To determine The angular acceleration of the link AB . Explanation Draw the schematic diagram of the system as shown in Figure (1). Write the expression for the velocity of the point A . v A = ( O A ¯ ) ω O A (I) Here, velocity of point A is v A , length of the link OA is O A ¯ and angular velocity of the link OA is ω O A . From Figure (1), consider the triangle Δ A D B and calculate the angle made by the link AB with horizontal. sin β = D B ¯ A B ¯ (II) Here, angle made by the link AB with horizontal is β , distance between the points D and B is D B ¯ and distance between the points A and B is A B ¯ . From Figure (1), consider the triangle Δ A D B and calculate the distance between the points A and D . cos β = A D ¯ A B ¯ A D ¯ = A B ¯ ( cos β ) (III) Here, distance between the points A and D is A D ¯ . From Figure (1), consider the triangle Δ C A D and calculate the distance between the points A and C . cos 30 ° = A D ¯ A C ¯ A C ¯ = A D ¯ cos 30 ° (IV) Here, distance between the points A and C is A C ¯ . Write the expression for the angular velocity of the link AB . ω A B = v A A C ¯ (V) Here, angular velocity of the link AB is ω A B . Write the expression for the position vector of point A with respect to point O . r A / O = ( O A ¯ ) ( cos 30 ° i − sin 30 ° j ) (VI) Here, position vector of point A with respect to point O is r A / O and unit vectors along x and y directions are i and j respectively. Write the expression for the acceleration of point A in vector form. a A = α O A × r A / O − ω O A 2 r A / O (VII) Here, acceleration of point A in vector form is a A and angular acceleration of the link OA in vector form is α O A . Write the expression for the position vector of point B with respect to point A . r B / A = ( A B ¯ ) ( cos β i + sin β j ) (VIII) Here, position vector of point B with respect to point A is r B / A . Write the expression for the acceleration of point B with respect to point A in vector form. a B / A = α × r B / A − ω A B 2 r B / A (IX) Here, acceleration of point B with respect to point A in vector form is a B / A , angular acceleration of the link AB in vector form is α . Write the expression for the acceleration of point B in vector form. a B = a A + a B / A (X) Here, acceleration of point B in vector form is a B and acceleration of point A in vector form is a A . Conclusion: Substitute 0 .80 m for O A ¯ and 9 rad / s for ω O A in Equation (I). v A = ( 0 .80 m ) ( 9 rad / s ) = 0.72 m / s Calculate the distance between the points D and B . D B ¯ = 0.1 m + 0 .08sin30 ° m = 0.14 m Substitute 0 .2 m for A B ¯ and 0.14 m for D B ¯ in Equation (II). sin β = 0.14 m 0 .2 m = 0.7 β = sin − 1 ( 0.7 ) = 44.42 ° Substitute 0 .2 m for A B ¯ and 44.42 ° for β in Equation (III). A D ¯ = ( 0 .2 m ) ( cos 44.42 ° ) = 0.1428 m Substitute 0 .1428 m for A D ¯ in Equation (IV). A C ¯ = 0 .1428 m cos 30 ° = 0 .1649 m Substitute 0 .1649 m for A C ¯ and 0.72 m / s for v A in Equation (V). ω A B = 0.72 m / s 0 .1649 m = 4.37 rad / s Substitute 0 .80 m for O A ¯ in Equation (VI)

8th Edition

ISBN: 9781118885840

Author: James L. Meriam, L. G. Kraige, J. N. Bolton

Publisher: WILEY

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Chapter 5.6, Problem 139P

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The angular acceleration of the link AB.

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Chapter 5.6, Problem 138P

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Chapter 5 Solutions

Engineering Mechanics: Dynamics

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