MATHEMATICS FOR ENGINEERING TUTORIAL 5 COORDINATE SYSTEMS

Transcription

1 MATHEMATICS FOR ENGINEERING TUTORIAL 5 COORDINATE SYSTEMS This tutorial is essential pre-requisite material for anyone studying mechanical engineering. This tutorial uses the principle of learning by example. The approach is practical rather than purely mathematical. On completion of this tutorial you should be able to do the following. Explain Cartesian, and Polar Coordinates for two dimensional systems. Explain Cartesian, Cylindrical and Polar Coordinates for three dimensional systems. Convert coordinates from one system to another. Define the Cartesians planes. It is presumed that students are already proficient at basic algebra and trigonometry. D.J.Dunn 1

2 INTRODUCTION We tend to think of a coordinate systems as a way of identifying the position of a point on a two dimensional plane or in a three dimensional space. In mathematics there can be as many dimensions as you need in order to identify a quantity. For example if time is a dimension then in three dimensional space you would need four dimensions to identify it. We will stay with the two and three dimensional world for the time being. 1. TWO DIMENSIONAL COORDINATE SYSTEMS CARTESIAN In a two dimensional system the vertical direction is usually y (positive up) and the horizontal is direction is x (positive to the right). Other letters may be used to designate an axis and they don t have to be vertical and horizontal. The origin o is where the axis cross at x = 0 and y = 0 A point p on this plane has coordinates x, y and this is usually written as p (x,y) Figure 1 POLAR If a line is drawn from the origin to point p it is a radius R and forms an angle θ with the x axis. The angle is positive measured from the x axis in a counter clockwise direction. CONVERSION Figure 2 The two systems are clearly linked as we can convert from one to the other using trigonometry and Pythagoras theorem. y = R sin θ x = R cos θ y/x = tan θ R = (x 2 + y 2 ) ½ Figure 3 WORKED EXAMPLE No. 1 The x, y coordinates of a point is 4, and 6. Calculate the polar coordinates. R = ( ) 1/2 = θ = tan -1 (6/4) = o D.J.Dunn 2

3 2. THREE DIMENSIONAL COORDINATE SYSTEMS CARTESIAN The Cartesian system has three directions x, y and z all at 90 o to each other. Most text sources draw the system with the z axis up and consequently have resulting formulae different form those developed here. As for two dimensional coordinates, the y axis is taken as vertically up and x as horizontal to the right. z must be horizontal into the plane of the page. The origin is the point where the three axes intercept. A point p in space as shown will be reached by travelling so far along the x direction, then so far in the y direction and then in the z direction. The order doesn t really matter. The point is designated p (x, y, z). Figure 4 C YLINDRICAL In the cylindrical system, we imagine a cylinder on the y axis with the centre of the bottom circle at the origin of the Cartesian system. The point p is on the surface of the cylinder at radius R and height y and rotated angle θ from the x axis. The point is defined as a radius R, a vertical height y and a horizontal angle θ. We could use any axis as the reference. The point is designated p(r, y, θ) SPHERICAL Figure 5 In this case we imagine the point as being on a sphere with the centre at the origin of the Cartesian system. Also imagine the point to be at the tip of a right angle triangle as shown with the radius being the hypotenuse. To define the position of the point we need the radius from the origin R, the angle the radius makes to the horizontal plane θ and the angle rotated from the x axis φ. The point is designated p (R, θ, φ). Note that most maths text books define the system differently but this one is best for use in Engineering. Figure 6 C ONVERSION BETWEEN SPHERICAL TO CARTESIAN If we formulate the radius in terms of the coordinates x, y and z by applying Pythagoras we get the following. R 2 = x 2 + y 2 + z 2 θ = tan -1 y/( x 2 + z 2 ) 1/2 φ = tan -1 (z/x) x = R cosθ cosφ y = R sinθ z = R cosθ sinφ The derivation is shown in the next example. D.J.Dunn 3

4 WORKED EXAMPLE No. 2 The x, y z coordinates of a point are 2, 3 and 4. Calculate the spherical coordinates. Figure 7 Consider the triangle formed by x and z R 1 2 = x 2 + z 2 The angle rotated in the z x plane is φ = tan -1 (4/2) = 64.4 o Next consider the triangle made by R, R 1 and y R 2 = y 2 + R 1 2 = y 2 + x 2 + z 2 = = 29 R = The angle of R to the z x plane is θ and found by considering the triangle made by R 1, R and y θ = tan -1 y/r 1 = tan -1 y/( x 2 + z 2 ) 1/2 = tan -1 3/( ) 1/2 = o The spherical coordinates are 5.385, o and 63.4 o Note that the solutions for conversion may be stated as follows. R 2 = R z 2 = x 2 + y 2 + z 2 θ = tan -1 y/( x 2 + z 2 ) 1/2 φ = tan -1 (z/x) WORKED EXAMPLE No. 3 The spherical coordinates of a point are p (7, 60 o -45 o ). Calculate the Cartesian coordinates. x = R sinθ cosφ y = R sinθ z = R cosθ sinφ x = 7 cos(60)cos(-45) =2.475 y = 7 sin(60) = 6.06 z = 7 cos(60)sin(-45) = R 2 = x 2 + y 2 + z 2 = 49 R = 7 so it is correct. D.J.Dunn 4

5 3. THE CARTESIAN PLANES The Cartesian planes are the planes between any two axis and defined by those axis as the x y plane, z x plane and z y plane. Figure 8 SELF ASSESSMENT EXERCISE 1. Find the polar coordinates of the point p(12,20) (23.32 and 59 o ) 2. Find the Cartesian coordinates of the point p (6-40 o ) (4.6, -3.86) 3. Find the spherical coordinates of the point p (8,4,2). (9.165, 25.9 o and 14 o ) 4. Find the spherical coordinates of the point p (4,-4, 4). (6.928, o and 45 o ) 5. Find the Cartesian coordinates of the point p (10, 60 o, 45 o ) (3.536, 8.66 and 3.536) 6. Find the Cartesian coordinates of the point p (3, -30 o, 50 o ) (1.67, -1.5 and 1.99) D.J.Dunn 5