Lecture 1: Coordinate Systems PDF

Lecture_1  Co-ordinate Systems: Choice of Co-ordinate systems :- Cartesian Co-ordinates: Unit vectors, Infinitesimal line, area and volume element Plane-Polar Co-ordinates: Unit vectors, transformations, Rate of change, velocity and acceleration, Infinitesimal line, area elem...

Lecture_1  Co-ordinate Systems: Choice of Co-ordinate systems :- Cartesian Co-ordinates: Unit vectors, Infinitesimal line, area and volume element Plane-Polar Co-ordinates: Unit vectors, transformations, Rate of change, velocity and acceleration, Infinitesimal line, area element Cylindrical Polar Co-ordinates: Unit vectors and its transformations, Rate of change, velocity and acceleration, Infinitesimal line, area and volume element Spherical Polar Co-ordinates: Unit vectors and its transformations, Rate of change, velocity and acceleration, Infinitesimal line, area and volume element  Co-ordinate Systems: Symbols 1. Cartesian or Rectangular Co-ordinates (Mathematician Rene Des Cartes) (x, y, z, 𝑒Ƹ𝑥 , 𝑒Ƹ𝑦 , 𝑒Ƹ𝑧 ) 2. Plane-Polar Co-ordinates (r, 𝜃, 𝑒Ƹ𝑟 , 𝑒Ƹ𝜃 ) 3. Cylindrical Polar Co-ordinates (𝜌, 𝜙, 𝑧 𝑒𝜌Ƹ , 𝑒Ƹ𝜙 , 𝑒Ƹ𝑧 ) 4. Spherical Polar Co-ordinates (r, 𝜃, 𝜙, 𝑒Ƹ𝑟 , 𝑒Ƹ𝜃 , 𝑒Ƹ𝜙 ) Why do we need different coordinate system? X Y Y 1 1 Origin? -1 1 Location of this point in space ! -1 -1 (0, 0) 𝑟 𝜃 1 -1 X (x, y) == (r, 𝜃) r 𝜃 1 45 1 135 Proper choice of a co-ordinate system can vastly 1 225 simplify a problem !! 1 315 ✓ Cartesian Co-ordinates: Unit vectors, Infinitesimal line, area and volume element A coordinate system consists of four basic elements: 1) Choice of origin 2) Choice of axes (Orthogonal axes intersecting at a common origin) 3) Choice of positive direction for each axis 4) Choice of unit vectors for each axis Vector defines the displacement of point rather than point/position? Write the displacement Vector ? P’ Specify range of coordinate variables x, y & z ? ✓ Cartesian Co-ordinates: Unit vectors, Infinitesimal line, area and volume element ✓ Cartesian Co-ordinates: Unit vectors, Infinitesimal line, area and volume element ✓ Cartesian Co-ordinates: Unit vectors, Infinitesimal line, area and volume element  Used to find the location of particle on a Plane ! Plane-Polar Co-ordinates: Location of a point- 𝑒ෞ 𝑦 𝑟Ԧ = 𝑥𝑒Ƹ𝑥 + y𝑒Ƹ𝑦 𝑒ෞ 𝜃 𝑒ෝ𝑟 −∞ < 𝑥 < ∞; −∞ < 𝑦 < ∞ 𝜃 𝑒ෞ𝑥 𝑟Ԧ = 𝑟𝑒Ƹ𝑟 0 ≤ 𝑟 < ∞; 0 ≤ 𝜃 < 2𝜋 Transformation between Unit vectors: 𝑒ෞ 𝑦 𝑒ෞ 𝜃 𝑒ෝ𝑟 𝜃 −𝑒 ෞ𝑥 𝑒ෞ𝑥 𝑒ෝ𝑟 cos 𝜃 sin 𝜃 𝑒Ƹ𝑥 = e Ƹ𝜃 − sin 𝜃 cos 𝜃 𝑒Ƹ𝑦 Where 𝑒ෝ𝑟 and 𝑒ෞ 𝜃 are perpendicular to each other? How do we check? Above vector is polar coordinates is represented by 𝑟Ԧ = 𝒓 𝑒ෝ𝑟 Motion in polar co-ordinates (Unit vector comparison):- 𝑒ෞ 𝑦 𝑒ෞ 𝜃 𝑒ෝ𝑟 𝜃 𝑒ෞ 𝑦 𝜃 𝑒ෞ 𝜃 −𝑒 ෞ𝑥 𝑒ෞ𝑥 𝑒ෝ𝑟 (x1, y1) −𝑒 ෞ𝑥 𝑒ෞ𝑥 (x2, y2) 𝜃1  Velocity in polar coordinates: Vector is polar coordinates is represented by 𝑟Ԧ = 𝒓 𝑒ෝ𝑟 How to include ‘𝜽′ component? Velocity in Polar coordinates: ෞ𝑟 ) 𝑑(𝒓 𝑒 𝑑𝑟 𝑑𝜃 𝑉= 𝑑𝑡 = 𝑒 ෝ 𝑑𝑡 𝑟 + 𝑟 𝑑𝑡 𝑒ෞ 𝜃 What is the meaning of this expression? 𝑉 = 𝑟ሶ 𝑒ෝ𝑟 + 𝑟𝜃ሶ 𝑒ෞ 𝜃  𝑉 = 𝑟ሶ 𝑒ෝ𝑟 + 𝑟𝜃ሶ 𝑒ෞ 𝜃 Let us find out meaning of each term involved here! Case 1: 𝜽 𝐢𝐬 𝐜𝐨𝐧𝐬𝐭𝐚𝐧𝐭, 𝐯𝐞𝐥𝐨𝐜𝐢𝐭𝐲 𝐢𝐬 𝐫𝐚𝐝𝐢𝐚𝐥. Case 2: r is constant. Velocity is tangential. 𝑉 = 𝑟𝜃ሶ 𝑒ෞ 𝜃 𝑉 = 𝑟ሶ 𝑒ෝ𝑟 For motion in general, both r and 𝜽 changes in time (acceleration generated is known as Coriolis acceleration or real acceleration).