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Lecture 5 - Last Lecture Physics 231 General University...

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Physics 231 General University Physics Spring 2007 Lecture 5 Last Lecture Constant acceleration Kinematic equations Today Properties of Vectors Components Adding Vectors Algebraically 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 Cartesian Coordinate System Also called rectangular coordinate system x - and y - axes intersect at the origin Points are labeled ( x , y ) Scales (m, cm, mi, etc.) Rene Descartes (cm) (cm) 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 ) Scales
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Polar to Cartesian Coordinates Based on forming a right triangle from r and θ x = r cos θ y = r sin θ Cartesian to Polar Coordinates r is the hypotenuse and θ an angle θ must be counter-clock- wise (ccw) from positive x axis for these equations to be valid 2 2 tan y x r x y θ = = + 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, and from Equation 3.3, 2 2 2 2 ( 3.50 m) ( 2.50 m) 4.30 m r x y = + = + − = 2.50 m tan 0.714 3.50 m 216 y x θ θ = = = = ° Note: not 36 D 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. 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 A G 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
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