Explanation Given: The angular velocity of the arm A B is ω A B = 5 rad/s (counter clockwise). The angular acceleration of the arm A B is α A B = 2 rad/s 2 . The angular velocity of the grip B C relative to fixed reference is ω ' = 6 rad/s (clockwise). The angular acceleration of the grip B C relative to fixed reference is α ' = 2 rad/s 2 . Draw the free body diagram of system as shown in Figure 1. Refer Figure (1). Write the formula for velocity of the object measured from the x , y , z reference. v C = v B + Ω × r C / B + ( v C / B ) x y z (I) Here, v C velocity vector of object at C measured from the X , Y , Z reference, v B is the velocity vector of the origin B (end of the arm where the object is connected) measured from the X , Y , Z reference, ( v C / B ) x y z is the velocity vector of object at the grip C with respect to arm end B measured by an observer attached to the rotating x , y , z reference, Ω is the angular velocity of the x , y , z reference measured from the X , Y , Z reference and r C / B is the position vector of the object at grip C with respect to arm end B . Write the formula for acceleration of the object measured from the x , y , z reference a C = a B + Ω ˙ × r C / B + Ω × ( Ω × r C / B ) + 2 Ω × ( v C / B ) x y z + ( a C / B ) x y z (II) Here, a C acceleration vector of object at C measured from the X , Y , Z reference, a B is the acceleration vector of the origin B (end of the arm where the object is connected) measured from the X , Y , Z reference, ( v C / B ) x y z , ( a C / B ) x y z is the velocity and acceleration vector of the object at grip C with respect to arm end B measured by an observer attached to the rotating x , y , z reference, Ω ˙ is the angular acceleration due to the rotation of x , y , z reference, Ω is the angular velocity of the x , y , z reference measured from the X , Y , Z reference and r C / B is the position vector of the object at grip C with respect to arm end B . Write the formula for velocity at B of the arm A B . v B = ω A B × r B / A (III) Here, ω A B is the angular velocity of the arm A B and r B / A is the position vector arm end B with respect to arm end A . Write the formula for acceleration at B of the arm A B . a B = α A B × r B / A − ω A B 2 r B / A (IV) Here, α A B is the angular acceleration of the arm A B . Conclusion: Refer Figure (1). Write the position vector ( r B / A ) directed from A to B . r B / A = ( 0.3 cos 30 ° ) i + ( 0.3 sin 30 ° ) j = 0 .2598i + 0.15 j Write the position vector ( r C / B ) directed from B to C . r C / B = ( 0.125 cos 15 ° ) i + ( 0.125 sin 15 ° ) j = 0.121 i + 0.0324 j The angular velocity vector of the arm A B is ω A B = 5 k (counter clockwise). The angular acceleration vector of the arm A B is α A B = 2 k (counter clockwise). Calculate the velocity at B of the arm A B . Substitute 5 k for ω A B and 0 .2598i + 0.15 j for r B / A in Equation (III). v B = 5 k × ( 0 .2598i + 0.15 j ) = | i j k 0 0 5 0.2598 0.15 0 | = i ( 0 − 0.75 ) − j ( 0 − 1.299 ) + k ( 0 − 0 ) = ( − 0.75 i + 1.299 j ) m/s Calculate the velocity at B of the arm A B . Substitute 2 k for α A B , 5 for ω A B and 0

Engineering Mechanics: Dynamics (14th Edition)

14th Edition

ISBN: 9780133915389

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 16.8, Problem 142P

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The velocity and acceleration of the object held at grip C .

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Chapter 16.8, Problem 141P

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The velocity and acceleration of the object held at grip C .

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