P7 - & & & 8 8 2 2 / 3 3 4 W W R R R R...

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Discussion 7: Name: solutions TAM212, Spring, 2003 The gear of radius R rolls on the horizontal rack. Pin G in the center of the gear engates a slot in the arm AB, which rotates at the constant angular velocity θ = ω counterclockwise as shown. Determine the angular velocity of the gear when ° = θ 60 . . If we let “W” refer the cylindrical wheel, “G” refer to its center, and “I” where it touches the surface, and we let “R” refer to the rod. () R v R v W G C G IC W I G ω = × ω + = × ω + = J K I r v v 0 v v v v We need W ω to find W ω . Since the two rigid bodies are not pinned together, we need the equation with the two frames of reference to relate the angular velocities of the two objects. Letting “F” be the frame where the pin at A, the surface, and the rack are stationary ( ) () ( ) ( ) ( ) J I K J I I r v v v R R v R R G W AC F R F A R G F G 2 3 2 0 2 / 1 2 / 3 / / / / / + × θ + + + = ω × ω + + = v I-direction: - ( ) θ = ω R v R R G W 2 2 / 3 / ; J-direction: () θ + = R v R G 3 2 2 / 1 0 / Combining these two equations: θ = R v R G 3 4 / , i.e. ( ) () ( ) ( ) J I J I v 3 2 6 2 / 1 2 / 3 3 4 / + θ = + θ = R R R G , and ( ) ( ) θ = ω θ = θ θ = ω
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Unformatted text preview: & & & 8 8 2 2 / 3 3 4 W W R R R R counter-clockwise. If time, determine the angular acceleration of the gear at this point as well. R a R v W G W G α − = ⇒ ω − = (since the rack is flat) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) J I J I K J I I r r a v a a R R R a R R G W AC AC F R F A R G F R R G F G 2 3 2 3 2 6 8 2 2 / 1 2 / 3 2 2 / 2 / / / / / / + θ − + + + θ − × θ + + = α − ⇒ ω − × α + + × ω + = & & & v v I-direction: ( ) ( ) 2 / 2 2 / 3 30 2 / 3 3 2 3 32 2 / 3 θ + = θ − θ + = α − & & & R a R R a R R G R G W ; J-direction: ( ) ( ) 2 / 2 2 / 98 2 / 1 2 96 2 / 1 θ − = θ − θ − = & & & R a R R a R G R G Combining these two equations: ( ) 2 / 98 2 / 1 θ = & R a R G , and ( ) 2 2 2 2 3 79 3 79 3 30 2 / 3 98 θ − = α ⇒ θ = θ + θ = α − & & & & W W R R R R , i.e. 2 3 79 θ = α ⇒ & W clockwise θ & J K I j k i...
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This note was uploaded on 04/09/2008 for the course TAM 212 taught by Professor Keane during the Spring '08 term at University of Illinois at Urbana–Champaign.

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