January 23, 2006Vectors1 Directions – Pointed Things January 23, 2005

January 23, 2006Vectors2 Today … We complete the topic of kinematics in one dimension with one last simple (??) problem.We complete the topic of kinematics in one dimension with one last simple (??) problem. We start the topic of vectors.We start the topic of vectors. –You already read all about them … right??? There is a WebAssign problem set assigned.There is a WebAssign problem set assigned. The exam date & content may be changed.The exam date & content may be changed. –Decision on Wednesday –Study for it anyway. “may” and “will” have different meanings!

January 23, 2006Vectors3 One more Kinematics Problem A freely falling object requires 1.50 s to travel the last 30.0 m before it hits the ground. From what height above the ground did it fall? 30 meters 1.5 seconds h=??

January 23, 2006 Vectors 4 The Consequences of Vectors You need to recall your trigonometry if you have forgotten it. We will be using sine, cosine, tangent, etc. thingys. You therefore will need calculators for all quizzes from this point on.  A cheap math calculator is sufficient,  Mine cost $11.00 and works just fine for most of my calculator needs.

January 23, 2006 Vectors 5 Hey … I’m lost. How do I get to lake Eola A B NO PROBLEM! Just drive 1000 meters and you will be there. HEY! Where the am I??

January 23, 2006 Vectors 6 The problem:

January 23, 2006 Vectors 7 IS this the way (with direecton now!) to Lake Eola? A B THIS IS A VECTOR

January 23, 2006 Vectors 8 YUM !!

January 23, 2006Vectors9 The following slides were supplied by the publisher … …………… thanks guys!

January 23, 2006 Vectors 10 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.

January 23, 2006 Vectors 11 Vector Notation When handwritten, use an arrow: When printed, will be in bold print: A When dealing with just the magnitude of a vector in print, an italic letter will be used: A or |A| The magnitude of the vector has physical units The magnitude of a vector is always a positive number

January 23, 2006 Vectors 12 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

January 23, 2006 Vectors 13 Equality of Two Vectors Two vectors are equal if they have the same magnitude and the same direction A = B if A = B and they point along parallel lines All of the vectors shown are equal

January 23, 2006 Vectors 14 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

January 23, 2006 Vectors 15 Cartesian Coordinate System Also called rectangular coordinate system x- and y- axes intersect at the origin Points are labeled (x,y)

January 23, 2006 Vectors 16 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,  )

January 23, 2006 Vectors 17 Polar to Cartesian Coordinates Based on forming a right triangle from r and  x = r cos  y = r sin 

January 23, 2006 Vectors 18 Cartesian to Polar Coordinates r is the hypotenuse and  an angle  must be ccw from positive x axis for these equations to be valid

January 23, 2006 Vectors 19 Example 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, and from Equation 3.3,

January 23, 2006 Vectors 20 If the rectangular coordinates of a point are given by (2, y) and its polar coordinates are (r, 30  ), determine y and r.

January 23, 2006 Vectors 21 Adding Vectors Graphically

January 23, 2006 Vectors 22 Adding Vectors Graphically

January 23, 2006 Vectors 23 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  A + B = B + A

January 23, 2006 Vectors 24 Adding Vectors 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  (A + B) + C = A + (B + C)

January 23, 2006 Vectors 25 Vector A has a magnitude of 8.00 units and makes an angle of 45.0° with the positive x axis. Vector B also has a magnitude of 8.00 units and is directed along the negative x axis. Using graphical methods, find (a) the vector sum A + B and (b) the vector difference A  B.

January 23, 2006 Vectors 26 Subtracting Vectors Special case of vector addition If A – B, then use A+(- B) Continue with standard vector addition procedure

January 23, 2006 Vectors 27 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

January 23, 2006 Vectors 28 Each of the displacement vectors A and B shown in Fig. P3.15 has a magnitude of 3.00 m. Find graphically (a) A + B, (b) A  B, (c) B  A, (d) A  2B. Report all angles counterclockwise from the positive x axis.

January 23, 2006 Vectors 29 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

January 23, 2006 Vectors 30 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

January 23, 2006 Vectors 31 Unit Vectors, cont. The symbols represent unit vectors They form a set of mutually perpendicular vectors

January 23, 2006 Vectors 32 Unit Vectors in Vector Notation A x is the same as A x and A y is the same as A y etc. The complete vector can be expressed as

January 23, 2006 Vectors 33 Adding Vectors with Unit Vectors

January 23, 2006 Vectors 34 Adding Vectors Using Unit Vectors – Three Directions Using R = A + B R x = A x + B x, R y = A y + B y and R z = A z + B z etc.

January 23, 2006 Vectors 35 Consider the two vectors and. Calculate (a) A + B, (b) A  B, (c) |A + B|, (d) |A  B|, and (e) the directions of A + B and A  B.

January 23, 2006 Vectors 36 Three displacement vectors of a croquet ball are shown in the Figure, where |A| = 20.0 units, |B| = 40.0 units, and |C| = 30.0 units. Find (a) the resultant in unit-vector notation and (b) the magnitude and direction of the resultant displacement.

January 23, 2006 Vectors 37 The helicopter view in Fig. P3.35 shows two people pulling on a stubborn mule. Find (a) the single force that is equivalent to the two forces shown, and (b) the force that a third person would have to exert on the mule to make the resultant force equal to zero. The forces are measured in units of newtons (abbreviated N).