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Chapter 1 – Math Review

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Surveyors use accurate measures of magnitudes and directions to create scaled maps of large regions. Vectors

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Objectives: After completing this module, you should be able to: Demonstrate that you meet mathematics expectations: unit analysis, algebra, scientific notation, and right-triangle trigonometry.Demonstrate that you meet mathematics expectations: unit analysis, algebra, scientific notation, and right-triangle trigonometry. Define and give examples of scalar and vector quantities.Define and give examples of scalar and vector quantities. Determine the components of a given vector.Determine the components of a given vector. Find the resultant of two or more vectors.Find the resultant of two or more vectors. Demonstrate that you meet mathematics expectations: unit analysis, algebra, scientific notation, and right-triangle trigonometry.Demonstrate that you meet mathematics expectations: unit analysis, algebra, scientific notation, and right-triangle trigonometry. Define and give examples of scalar and vector quantities.Define and give examples of scalar and vector quantities. Determine the components of a given vector.Determine the components of a given vector. Find the resultant of two or more vectors.Find the resultant of two or more vectors.

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Expectations You must be able convert units of measure for physical quantities.You must be able convert units of measure for physical quantities. Convert 40 m/s into kilometers per hour. 40--- x ---------- x -------- = 144 km/h m s 1 km 1000 m 3600 s 1 h

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Expectations (Continued) You must be able to work in scientific notation.You must be able to work in scientific notation. Evaluate the following: (6.67 x 10 -11 )(4 x 10 -3 )(2) (8.77 x 10 -3 ) 2 F = -------- = ------------ Gmm’ r 2 F = 6.94 x 10 -9 N = 6.94 nN

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Expectations (Continued) You must be familiar with SI prefixesYou must be familiar with SI prefixes The meter (m) 1 m = 1 x 10 0 m 1 Gm = 1 x 10 9 m 1 nm = 1 x 10 -9 m 1 Mm = 1 x 10 6 m 1 m = 1 x 10 -6 m 1 km = 1 x 10 3 m 1 mm = 1 x 10 -3 m

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Expectations (Continued) You must have mastered right-triangle trigonometry.You must have mastered right-triangle trigonometry. y x R y = R sin x = R cos R 2 = x 2 + y 2

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Science of Measurement We begin with the measurement of length: its magnitude and its direction. Length Weight Time

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Some Physics Quantities Vector - quantity with both magnitude (size) and direction Scalar - quantity with magnitude only Scalar - quantity with magnitude only Vectors: Displacement Displacement Velocity Velocity Acceleration Acceleration Momentum Momentum Force ForceScalars: Distance Distance Speed Speed Time Time Mass Mass Energy Energy

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Mass vs. Weight On the moon, your mass would be the same, but the magnitude of your weight would be less. Mass Scalar (no direction) Scalar (no direction) Measures the amount of matter in an object Measures the amount of matter in an object Weight Vector (points toward center of Earth) Vector (points toward center of Earth) Force of gravity on an object Force of gravity on an object

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Vectors The length of the arrow represents the magnitude (how far, how fast, how strong, etc, depending on the type of vector).The length of the arrow represents the magnitude (how far, how fast, how strong, etc, depending on the type of vector). The arrow points in the directions of the force, motion, displacement, etc. It is often specified by an angle. Vectors are represented with arrows 42° 5 m/s

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Units Quantity... Unit (symbol) Displacement & Distance... meter (m)Displacement & Distance... meter (m) Time... second (s)Time... second (s) Velocity & Speed... (m/s)Velocity & Speed... (m/s) Acceleration... (m/s 2 )Acceleration... (m/s 2 ) Mass... kilogram (kg)Mass... kilogram (kg) Momentum... (kg · m/s)Momentum... (kg · m/s) Force...Newton (N)Force...Newton (N) Energy... Joule (J)Energy... Joule (J) Units are not the same as quantities!

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SI Prefixes Little Guys Big Guys

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Distance: A Scalar Quantity A scalar quantity: Contains magnitude only and consists of a number and a unit. (20 m, 40 mi/h, 10 gal) A B Distance is the length of the actual path taken by an object. s = 20 m

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Displacement—A Vector Quantity A vector quantity: Contains magnitude AND direction, a number, unit & angle. (12 m, 30 0 ; 8 km/h, N) A B D = 12 m, 20 o Displacement is the straight-line separation of two points in a specified direction.Displacement is the straight-line separation of two points in a specified direction.

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Distance and Displacement Net displacement: 4 m,E 6 m,W D What is the distance traveled? 10 m !! D = 2 m, W Displacement is the x or y coordinate of position. Consider a car that travels 4 m, E then 6 m, W.Displacement is the x or y coordinate of position. Consider a car that travels 4 m, E then 6 m, W. x= +4 x = +4 x= -2 x = -2

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Identifying Direction A common way of identifying direction is by reference to East, North, West, and South. (Locate points below.) 40 m, 50 o N of E EW S N 40 m, 60 o N of W 40 m, 60 o W of S 40 m, 60 o S of E Length = 40 m 50 o 60 o

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Identifying Direction Write the angles shown below by using references to east, south, west, north. EW S N 45 o EW N 50 o S Click to see the Answers... 50 0 S of E 45 0 W of N

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Vectors and Polar Coordinates Polar coordinates (R, ) are an excellent way to express vectors. Consider the vector 40 m, 50 0 N of E, for example. 0o0o 180 o 270 o 90 o 0o0o 180 o 270 o 90 o R R is the magnitude and is the direction. 40 m 50 o

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Vectors and Polar Coordinates (R, ) = 40 m, 50 o (R, ) = 40 m, 120 o (R, ) = 40 m, 210 o (R, ) = 40 m, 300 o 50 o 60 o 0o0o 180 o 270 o 90 o 120 o Polar coordinates (R, ) are given for each of four possible quadrants: 210 o 300 0

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Example 1: Find the height of a building if it casts a shadow 90 m long and the indicated angle is 30 o. 90 m 30 0 The height h is opposite 30 0 and the known adjacent side is 90 m. h h = (90 m) tan 30 o h = 57.7 m

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Finding Components of Vectors A component is the effect of a vector along other directions. The x and y components of the vector (R, are illustrated below. x y R x = R cos y = R sin Finding components: Polar to Rectangular Conversions

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Example 2: A person walks 400 m in a direction of 30 o N of E. How far is the displacement east and how far north? x y R x = ? y = ? 400 m E N The y-component (N) is OPP: The x-component (E) is ADJ: x = R cos y = R sin E N

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Example 2 (Cont.): A 400-m walk in a direction of 30 o N of E. How far is the displacement east and how far north? x = R cos x = (400 m) cos 30 o = +346 m, E x = ? y = ? 400 m E N Note: x is the side adjacent to angle 30 0 ADJ = HYP x Cos 30 0 The x-component is: R x = +346 m

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Example 2 (Cont.): A 400-m walk in a direction of 30 o N of E. How far is the displacement east and how far north? y = R sin y = (400 m) sin 30 o = + 200 m, N x = ? y = ? 400 m E N OPP = HYP x Sin 30 0 The y-component is: R y = +200 m Note: y is the side opposite to angle 30 0

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Example 2 (Cont.): A 400-m walk in a direction of 30 o N of E. How far is the displacement east and how far north? R x = +346 m R y = +200 m 400 m E N The x- and y- components are each + in the first quadrant Solution: The person is displaced 346 m east and 200 m north of the original position.

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Signs for Rectangular Coordinates First Quadrant: R is positive (+) 0 o > < 90 o x = +; y = + x = R cos y = R sin + + 0o0o 90 o R

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Signs for Rectangular Coordinates Second Quadrant: R is positive (+) 90 o > < 180 o x = - ; y = + x = R cos y = R sin + R 180 o 90 o

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Signs for Rectangular Coordinates Third Quadrant: R is positive (+) 180 o > < 270 o x = - y = - x = R cos y = R sin - R 180 o 270 o

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Signs for Rectangular Coordinates Fourth Quadrant: R is positive (+) 270 o > < 360 o x = + y = - x = R cos y = R sin 360 o + R 270 o

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Resultant of Perpendicular Vectors Finding resultant of two perpendicular vectors is like changing from rectangular to polar coord. R is always positive; is from + x axis x y R

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