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Research Paper > ACS College of Engineering - MATH. 1014. Polar coordinate system

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ACS College of Engineering - MATH. 101 4. Polar coordinate system Polar coordinate system • Answers and detailed solutions to all problems are provided in iOS/Android "PhysOlymp" app • With an ... y suggestions please write to [email protected] Selection of fit for purpose coordinate system can significantly simplify solution of the problem. For the cases when trajectory of the particle is close to circular or spiral like contour in a plane, it is usually convenient to use polar coordinates Polar coordinate system is characterized by a distance r from the reference point O and angle ’ from some reference direction (Ox) Location of some specific point A in Cartesian system in terms of known polar coordinates can be expressed as xA = r cos’ yA = r sin’ Components of velocity ~v in Cartesian coordinates are described with two unit vectors ~i and ~j, which are parallel to x and y axises respectively, with a direction alongside of increasing values of coordinates x and y ~v x = d x dt ~i ~v y = d y dt ~j Total velocity in Cartesian coordinate system is a vector sum of its components: ~v = ~v x + ~vy = d x dt ~i + d y dt ~j Similarly, directions in polar coordinate system are characterized with two unit vectors: one with a normal n~ outward direction and another one perpendicular to vector n~ with a counterclockwise direction ~τ By definition of velocity, it is a displacement in unit of time. Let’s consider a motion in plane from the point A to the position A0 Step by step to the Golden medal at Physics Olympiad 2 C Copyright 2018, www.physolymp.com Normal component of velocity vn can be defined as displacement ∆r from the origin of coordinates in a small time interval ∆t v n = ∆r ∆t With accounting to direction described by unit vector n~ this can be rewritten as ~vn = ∆t n~ = dt n~ = ˙rn~ | (1) ∆r | drSimilarly tangential component of velocity can be defined as displacement r∆’ obtained by rotating at small angle ∆’ during time interval ∆t v τ = r∆’ ∆t Direction of tangential component of velocity coincides with unit vector τ. Then, ~vτ = r∆’ ∆t ~τ = r d’ dt ~τ = r’ ~τ ˙ (2) Equations (1) and (2) can be easily comprehended by considering limiting cases, with trajectory of the moving particle being either a straight line or a circle For a motion along a straight line equation (1) becomes a regular definition of velocity as a measure of change of coordinate r per unit of time v n = dr dt = ˙r For another limiting case of the motion around a fixed center with constant radius r, velocity of the object can be defined by its angular velocity ! = ’˙ as v τ = !r = ’˙r which corresponds to a general form of tangential component of [Show More]

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