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Lecture 25 ENGR-1100 Introduction to Engineering Analysis

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Today’s Lecture Outline Belt Friction Flat belts V-belts

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Belt Friction

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Flat Belt F r = 0 - T sin -(T + T) sin =0 PP P= 2T sin + T sin (a)

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(T + T) cos - T cos - F = 0 F = 0 T cos = F (b)

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Assuming slip is impending F = s P Tcos( /2)= s sin( /2)+ s sin( /2) Combining equation (a) and (b) yields: P= 2T sin + T sin (a) T cos = F (b) cos ( /2) = s T + Dividing by

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cos 1 1 dT/d = s T +O( T) dT/T = = s d Rearranging dT/d = s T TT dT d ln ( T 2 / T 1 ) = s ( 2 - 1 ) = s Integrating Where: = 2 - 1 Or T 2 = T 1 e s (c) T1 T2

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V-Belt T 2 = T 1 e ( s ) enh Where ( s ) enh =[ s /sin( /2)] is the enhanced coefficient of friction.

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Example 9-85 AA rope attached to a 500 lb block passes over a frictionless pulley and is wrapped for one full turn around a fixed post as shown in Fig. P9-85. If the coefficient of friction between the rope and the post is 0.25, determine: T(a) The minimum force P that must be used to keep the block from falling. ((b) The minimum force P that must be used to begin to raise the block.

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Solution A free-body diagram for the block: F y = T - W = 0 T 2 = T 1 e =T 1 e 0.25(2 ) = 4.8104 T 1 (a) During lowering of the block: T= 500 lb From equation (c): P min = T 1 = = 103.9 lb T 2 = 500 lb

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(b) During raising of the block: P min = T 2 = 4.8104 T 1 = 4.8104(500)= 2405.2 lb 2410 lb T 1 = 500 lb

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Class Assignment: Exercise set P9-86 please submit to TA at the end of the lecture AA rope attached to a 220-kg block passes over a fixed drum as shown in Fig. P9-86. If the coefficient of friction between the rope and the drum is 0.30, determine (a) The minimum force P that must be used to keep the block from falling. (b) The minimum force P that must be used to begin to raise the block. a)P min = 1347 N b) P min = 3460 N

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Solution F y = T - mg = 0 T = mg =220 (9.807) = 2158 N A free-body diagram for the block: T 2 = T 1 e =T 1 e 0.30( /2) = 1.6020T 1 From equation (c): (a) During lowering of the block:T 2 = 2158 N P min = T 1 = = 1347 N

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(b) During raising of the block: T 1 = 2158 N P min = T 2 = 1.6020 T 1 = 1.6020 (2158) 3460 N

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The band brake of Fig. P-90 is used to control the rotation of a drum. The coefficient of friction between the belt and the drum is 0.35 and the weight of the handle is 15 N. If a force of 200 N is applied to the end of the handle, determine the maximum torque for which no motion occurs if the torque is applied (a) Clockwise (b) Counterclockwise Class assignment: Exercise set P9-90 please submit to TA at the end of the lecture T max = 190.3 N m T max = 686 N m

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Solution = (2 ) = 3.665 rad = 0.35 (3.665) = 1.2828 T 2 = T 1 e =T 1 e 1.2828 = 3.607T 1 From equation (c): (a) From a free-body diagramof the handle M A = 200(675) - 15(225) - T (75) = 0 T = 1755.0 N T

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T 1 = =486.6 N T max = (T 2 - T 1 )R = (1755.0 - 486.6) (0.150) = 190.3 N m (a) From a free-body diagramof the drum when the rotation of the drum is clockwise: T 2 =T T1 T2 T

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T 1 = 1755.0 N T 2 = 3.6067 T 1 = 3.6067(1755.0) = 6329.8 N T max = (T 2 - T 1 )R = (6329.8 - 1755.0) (0.150) = 686 N m (b) From a free-body diagramof the drum when the rotation of the drum is counterclockwise: T2 T1 T T 1 =T

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Class assignment: Exercise set P9-89 please submit to TA at the end of the lecture Solution: a min =7.53 in.

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Class assignment: Exercise set 9-96 A 100-N crate is sitting on a 200-N crate as shown in the figure. The rope that join the crates passes around a frictionless pulley and a fixed drum. The coefficient of friction is 0.3 at all surfaces. Determine the minimum force P that must be applied to the 200-N crate to start motion. P min =159 N

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