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Lecture-07-08[1]

# 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, θ ) [position] » (v R , ω ) [velocity] l In UCM: h R is constant (hence v R = 0 ). h ω (angular velocity) is constant. h Polar coordinates are a natural way to describe UCM! R v x y (x,y) θ d t dt θ θ ω Δ = = Δ Polar Coordinates: Polar Coordinates: l The arc length s (distance along the circumference) is related to the angle in a simple way: s = R θ , where θ is the angular displacement . h units of θ are called radians . l For one complete revolution: 2 π R = R θ c h θ c = 2 π θ has period 2 π . 1 revolution = 2 π radians R v x y (x,y) s θ Polar Coordinates... Polar Coordinates... x = R cos θ y = R sin θ π/ 2 π 3 π /2 2 π-1 1 sin cos R x y (x,y) θ θ Polar Coordinates... Polar Coordinates... l In Cartesian coordinates, we say velocity dx/dt = v . h x = vt l In polar coordinates, angular velocity d θ /dt = ω . h θ = ω t h ω has units of radians/second . l Displacement s = vt . but s = R θ = R ω t, so: R v x y S=vt θ=ω t v = ω R Period and Frequency Period and Frequency l Recall that 1 revolution = 2 π radians h frequency (f) = revolutions / second (a) h angular velocity ( ω ) = radians / second (b) l By combining (a) and (b) h ω = 2 π f l Realize that: h period (T) = seconds / revolution h So T = 1 / f = 2 π / ω R v s ω = 2 π / T = 2 π f ω Recap: Recap: R v s θ=ω t (x,y) x = R cos( θ ) = R cos( ω t) y = R sin( θ ) = R sin( ω t) θ = arctan (y/x) θ = ω t s = v t s = R θ = R ω t v = ω 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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