W9.1 - Physics for Scientists Engineers 1 Lecture 31...

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March 10, 2008 Physics for Scientists&Engineers 1 1 Physics for Scientists & Physics for Scientists & Engineers 1 Engineers 1 Lecture 31

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March 10, 2008 Physics for Scientists&Engineers 1 2 Angular Velocity Angular Velocity Linear motion: rate of change of displacement is velocity Angular motion: rate of change of angular displacement is angular velocity Average: Instantaneous: Unit: Direction: right-hand rule ! = " 2 # " 1 t 2 # t 1 = \$ " \$ t ! = lim " t # 0 ! = d \$ dt ! [ ] = rad/s
March 10, 2008 Physics for Scientists&Engineers 1 3 Frequency Frequency Frequency, f , measures numbers of turns around the circle Example: rpm on tachometer Since 1 turn = 2 π radians: Unit: [ f ]=1/s In honor of Heinrich Rudolf Hertz (1857-1894): 1/s = 1 Hz Period, T : Relationship with angular velocity: f = ! 2 " # ! = 2 " f T = 1 f ! = 2 " f = 2 " T T = 2 ! "

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March 10, 2008 Physics for Scientists&Engineers 1 4 Linear and Angular Velocity Linear and Angular Velocity Write r in Cartesian coordinates, then use transformation to polar coordinates and take derivatives (remember: r = const ., and d θ /dt = ω ) Result: linear velocity vector points in tangential direction Magnitude: ! r = ( x , y ) = ( r cos ! , r sin ! ) = r (cos ! ,sin ! ) = r ˆ r ! v = r ! ˆ t ! v = d ! r dt = d dt ( r cos ! , r sin ! ) = d dt ( r cos ! ), d dt ( r sin ! ) " # \$ % & = ! r sin " d " dt , r cos " d " dt # \$ % & ( = r d " dt ! sin " ,cos " ( ) = r ) ˆ t v = r !
March 10, 2008 Physics for Scientists&Engineers 1 5 Example: Earth Example: Earth Question: The Earth orbits around the Sun and revolves around its own pole-to-pole axis.

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