Introduction and Vectors

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1 Introduction and Vectors
Chapter 1 Introduction and Vectors

2 About Physics Provides a quantitative understanding of certain phenomena that occur in our universe Based on experimental observations and mathematical analysis Used to develop theories that explain the phenomena being studied and that relate to other established theories

3 Theory and Experiments
Should complement each other When a discrepancy occurs, theory may be modified Theory may apply to limited conditions Example: Newtonian Mechanics is confined to objects traveling slowly with respect to the speed of light Used to try to develop a more general theory

4 1.1 Standards of Length, Mass, and Time
SI system– Systéme International Main system used in this text Consists of a system of definitions and standards to describe fundamental physical quantities

5 Length Units Historically length has had many definitions
SI – meter, m Historically length has had many definitions Length is now defined in terms of a meter – the distance traveled by light in a vacuum during a given time See table 1.1 for some examples of lengths

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7 Mass Units SI – kilogram, kg Defined in terms of kilogram, based on a specific cylinder kept at the International Bureau of Weights and Measures See table 1.2 for masses of various objects

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9 Time Units Seconds, s Historically defined in terms of a solar day, as well as others Now defined in terms of the oscillation of radiation from a cesium atom See table 1.3 for some approximate time intervals

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11 Number Notation When writing out numbers with many digits, spacing in groups of three will be used No commas Examples: 25 100

12 Reasonableness of Results
When solving problem, you need to check your answer to see if it seems reasonable Reviewing the tables of approximate values for length, mass, and time will help you test for reasonableness

13 Systems of Measurements, SI Summary
SI System Most often used in the text Almost universally used in science and industry Length is measured in meters (m) Time is measured in seconds (s) Mass is measured in kilograms (kg)

14 Systems of Measurements, US Customary
Everyday units Length is measured in feet Time is measured in seconds Mass is measured in slugs

15 Prefixes(1) Prefixes correspond to powers of 10
Each prefix has a specific name Each prefix has a specific abbreviation

16 Prefixes(2) The prefixes can be used with any base units
They are multipliers of the base unit Examples: 1 mm = 10-3 m 1 mg = 10-3 g

17 1.2 Dimensional Analysis In mechanics, three fundamental quantities are used Length Mass Time Will also use derived quantities These are other quantities that can be expressed as a mathematical combination of fundamental quantities

18 Density Density is an example of a derived quantity
It is defined as mass per unit volume Units are kg/m3

19 Dimensional Analysis(1)
Technique to check the correctness of an equation or to assist in deriving an equation Dimensions (length, mass, time, combinations) can be treated as algebraic quantities Add, subtract, multiply, divide Both sides of equation must have the same dimensions

20 Dimensional Analysis(2)
Dimension has a specific meaning – it denotes the physical nature of a quantity Dimensions are denoted with square brackets Length – L Mass – M Time – T

21 Dimensional Analysis(3)
Cannot give numerical factors: this is its limitation Dimensions of some common quantities are given below

22 Dimensional Analysis(example)
Given the equation: x = 1/2 a t2 Check dimensions on each side: The T2’s cancel, leaving L for the dimensions of each side The equation is dimensionally correct There are no dimensions for the constant

23 1.3 Conversion of Units When units are not consistent, you may need to convert to appropriate ones Units can be treated like algebraic quantities that can cancel each other out See Appendix A for an extensive list of conversion factors

24 Conversion Always include units for every quantity, you can carry the units through the entire calculation Multiply original value by a ratio equal to one The ratio is called a conversion factor Example

25 1.4 Order of Magnitude Approximation based on a number of assumptions
May need to modify assumptions if more precise results are needed Order of magnitude is the power of 10 that applies In order of magnitude calculations, the results are reliable to within about a factor of 10

26 Uncertainty in Measurements
There is uncertainty in every measurement, this uncertainty carries over through the calculations Need a technique to account for this uncertainty We will use rules for significant figures to approximate the uncertainty in results of calculations

27 1.5 Significant Figures A significant figure is one that is reliably known Zeros may or may not be significant Those used to position the decimal point are not significant To remove ambiguity, use scientific notation In a measurement, the significant figures include the first estimated digit

28 Significant Figures, examples
m has 2 significant figures The leading zeroes are placeholders only Can write in scientific notation to show more clearly: 7.5 x 10-3 m for 2 significant figures 10.0 m has 3 significant figures The decimal point gives information about the reliability of the measurement 1500 m is ambiguous Use 1.5 x 103 m for 2 significant figures Use 1.50 x 103 m for 3 significant figures Use x 103 m for 4 significant figures

29 Operations with Significant Figures – Multiplying or Dividing
When multiplying or dividing, the number of significant figures in the final answer is the same as the number of significant figures in the quantity having the lowest number of significant figures. Example: m x 2.45 m = 62.6 m2 The 2.45 m limits your result to 3 significant figures

30 Operations with Significant Figures – Adding or Subtracting
When adding or subtracting, the number of decimal places in the result should equal the smallest number of decimal places in any term in the sum. Example: 135 cm cm = 138 cm The 135 cm limits your answer to the units decimal value

31 Operations With Significant Figures – Summary
The rule for addition and subtraction are different than the rule for multiplication and division For adding and subtracting, the number of decimal places is the important consideration For multiplying and dividing, the number of significant figures is the important consideration

32 Rounding Last retained digit is increased by 1 if the last digit dropped is 5 or above Last retained digit is remains as it is if the last digit dropped is less than 5 If the last digit dropped is equal to 5, the retained should be rounded to the nearest even number Saving rounding until the final result will help eliminate accumulation of errors

33 1.6 Coordinate Systems Used to describe the position of a point in space Coordinate system consists of A fixed reference point called the origin Specific axes with scales and labels Instructions on how to label a point relative to the origin and the axes

34 Cartesian Coordinate System
Also called rectangular coordinate system x- and y- axes intersect at the origin Points are labeled (x,y)

35 Polar Coordinate System
Origin and reference line are noted Point is distance r from the origin in the direction of angle , ccw from reference line Points are labeled (r,)

36 Polar to Cartesian Coordinates
Based on forming a right triangle from r and q x = r cos q y = r sin q

37 Cartesian to Polar Coordinates
r is the hypotenuse and q an angle q must be ccw from positive x axis for these equations to be valid

38 1.7 Vectors and Scalars A scalar is a quantity that is completely specified by a positive or negative number with an appropriate unit and has no direction. A vector is a physical quantity that must be described by a magnitude (number) and appropriate units plus a direction.

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40 Some Notes About Scalars
Some examples Temperature Volume Mass Time intervals Rules of ordinary arithmetic are used to manipulate scalar quantities

41 Vector Example A particle travels from A to B along the path shown by the dotted red line This is the distance traveled and is a scalar The displacement is the solid line from A to B The displacement is independent of the path taken between the two points Displacement is a vector

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43 Other Examples of Vectors
Many other quantities are also vectors Some of these include Velocity Acceleration Force Momentum

44 Vector Notation When handwritten, use an arrow:
When printed, will be in bold print with an arrow: When dealing with just the magnitude of a vector in print, an italic letter will be used: A or | | The magnitude of the vector has physical units The magnitude of a vector is always a positive number

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46 1.8 Some Properties of Vectors
Equality of two vectors Two vectors are equal if they have the same magnitude and the same direction if A = B and they point along parallel lines All of the vectors shown are equal

47 Adding Vectors When adding vectors, their directions must be taken into account Units must be the same Graphical Methods Use scale drawings Algebraic Methods More convenient

48 Adding Vectors Graphically(1)
Choose a scale Draw the first vector with the appropriate length and in the direction specified, with respect to a coordinate system Draw the next vector with the appropriate length and in the direction specified, with respect to a coordinate system whose origin is the end of vector and parallel to the coordinate system used for

49 Adding Vectors Graphically(2)
Continue drawing the vectors “tip-to-tail” The resultant is drawn from the origin of to the end of the last vector Measure the length of and its angle Use the scale factor to convert length to actual magnitude

50 Adding Vectors Graphically(3)
When you have many vectors, just keep repeating the process until all are included The resultant is still drawn from the origin of the first vector to the end of the last vector

51 Adding Vectors, Rules(1)
When two vectors are added, the sum is independent of the order of the addition. This is the commutative law of addition

52 Adding Vectors, Rules (2)
When adding three or more vectors, their sum is independent of the way in which the individual vectors are grouped This is called the Associative Property of Addition

53 Adding Vectors, Rules (3)
When adding vectors, all of the vectors must have the same units All of the vectors must be of the same type of quantity For example, you cannot add a displacement to a velocity

54 Negative of a Vector The negative of a vector is defined as the vector that, when added to the original vector, gives a resultant of zero Represented as – The negative of the vector will have the same magnitude, but point in the opposite direction

55 Subtracting Vectors Special case of vector addition
Continue with standard vector addition procedure

56 Multiplying or Dividing a Vector by a Scalar
The result of the multiplication or division is a vector The magnitude of the vector is multiplied or divided by the scalar If the scalar is positive, the direction of the result is the same as of the original vector If the scalar is negative, the direction of the result is opposite that of the original vector

57 Multiplying Vectors Two vectors can be multiplied in two different ways One is the scalar product Also called the dot product The other is the vector product Also called the cross product These products will be discussed as they arise in the text

58 1.9 Components of a Vector A component is a part
It is useful to use rectangular components These are the projections of the vector along the x- and y-axes

59 Vector Component Terminology
are the component vectors of They are vectors and follow all the rules for vectors Ax and Ay are scalars, and will be referred to as the components of A The combination of the component vectors is a valid substitution for the actual vector

60 Components of a Vector(1)
The x-component of a vector is the projection along the x-axis The y-component of a vector is the projection along the y-axis When using this form of the equations, q must be measured ccw from the positive x-axis

61 Components of a Vector(2)
The y-component is moved to the end of the x-component This is due to the fact that any vector can be moved parallel to itself without being affected This completes the triangle

62 Components of a Vector(3)
The components are the legs of the right triangle whose hypotenuse is May still have to find θ with respect to the positive x-axis Use the signs of Ax and Ay

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64 Components of a Vector (4)
The components can be positive or negative and will have the same units as the original vector The signs of the components will depend on the angle

65 Unit Vectors (1) A unit vector is a dimensionless vector with a magnitude of exactly 1. Unit vectors are used to specify a direction and have no other physical significance

66 Unit Vectors (2) The symbols
represent unit vectors in the x, y and z directions They form a set of mutually perpendicular vectors

67 Unit Vectors in Vector Notation
is the same as Ax and is the same as Ay etc. The complete vector can be expressed as

68 Adding Vectors Using Unit Vectors
Then Then Rx = Ax + Bx and Ry = Ay + By

69 Adding Vectors with Unit Vectors – Diagram

70 Adding Vectors Using Unit Vectors – Three Directions
Rx = Ax + Bx , Ry = Ay + By and Rz = Az + Bz etc.

71 Trig Function Warning The component equations (Ax = A cos q and Ay = A sin q) apply only when the angle is measured with respect to the x-axis (preferably ccw from the positive x-axis). The resultant angle (tan q = Ay / Ax) gives the angle with respect to the x-axis. You can always think about the actual triangle being formed and what angle you know and apply the appropriate trig functions

72 1.10 Model Building A model is a simplified substitution for the real problem that allows us to solve the problem in a relatively simple way Make predictions about the behavior of the system The predictions will be based on interactions among the components and/or Based on the interactions between the components and the environment As long as the predictions of the model agree with the actual behavior of the real system, the model is valid

73 Particle Model The particle model allows the replacement of an extended object with a particle which has mass, but zero size Two conditions for using the particle model are The size of the actual object is of no consequence in the analysis of its motion Any internal processes occurring in the object are of no consequence in the analysis of its motion

74 Categories of Models There are four categories of models that will be used in this book Geometric model Simplification model Analysis model Structural model

75 Geometric Models Form a geometric construction that represents the real situation Perform the analysis of the geometric construction

76 Simplification Model The details that are not significant in determining the outcome of the problem are ignored The particle model is an example Another example is to assume friction is negligible in many cases

77 Analysis Model Based on general types of problems that have been solved before You cast a new problem into the form of one seen (and solved) before

78 Structural Model Used to understand the behavior of a system that is different in scale from our macroscopic world Can be used for actual systems much larger or much smaller Used for systems you cannot interact with directly

79 Representation Related to modeling is that of forming alternative representations of the problem A representation is a method of viewing or presenting the information relating to the problem Considering alternative representations can help you think about the information in the problem in several different ways to help you understand and solve it.

80 Types of Representations
Mental representation Pictorial representation Simplified pictorial representation Graphical representation Tabular representation Mathematical representation

81 Mental Representation
Imagine the scene described Let time progress so you understand the problem and can make predictions about changes that will occur over time A critical step in solving every problem

82 Pictorial Representation
Drawing a picture of the situation The representation describes what you would see if you were observing the situation described in the problem Any coordinate system included would be the x- and y-axes

83 Simplified Pictorial Representation
Redraw the pictorial representation without complicating details Using the particle model could be part of the simplification This representation will be used extensively throughout the text

84 Graphical Representation
A graph can be very useful in describing a situation The axes of the graph may be any two related variables A graphical representation is generally not something you would see when actually observing the situation

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86 Tabular Representation
Organizing information in a table form may make it clearer Periodic table is an example A table of known and unknown quantities may be helpful in solving a problem

87 Mathematical Representation
The ultimate goal in solving problems is often the mathematical representation Through various representations that help in understanding the problem, you can arrive at the one or more equations that represent the situation The result can be solved for mathematically

88 General Problem Solving Strategy
Conceptualize Categorize Analyze Finalize

89 Problem Solving –Conceptualize
Read the problem At least twice Be sure you understand the situation and the nature of the problem Make a quick drawing of the situation This is the pictorial representation Generate other representations that would be helpful Focus on the expected result Think about units Think about what a reasonable answer should be

90 Problem Solving – Categorize
Simplify the problem Draw a simplified representation Can you ignore air resistance? Model objects as particles Classify the type of problem Plug-in problem – just substitute numbers Try to identify similar problems you have already solved The analysis representation

91 Problem Solving – Analyze
Select the relevant equation(s) to apply Solve for the unknown variable Substitute appropriate numbers Calculate the results Include units Round the result to the appropriate number of significant figures

92 Problem Solving – Finalize
Check your result Does it have the correct units? Does it agree with your conceptualized ideas? Look at limiting situations to be sure the results are reasonable Compare the result with those of similar problems

93 Problem Solving – Some Final Ideas
When solving complex problems, you may need to identify sub-problems and apply the problem-solving strategy to each sub-part. These steps can be a guide for solving problems in this course