Lecture-07-08[1] - Uniform Circular Motion Uniform Circular...

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Unformatted text preview: Uniform Circular Motion Uniform Circular Motion l What does it mean? l How do we describe it? l What can we learn about it? Motion in a plane Circular Motion Circular Motion Uniform Uniform What is UCM? What is UCM? l Motion in a circle with: h Constant Radius R h Constant Speed v = | v | R v x y (x,y) Puck on ice How can we describe UCM? How can we describe UCM? l In general, one coordinate system is as good as any other: h Cartesian: (x,y) [position] (v x ,v y ) [velocity] h Polar: (R, ) [position] (v R , ) [velocity] l In UCM: h R is constant (hence v R = 0 ). h (angular velocity) is constant. h Polar coordinates are a natural way to describe UCM! R v x y (x,y) d t dt = = Polar Coordinates: Polar Coordinates: l The arc length s (distance along the circumference) is related to the angle in a simple way: s = R , where is the angular displacement . h units of are called radians . l For one complete revolution: 2 R = R c h c = 2 has period 2 . 1 revolution = 2 radians R v x y (x,y) s Polar Coordinates... Polar Coordinates... x = R cos y = R sin / 2 3 /2 2 -1 1 sin cos R x y (x,y) Polar Coordinates... Polar Coordinates... l In Cartesian coordinates, we say velocity dx/dt = v . h x = vt l In polar coordinates, angular velocity d /dt = . h = t h has units of radians/second . l Displacement s = vt . but s = R = R t, so: R v x y S=vt = t v = R Period and Frequency Period and Frequency l Recall that 1 revolution = 2 radians h frequency (f) = revolutions / second (a) h angular velocity ( ) = radians / second (b) l By combining (a) and (b) h = 2 f l Realize that: h period (T) = seconds / revolution h So T = 1 / f = 2 / R v s = 2 / T = 2 f Recap: Recap: R v s = t (x,y) x = R cos( ) = R cos( t) y = R sin( ) = R sin( t) = arctan (y/x) = t s = v t s = R = R t v = R Acceleration in UCM: Acceleration in UCM: l Even though the speed is constant, velocity is not constant since the direction is changing: must be some acceleration !...
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This note was uploaded on 02/21/2009 for the course PHY A10 taught by Professor Tawfiq during the Fall '08 term at University of Toronto- Toronto.

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Lecture-07-08[1] - Uniform Circular Motion Uniform Circular...

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