CH3%20Vectors - CH3 Vectors.notebook Chapter Outline...

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Unformatted text preview: CH3 Vectors.notebook September 20, 2010 Chapter Outline Chapter 3 Vectors Sep 15­5:34 PM 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 Sep 15­5:34 PM • • • • 3.1 Coordinate Systems 3.2 Vector and Scalar Quantities 3.3 Some Properties of Vectors 3.4 Components of a Vector and Unit Vectors Sep 15­5:34 PM Cartesian Coordinate System • Also called rectangular coordinate system • x­ and y­ axes intersect at the origin • Points are labeled (x,y) Sep 15­5:34 PM 1 CH3 Vectors.notebook September 20, 2010 Polar Coordinate System • Origin and reference line are noted • Point is distance r from the origin in the direction of angle θ, ccw from reference line • Points are labeled (r,θ) Sep 15­5:34 PM Trigonometry Review • Given various radius vectors, find • Length and angle • x­ and y­components • Trigonometric functions: sin, cos, tan Sep 15­5:34 PM Sep 20­6:08 PM Cartesian to Polar Coordinates • r is the hypotenuse and θ an angle • θ must be ccw from positive x axis for these equations to be valid Sep 15­5:34 PM 2 CH3 Vectors.notebook Example 3.1 • The Cartesian coordinates of a point in the xy plane are (x,y) = (­3.50, ­2.50) m, as shown in the figure. Find the polar coordinates of this point. Solution: From Equation 3.4, September 20, 2010 Example 3.1, cont. • Change the point in the Please insert x­y plane active fig. 3.3 • Note its Cartesian here coordinates • Note its polar coordinates and from Equation 3.3, Sep 15­5:34 PM Vectors and Scalars • A scalar quantity is completely specified by a single value with an appropriate unit and has no direction. • A vector quantity is completely described by a number and appropriate units plus a direction. Sep 15­5:34 PM Sep 15­5:34 PM 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 Sep 15­5:34 PM 3 CH3 Vectors.notebook Vector Notation • Text uses bold with arrow to denote a vector: • Also used for printing is simple bold print: A • 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 • When handwritten, use an arrow: Sep 15­5:34 PM 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 Sep 15­5:34 PM September 20, 2010 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 Sep 15­5:34 PM Adding Vectors Graphically • 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 Sep 15­5:34 PM 4 CH3 Vectors.notebook Adding Vectors Graphically, cont. • Continue drawing the vectors “tip­to­tail” • 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 September 20, 2010 Adding Vectors Graphically, final • When you have many vectors, just keep repeating the process until all are included • The resultant is still drawn from the tail of the first vector to the tip of the last vector Sep 15­5:34 PM Adding Vectors, Rules • When two vectors are added, the sum is independent of the order of the addition. • This is the Commutative Law of Addition Sep 15­5:34 PM Adding Vectors, Rules cont. • 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 • • Sep 15­5:34 PM Sep 15­5:34 PM 5 CH3 Vectors.notebook Adding Vectors, Rules final • 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 Sep 15­5:34 PM Subtracting Vectors • Special case of vector addition • If , then use • Continue with standard vector addition procedure September 20, 2010 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 Sep 15­5:34 PM Subtracting Vectors, Method 2 • Another way to look at subtraction is to find the vector that, added to the second vector gives you the first vector • • As shown, the resultant vector points from the tip of the second to the tip of the first Sep 15­5:34 PM Sep 15­5:34 PM 6 CH3 Vectors.notebook Multiplying or Dividing a Vector by a Scalar • The result of the multiplication or division of a vector by a scalar 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 Sep 15­5:34 PM Components of a Vector, Introduction • A component is a projection of a vector along an axis • Any vector can be completely described by its components • It is useful to use rectangular components September 20, 2010 Component Method of Adding Vectors • Graphical addition is not recommended when • High accuracy is required • If you have a three­dimensional problem • Component method is an alternative method • It uses projections of vectors along coordinate axes Sep 15­5:34 PM 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 • These are the projections of the vector along the x­ and y­ axes Sep 15­5:34 PM Sep 15­5:34 PM 7 CH3 Vectors.notebook September 20, 2010 Components of a Vector, 2 Components of a Vector • Assume you are given a vector • It can be expressed in terms of two other vectors, and • These three vectors form a right triangle • Sep 15­5:34 PM Components of a Vector, 3 • The x­component of a vector is the projection along the x­axis • The y­component of a vector is the projection along the y­axis • The y­component 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 Sep 15­5:34 PM Components of a Vector, 4 • The components are the legs of the right triangle whose hypotenuse is the length of A • May still have to find θ with respect to the positive x­axis • This assumes the angle θ is measured with respect to the x­axis • If not, do not use these equations, use the sides of the triangle directly Sep 15­5:34 PM Sep 15­5:34 PM 8 CH3 Vectors.notebook Components of a Vector, final • 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 September 20, 2010 Unit Vectors • 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 Sep 15­5:34 PM Unit Vectors, cont. • The symbols represent unit vectors • They form a set of mutually perpendicular vectors in a right­handed coordinate system • Remember, Sep 15­5:34 PM Sep 15­5:34 PM Viewing a Vector and Its Projections • Rotate the axes for various views • Study the projection of a vector on various planes • x, y • x, z • y, z Sep 15­5:34 PM 9 CH3 Vectors.notebook Unit Vectors in Vector Notation • Ax is the same as Ax and Ay is the same as Ay etc. • The complete vector can be expressed as September 20, 2010 Adding Vectors Using Unit Vectors • Using • Then • and so Rx = Ax + Bx and Ry = Ay + By Sep 15­5:34 PM Adding Vectors with Unit Vectors • Note the relationships among the components of the resultant and the components of the original vectors • Rx = Ax + Bx • Ry = Ay + By Sep 15­5:34 PM Sep 15­5:34 PM Three­Dimensional Extension • Using • Then • and so Rx= Ax+Bx, Ry= Ay+By, and Rz =Ax+Bz Sep 15­5:34 PM 10 CH3 Vectors.notebook Example 3.5 – Taking a Hike • A hiker begins a trip by first walking 25.0 km southeast from her car. She stops and sets up her tent for the night. On the second day, she walks 40.0 km in a direction 60.0° north of east, at which point she discovers a forest ranger’s tower. September 20, 2010 Example 3.5 • (A) Determine the components of the hiker’s displacement for each day. Solution: We conceptualize the problem by drawing a sketch as in the figure above. If we denote the displacement vectors on the first and second days by and respectively, and use the car as the origin of coordinates, we obtain the vectors shown in the figure. Drawing the resultant , we can now categorize this problem as an addition of two vectors. Sep 15­5:34 PM Example 3.5 Sep 15­5:34 PM Example 3.5 • We will analyze this problem by using our new knowledge of vector components. Displacement has a magnitude of 25.0 km and is directed 45.0° below the positive x axis. • The second displacement has a magnitude of 40.0 km and is 60.0° north of east. From Equations 3.8 and 3.9, its components are: Its components are: The negative value of Ay indicates that the hiker walks in the negative y direction on the first day. The signs of Ax and Ay also are evident from the figure above. Sep 15­5:34 PM Sep 15­5:34 PM 11 CH3 Vectors.notebook September 20, 2010 Example 3.5 Example 3.5 • (B) Determine the components of the hiker’s resultant displacement for the trip. Find an expression for in terms of unit vectors. Solution: The resultant displacement for the trip has components given by Equation 3.15: Rx = Ax + Bx = 17.7 km + 20.0 km = 37.7 km Ry = Ay + By = ­17.7 km + 34.6 km = 16.9 km In unit­vector form, we can write the total displacement as Sep 15­5:34 PM • Using Equations 3.16 and 3.17, we find that the resultant vector has a magnitude of 41.3 km and is directed 24.1° north of east. Let us finalize. The units of are km, which is reasonable for a displacement. Looking at the graphical representation in the figure above, we estimate that the final position of the hiker is at about (38 km, 17 km) which is consistent with the components of in our final result. Also, both components of are positive, putting the final position in the first quadrant of the coordinate system, which is also consistent with the figure. Sep 15­5:34 PM Some Practice problems Section/s Problem No. 3.1­3.3 5,14,21,36,42,45 Additional Problems 52,55,59,62 Sep 15­5:34 PM Sep 20­6:21 PM 12 CH3 Vectors.notebook Sep 20­6:29 PM September 20, 2010 Sep 20­6:37 PM Sep 20­6:43 PM 13 ...
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This note was uploaded on 09/01/2011 for the course PHY 303 taught by Professor Erskine/tsoi during the Spring '08 term at University of Texas.

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