1. Introduction and Mathematical Concepts
Chapter 1
Introduction and Mathematical Concepts

2. Table of Contents The Nature of Physics Units
Role of Units in Problem Solving
Trigonometry
Scalars and Vectors
Vector Addition and Subtraction
Components of a Vector
Addition of Vectors by Means of Components
Other Stuff

3. Chapter 1: Introduction and Mathematical Concepts
Section 1 – The Nature of Physics

4. What is Physics? The “Fundamental Science”
Study of matter and how it moves through space-time
Applications of concepts such as Energy, and Force
The general analysis of the natural world
“understand” and predict how our universe behaves

5. Chapter 1: Introduction and Mathematical Concepts
Section 2 - Units

6. Units To “understand” nature, we must first study what it does
Must have/use a universal way of describing what nature does
Systems of measurement
“British” (American)
Metric SI

7. Base Units Most fundament forms of measurement Mass – kilogram (kg)
Length – meter (m)
Time – second (s)
Count – mole (mol)
Temperature – kelvin (K)
Current – ampere (A)
Luminous Intensity – candela (cd)

8. SI Features Derived Units Common combinations of base units
e.g.: area, force, pressure
Prefixes
Adjust scale of measurement
Metric – powers of 10
SI – powers of 1000

9. SI Prefixes 1024 yotta (Y) 1021 zetta (Z) 1018 exa (E) 1015 peta (P)
1012 tera (T)
109 giga (G)
106 mega (M)
103 kilo (k)
10-3 milli (m)
10-6 micro (µ)
10-9 nano (n)
10-12 pico (p)
10-15 femto (f)
10-18 atto (a)
10-21 zepto (z)
10-24 yocto (y)

10. Chapter 1: Introduction and Mathematical Concepts
Section 3
The Role of Units in Problem Solving

11. Conversion of Units Remember from algebra…
Multiplying by 1 does not change number
If 1 m = 1000 mm, then
1 m/1000 mm = 1

12. Question #1
When we measure physical quantities, the units may be anything that is reasonable as long as they are well defined. It’s usually best to use the international standard units. Density may be defined as the mass of an object divided by its volume. Which of the following units would probably not be acceptable units of density?
a)gallons/liter b)kilograms/m3 c)pounds/ft3
d)slugs/yd3 e)grams/milliliter

13. Question #2
A car starts from rest on a circular track with a radius of 150 m. Relative to the starting position, what angle has the car swept out when it has traveled 150 m along the circular track?
a) 1 radian b) /2 radians c)  radians
d) 3/2 radians e) 2 radians

14. Question #3
A section of a river can be approximated as a rectangle that is 48 m wide and 172 m long. Express the area of this river in square kilometers.
a) × 103 km2 b) km2
c) × 103 km2 d) km2
e) × 102 km2

15. Question #4
If one inch is equal to 2.54 cm, express 9.68 inches in meters.
a) m b) m
c) m d) m
e) m

16. Dimensional Analysis When in doubt, look at the units
Since units are part of the number, units must balance out for a valid equation
By analyzing the units, you can determine if your solution is correct.
If the units from your calculation do not give you the units you need, you have an error

17. Example DIMENSIONAL ANALYSIS [L] = length [M] = mass [T] = time
Is the following equation dimensionally correct?

18. Question #5
Using the dimensions given for the variables in the table, determine which one of the following expressions is correct. a) b) c)
d) e)

19. Question #6
Given the following equation: y = cnat2, where n is an integer with no units, c is a number between zero and one with no units, the variable t has units of seconds and y is expressed in meters, determine which of the following statements is true.
a) a has units of m/s and n =1.
b) a has units of m/s and n =2.
c) a has units of m/s2 and n =1.
d) a has units of m/s2 and n =2.
e) a has units of m/s2, but value of n cannot be determined through dimensional analysis.

20. Question #7 Approximately how many seconds are there in a century?
a) 86,400 s
b) 5.0 × 106 s
c) 3.3 × 1018 s
d) 3.2 × 109 s
e) 8.6 × 104 s

21. Chapter 1: Introduction and Mathematical Concepts
Section 4 - Trigonometry

22. Basics you should remember…

23. Basics you should remember…

24. Question #8 Determine the angle  in the right triangle shown.
b) 62.0
c) 35.5
d) 28.0
e) 41.3

25. Question #9
Determine the length of the side of the right triangle labeled x.
a) m
b) m
c) m
d) m
e) m

26. Question #10
Determine the length of the side of the right triangle labeled x.
a) km
b) km
c) km
d) km
e) km

27. Chapter 1: Introduction and Mathematical Concepts
Section 5 – Scalar & Vectors

28. Scalar & Vector
A scalar quantity is one that can be described by a single number:
temperature, speed, mass
A vector quantity deals inherently with both magnitude and direction:
velocity, force, displacement

29. More on Vectors
Arrows are used to represent vectors. The direction of the arrow gives the direction of the vector.
By convention, the length of a vector arrow is proportional to the magnitude of the vector.
8 lb
4 lb

30. Question #11
Which one of the following statements is true concerning scalar quantities?
a) Scalar quantities must be represented by base units.
b) Scalar quantities have both magnitude and direction.
c) Scalar quantities can be added to vector quantities using rules of trigonometry.
d) Scalar quantities can be added to other scalar quantities using rules of trigonometry.
e) Scalar quantities can be added to other scalar quantities using rules of ordinary addition.

31. Chapter 1: Introduction and Mathematical Concepts
Section 6
Vector Addition and Subtraction

32. Graphical Addition of vectors
Remember length of arrow is proportional to magnitude
Angle of arrow proportional to direction
Place tail of 2nd vector at tip of 1st
Resultant starts at 1st and ends at 2nd q

33. Graphical Subtraction of Vectors
Same as addition, multiply value by (-1)
Resultant is still tail to tip q

34. Question #12
Which expression is false concerning the vectors shown in the sketch? a) b) c)
d) C < A + B
e) A2 + B2 = C2

35. Chapter 1: Introduction and Mathematical Concepts
Section 7
Components of a Vector

36. Vector Component

37. Scalar Components
It is often easier to work with the scalar components
rather than the vector components.
In math, they are called i and j

38. Example Problem
A displacement vector has a magnitude of 175 m and points at an angle of 50.0 degrees relative to the x axis. Find the x and y components of this vector.

39. Question #13
During the execution of a play, a football player carries the ball for a distance of 33 m in the direction 76° north of east. To determine the number of meters gained on the play, find the northward component of the ball’s displacement.
a) 8.0 m b) 16 m
c) 24 m d) 28 m
e) 32 m

40. Question #14
Vector has components ax = 15.0 and ay = What is the approximate magnitude of vector ?
a) b) 24.0
c) d) 6.87
e) 17.5

41. Question #15 a) 4.46 m b) 11.7 m c) 5.02 m d) 7.97 m e) 14.3 m
Vector has a horizontal component ax = 15.0 m and makes an angle  = 38.0 with respect to the positive x direction. What is the magnitude of ay, the vertical component of vector ?
a) m b) m
c) m d) m
e) m

42. Chapter 1: Introduction and Mathematical Concepts
Section 8
Addition of Vectors by Means of Components

43. Addition using components

44. Quesiton #16,17
The drawing above shows two vectors A and B, and the drawing on the right shows their components. Each of the angles θ = 31°.
When the vectors A and B are added, the resultant vector is R, so that R = A + B. What are the values for Rx and Ry, the x- and y-components of R?
Rx =     m
Ry =     m

45. Question #18,19
The displacement vectors A and B, when added together, give the resultant vector R, so that R = A + B. Use the data in the drawing and the fact that φ = 27° to find the magnitude R of the resultant vector and the angle θ that it makes with the +x axis.
R =     m
θ =     degrees

46. Question #20
Use the component method of vector addition to find the resultant of the following three vectors:            = 56 km, east            = 11 km, 22° south of east            = 88 km, 44° west of south
A) 66 km, 7.1° west of south B) 97 km, 62° south of east
C) 68 km, 86° south of east D) 52 km, 66° south of east
E) 81 km, 14° west of south

47. Adding Multiple Vectors
Adding Vectors F2 F3 F4 F1

48. Adding Multiple Vectorsq2 F2 F1
43.3
25.0 q1
-70.7
70.7
-10.3
-28.2 q3 q4 F4
20.0
-34.6 F3
-17.7
32.9
F1 = 50 N q1 = 30o
F2 = 100 N q2 = 135o
F3 = 30 N q3 = 250o
F4 = 40 N q4 = 300o

49. Adding Multiple Vectors
FR = 37.4 N qR f
-17.7
32.9

50. Now You Try: q2 F2 F1 q1 q3 q4 F4 F3 F1 = 90 N q1 = 45o

51. Chapter 1: Introduction and Mathematical Concepts
“Section 9”
Additional Stuff You Should Know

52. Basic Rules Multiplication of 1
Multiplying a number by 1 doesn’t change it
Addition Property of Equality
Add the same thing to both sides
Multiplication Property of Equality
Multiply both sides of equation by same thing
“undo” function on both sides

53. Inverse “Functions” for algebra
Addition
Multiplication
Square
Sine
log ln
Add opposite (“-”)
Multiply by inverse
Square root
Arcsine
10x ex
(“ “)

54. Graphing Linear equations y = mx + b Quadratic equations
y = ax2 + bx + c
y = a(x-h)2 + k
Wave equations
y = A sin (w x + q) + d

59. END