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Unformatted text preview: Chapter 6 Kinematics of Rigid Bodies Undergoing Planar Motion 6.1 Relative Velocities on a Rigid Body 101 6.1.1 GOAL: Determine the angular velocity of the rollers supporting a bikes rear wheel. GIVEN: Size and configuration of the wheel and rollers. DRAW: FORMULATE EQUATIONS: We know from geometry that both rollers will have the same angular velocity and thus well concentrate on R 2 . We know that if the tire and roller roll without slip on each other that the velocity of the contact point is the same for each body. * v C = * v A + 1 * k * r C / A = 0 + 1 * b 3 ( r 1 * b 1 ) = r 1 1 * b 2 * v C = * v B + 2 * b 3 * r C / B = 0 + 2 * b 3 ( r 2 * b 1 ) = r 2 2 * b 2 Equating these two velocity expressions for C gives us r 1 1 = r 2 2 2 = r 1 r 2 ! 1 The angular velocity of the rollers is therefore given by (24 rad/s) r 1 r 2 * k 102 6.1.2 GOAL: Find  * v A  at t = 3 s. GIVEN: ( t ) and * r A / O . DRAW: FORMULATE EQUATIONS: * v B = * v A + * k * r A / O SOLVE: To solve the problem well need to derive an expression for . = (1 e at ) rad = ae at rad We can now use this in our expression for velocity. * v A = r * e r + r * e = are at rad Using our given data gives us * v A = ( . 8 s 1 )(1 m) e ( . 8 s 1 )(3 . s ) rad = 7 . 26 10 2 * k m/s  * v A  = 7 . 26 10 2 m/s 103 6.1.3 GOAL: Find the rotational speed of the wheels in a microwave oven turntable. GIVEN: Dimensions and orientations of wheels and support plate. DRAW: FORMULATE EQUATIONS: Well use the formula for speed on a rotating body: * v B = * v A + * * r B / A ASSUME: At the point of contact between the wheels and the support plate we have equal velocities, a consequence of the noslip condition. SOLVE: First well determine the angular speed of the support plate from knowing its period of rotation: P = 2 10 s = 5 rad/s Now well equate the velocity of the contact point C as found from the center of the supporting wheel ( A ) and from the center of the support plate ( O ). * v C = * v C * v A + W * (0 . 5 cm) * k = * v O + P * k (12 cm) * (0 . 5 cm) W * = (12 cm) P * W = 24 P W = 24 P = 24 5 rad/s 104 6.1.4 GOAL: Determine the velocity of a linear rack thats connected by two gears to another rack. GIVEN: Size and configuration of the two meshing gears and racks. DRAW: FORMULATE EQUATIONS: We know that where the gear teeth mesh they move at the same velocity. C 1 , C 2 and C 2 are our three points of contact and were given that * v C 1 = v A * * v C 1 = * v O 1 + 1 * k * r C 1 /O 1 = 0 + 1 * k ( r 1 * ) = r 1 1 * Equating these two velocity expressions for C 1 gives us v A = r 1 1 1 = v A r 1 Next look at the point C 2 : * v C 2 = * v O 1 + 1 * k * r C 2 /O 1 = 0 + v A r 1 !...
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 Spring '06
 KAM

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