PH101 Lecture 2 Coordinate systems Cartesian coordinate System in plane ͭȤ P Y In Cartesian coordinate position is ͬȤ represented by ( ͬ, ͭ). Q ͭȤ P ( ͬ, ͭ) ͬ «¬ = È = Î Îȥ + Ï Ïȥ ͬȤ ͭ Note: O X • ͬȤ and ͭȤ are unit vectors pointing the increasing direction of ͬ and ͭ. Cartesian Coordinate System • ͬȤ and ͭȤ are orthogonal and points in the same direction everywhere or for any location (x, y). Another way of looking unit vector Cartesian coordinate in plane ͬȤ is the unit vector perpendicular to ͬ = ͕͗ͣͧͨͨ͢͢ line (surface) ͭȤ is the unit vector perpendicular to ͭ = ͕͗ͣͧͨͨ͢͢ line (surface) Notations We may interchangeably use the notations: ͬȤ= ̓̂ ͭȤ= ̈́̂ ͮ̂ = ͟Ȩ Standard Notations: ͬ͘ ͦ͘ = ͬʖ ͉ͦ = ͦʖ ͨ͘ ͨ͘ ͦ͘ = ʗ ͨͦ͘ For time derivatives (only)! Velocity and acceleration in Cartesian -ľ Velocity ͪľ= / ºÈ º Velocity in Cartesian: Ì = = Î Îȥ + Ï Ïȥ ºÊ ºÊ ͬ͘Ȥ ͭ͘Ȥ = ͬʖͬȤ+ ͬ + ͭʖͭȤ+ ͭ ͨ͘ ͨ͘ Since, 3Ȥ 4Ȥ ͪľ=ͬʖͬȤ+ ͭʖͭȤ = = 0 / / Acceleration, ºÌ ·= = Îʗ Îȥ + Ïʗ Ïȥ ºÊ Newton’s second law in vector form, ºÌ ¢=¢ Îȥ + ¢ Ïȥ= à = Ã(Îʗ Îȥ + Ïʗ Ïȥ) Î Ï ºÊ I. Plane polar coordinate Y Each point P ( ͬ, ͭ) on the plane can also be represented by its distance ( ͦ) P ( ͬ, ͭ) (ͦ, ) from the origin O and the angle ( ) OP makes with X-axis. ò O X Relationship with Cartesian coordinates ͬ = ͦ cos & ͭ = ͦ sin u Thus , ͦ = (ͬͦ + ͭͦ) ⁄v ͭ = tan ͯͥ ͬ Unit vector in plane polar coordinate • For plane polar unit vectors: Y ÈȤ and òȩ ȩ ò ÈȤ associated to each point in the plane. ͦ • ͦ̂ and Ȩ are unit vector along ò increasing direction of coordinate ͦ X O and . ͦ̅= ͦ ̣̯̳ ò Îȥ + ͦ ̳̩̮ ò Ïȥ "ͦ̅ ͦ̂ = ÈȤ = ̣̯̳ ò Îȥ+ ̳̩̮ ò Ïȥ "ͦ òȩ = - ̳̩̮ ò Îȥ+ ̣̯̳ ò Ïȥ "ͦ̂ ͦ̂ and ɸ are orthogonal: ÈȤ⋅⋅⋅òȩ=0 Ȩ = but their directions depend on location. " Unit vector in plane polar coordinate The unit vectors in the polar coordinate can also be viewed in another way. ͦ̂ is the unit vector perpendicular to ͦ = ͕͗ͣͧͨͨ͢͢ surface and points in the increasing direction of ͦ . Similarly, ɸ is the unit vector perpendicular to = ͕͗ͣͧͨͨ͢͢ surface (i,e tangential to ͦ = ͕͗ͣͧͨͨ͢͢) and points in the increasing direction of . ÈȤ òȩ = ͕͗ͣͧͨͨ͢͢ ͦ = ͕͗ͣͧͨͨ͢͢ Unit vector in plane polar coordinate Y ͦľ = (radial distance) (unit vector along the vector) ͦľ = ͦ ͦ̂ ɸ ͦ̂ ͬȤ= ̓̂ ͦ ͭȤ= ̈́̂ ò ͮ̂ = ͟Ȩ O X Unit vectors in polar coordinate are function of only. ĄÈȤ i = ͬȤcos + ͭȤsin =ƎͬȤ sin + ͭȤcos = òȩ Ąò iW Ąòȩ " = ƎͬȤsin + ͭȤcos = Ǝ ͬȤcos + ͭȤsin = ƎÈȤ Ąò " Velocity in plane polar coordinate -ľ Velocity ͪľ= / -ľ ͦ͘ ͦ̂͘ ͪľ= = ͦ ͦ̂ = ͦ̂ + ͦ / / ͨ͘ ͨ͘ "ͦ̂ ͘ = ͦʖͦ̂ + ͦ Since, " ͨ͘ "ͦ̂ = Ȩ " Ì = ÈʖÈȤ + Èòʖ òȩ Radial component Èʖ and Tangential/transverse component Èòʖ Acceleration in plane polar coordinate ĄÈȤ Ąòȩ ºÌ Note: =òȩ & = ƎÈȤ · = Ąò Ąò ºÊ º ºÈʖ ºÈȤ ºòȩ = (ÈʖÈȤ + Èòʖ òȩ) = ÈȤ + Èʖ + Èʖòʖ òȩ + Èòʗ òȩ + Èòʖ ºÊ ºÊ ºÊ ºÊ ĄÈȤ ºò Ąòȩ ºò =ÈʗÈȤ + Èʖ + Èʖòʖ òȩ + Èòʗ òȩ + Èòʖ Ąò ºÊ Ąò ºÊ =ÈʗÈȤ + Èʖòʖ òȩ + Èʖòʖ òȩ + Èòʗ òȩ Ǝ Èòʖ ̋ÈȤ ·=( Èʗ Ǝ Èòʖ ̋)ÈȤ + (̋Èʖòʖ + Èòʗ )òȩ Radial component of acceleration : Èʗ Ǝ Èòʖ ̋ (Note: -Èòʖ ̋ is the familiar Centripetal contribution !) Tangential component: ̋Èʖ òʖ + Èòʗ (Note: ̋Èʖ òʖ is called the Coriolis contribution!) Newton’s law in plane polar coordinate ͪ͘ľ ̀ľ = ͡ ͨ͘ ͦ ̀ľ = ̀-ͦ̂ + ̀Wɸ = ͡[(ͦʗ Ǝ ͦʖ )ÈȤ + (2ͦʖʖ + ͦʗ )òȩ] ̋ Newton’s law for radial direction: ¢È= m(Èʗ Ǝ Èòʖ ) Newton’s law for tangential direction: ¢ò = Ã(̋Èʖ òʖ + Èòʗ ) Note: Newton’s law in polar coordinates do not follow its Cartesian form as, ¢È ≠ ÃÈʗ or ¢ ≠ Ãʗ Highlights • Transformation relation between Cartesian and polar coordinate is given by, ͬ = ͦ cos ͭ = ͦ sin Reverse transformation u ͦ = (ͬͦ + ͭͦ) ⁄v ͭ = tan ͯͥ ͬ • Directions of unit vectors ( ͬȤ, ͭȤ) in Cartesian system remain fixed irrespective of the location ( x, ͭ). • Directions of unit vectors (ÈȤ, òȩ) in plane polar coordinates depend on the location . • Caution: Form of Newton’s law is different in different coordinate systems. Questions please.