Explanation Given: The given velocity of gear, υ = 2 m / s . The given acceleration of gear, a = 3 m / s . The length of the fixed gear rack, l = ( 500 mm ) or 0 .5 m . The smaller radius, r i = ( 150 mm ) or 0 .15 m . The radius of gear rack, r o = ( 200 mm ) or 0 .2 m . The given angle formed by gear rack and fixed end, θ = 60 ° Show the location of I C for gear C as in Figure (1). Show the I C of link A B as in Figure (2). Show the free body diagram of gear C as in Figure (3). Show the free body diagram of link A B as in Figure (4). Conclusion: Take the velocity of gear about point A and B as equal. υ = 2 m / s = υ A = υ B Express the velocity of gear at point B . υ B = ω C r B / ( I C ) 1 = ω C ( r o − r i ) (I) Here, angular velocity of at C is ω C and position vector from B to ( I C ) 1 is r B / ( I C ) 1 . Substitute 2 m / s for υ B , 0 .15 m for r i and 0 .2 m for r o in Equation (I). 2 = ω C ( 0.2 − 0.15 ) = 0.05 ω C ω C = 4 0 rad / s ( ccw ) Hence, the angular velocity ω C of the gear is 4 0 rad / s ( ccw ) . From Figure (2), take the I C of link A B using υ A and υ B at infinity. r A / ( I C ) 2 = ∞ Express the angular velocity of link A B . ω A B = υ A r A / ( I C ) 2 (II) Substitute 2 m / s for υ A and ∞ for r A / ( I C ) 2 in Equation (II). ω A B = 2 ∞ = 0 Express the acceleration at its mass center. a O = α C r o (III) Substitute 0 .2 m for r o in Equation (III). a O = α C ( 0.2 ) ↓ Refer Figure (3), and apply the relative acceleration to gear C . a B = − ( a O ) j + α C k × r B / O i − ω C 2 ( r B / O ) i = − ( a O ) j + α C k × r i i − ω C 2 ( r i ) i (IV) Substitute α C ( 0.2 ) for a O , 0 .15 m for r i and 4 0 rad / s for ω C in Equation (IV). a B = − ( a O ) j + α C k × r B / O i − ω C 2 ( r B / O ) i = − α C ( 0

Engineering Mechanics: Statics & Dynamics (14th Edition)

14th Edition

ISBN: 9780133915426

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 16.7, Problem 120P

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The angular velocity ωC and angular acceleration αC of the gear.

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Chapter 16.7, Problem 119P

Chapter 16.7, Problem 121P

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The angular velocity ω C and angular acceleration α C of the gear.

Solution Summary: The author explains the angular velocity and acceleration of the gear, and the radius of gear rack and fixed end.

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The angular velocity ωC and angular acceleration αC of the gear.

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