velocity

7. Numbers and Units Unit: a reference standard with an agreed-upon definition that allows quantities to be specified by comparison to it. examples: meter, second, pound

8. Types of Units Base unit: the units of the quantities length, mass, and time. examples: meter, kilogram, second Derived units: units made by combining other units examples: meters / second, kilogram·meters / (second) 2

9. Systems of Units SI (Système International d’Unités) Base units: length: meter (m) mass: kilogram (kg) time: second (s) CGS (“small metric”) Base units: length: centimeter (cm) mass: gram (g) time: second (s)

10. Systems of Units BE (“British engineering”) Base units: length: foot (ft) mass: slug (sl) time: second (s) In this class, we’ll use SI units almost exclusively.

11. Significant Figures The uncertainty in our knowledge of the numerical value of a physical quantity is indicated by the number of significant figures we use to express that value. To determine how many significant figures are in a number (example: 0.0000149 m)  Write the number in proper scientific notation: 1.49×10 -5 m.  Count the digits in this part of the number: 3 digits Note: for “proper” scientific notation: a × 10 n, 1 ≤ a < 10

12. Unit Conversions Basic principles:  1 is the multiplicative identity. Multiplying any quantity by one does not change the value of that quantity.  If a fraction’s numerator and denominator are equal, the fraction is equal to one. Example: convert 3.45 years to seconds fractions equal to one

13. Dimensional Analysis A consistency check for mathematical relationships in physics. A formula or equation that “passes” a dimensional analysis may or may not be correct. However … A formula or equation that “fails” a dimensional analysis cannot be correct.

14. Dimensional Analysis What is a dimensional analysis? How can I do one? Substitute the dimensions represented by each variable for that variable in the equation to be analyzed. Algebraically simplify the equation: exponentiate, multiply, divide, add, subtract, cancel. In simplest terms, both sides of the equation should have the same dimensions.

15. Dimensional Analysis (Simple) example: your geometry is a little fuzzy. But you’re pretty sure that the surface area of a sphere is given by: Check it: … and the analysis fails. That formula can’t be correct.

16. Common Powers-of-Ten power of ten prefixsymbolexample 10 9 giga -GGHz 10 6 mega -MMW 10 3 kilo -kkm 10 -2 centi -ccm 10 -3 milli -mmm 10 -6 micro - mm 10 -9 nano -nnm

17. Right-Triangle Trigonometry Basic relationships

18. Right-Triangle Trigonometry Basic relationships

19. Vector and Scalar Quantities Scalar: completely specified by a magnitude (size) Vector: completely specified by both a magnitude and a direction Examples: Distance (scalar): the airport is 15 km away from here. Displacement (vector): the airport is 15 km southwest from here.

20. Vector and Scalar Quantities Scalar: speed, temperature, time, mass, energy, volume, area, length Vector: velocity, acceleration, momentum, force Note that the ability to take on + or – values does not make a quantity a vector. Example: Celsius or Fahrenheit temperature.

21. Vector Properties Symbol: an arrow (line segment with a point)  arrow length shows vector magnitude  arrow points in vector direction Mathematical notation:  bold-font letter A  arrow on top of letter  “hat” on top of letter (usually a unit vector)

22. Vector Mathematics Vectors can be:  Added  Multiplied  by a scalar  by another vector (in two different ways)  Subtracted  Divided

23. Vector Addition Here’s a graphical look at vector addition: we want to add A and B.

24. Vector Addition First, we note that we can translate a vector to any other location without changing it (either magnitude or direction).

25. Vector Addition So, we translate B so that its “tail” coincides with A’s “point.”

26. Vector Addition Now, we draw a third vector from the beginning point of A (its “tail”) to the ending point of B (its “point”). That third vector is the sum: A + B.

27. Vector Multiplication There are three kinds of multiplication that can be done with vectors. First: multiplication by a scalar. Magnitude of the product vector: magnitude of the factor vector times the scalar. Product vector direction: same or opposite the factor vector direction

28. Vector Multiplication There are three kinds of multiplication that can be done with vectors. Second: scalar product of two vectors (“dot product”). The scalar product is zero if the vectors are perpendicular; a maximum value when they are parallel This kind of vector multiplication is commutative

29. Vector Multiplication There are three kinds of multiplication that can be done with vectors. Third: vector product of two vectors (“cross product”). Direction: perpendicular to both A and B, and in accordance with the right-hand rule The vector product is zero if the vectors are parallel; a maximum value when they are perpendicular This kind of vector multiplication is NOT commutative

30. Vector Subtraction This time, we want A – B. Graphically:

31. Vector Subtraction Our first step is to muliply B by the scalar -1, producing – B:

32. Vector Subtraction And now we move – B to the point of A, just as we did before:

33. Vector Subtraction And we draw in the sum: A + (-B) =A – B.

34. Vector Addition by Components Any vector can be expressed as the sum of two vectors, both orthogonal to the coordinate axes. One is the X component, and one is the Y component.

35. Vector Addition by Components Simple right-triangle trigonometry allows us to calculate the magnitudes of these components:

36. Vector Addition by Components Example: we want to add vectors A and B.

37. Vector Addition by Components First: resolve A and B into components. (Replace A and B with component vectors A X, A Y, B X, and B Y, all orthogonal to the coordinate system.)

38. Vector Addition by Components The components of the sum, C, are the sums of the components of A and B. Since the X components are either parallel or antiparallel, their magnitudes add algebraically. The same is true of the Y components.

39. Vector Addition by Components (magnitude)

40. Vector Addition by Components (magnitude)

41. Vector Addition by Components Pythagoras’ theorem yields the magnitude of C: The direction of C:

42. Vector Addition by Components A couple of things to remember:  You are free to define your coordinate system so that it makes your life easier.  These are always correct: as long as you measure  counterclockwise from the +X direction.