Explanation Given: The constant angular velocity of the arm A B about B is ω f = 1 rad/s . The constant angular velocity of the car relative to the arm is ω A / f = 2 rad/s . Draw the free body diagram of system as shown in Figure 1. Refer Figure (1). Write the formula for velocity of the car measured from the x , y , z reference. v C = v A + Ω × r C / A + ( v C / A ) x y z (I) Here, v C velocity vector of car C measured from the X , Y , Z reference, v A is the velocity vector of the origin A (end of the arm where the car is connected) measured from the X , Y , Z reference, ( v C / A ) x y z is the velocity vector of passenger at the car C with respect to arm end A 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 / A is the position vector of the passenger at car C with respect to arm end A . Write the formula for acceleration of the car measured from the x , y , z reference a C = a A + Ω ˙ × r C / A + Ω × ( Ω × r C / A ) + 2 Ω × ( v C / A ) x y z + ( a C / A ) x y z (II) Here, a C acceleration vector of car C measured from the X , Y , Z reference, a A is the acceleration vector of the origin A (end of the arm where the car is connected) measured from the X , Y , Z reference, ( v C / A ) x y z , ( a C / A ) x y z is the velocity and acceleration vector of the passenger at car C with respect to arm end A 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 / A is the position vector of the passenger at car C with respect to arm end A . Write the formula for velocity at A of the arm A B . v A = ω f × r A / B (III) Here, ω f is the angular velocity of the arm A B and r A / B is the position vector arm end A with respect to arm end B . Write the formula for acceleration at A of the arm A B . a A = α f × r A / B − ω f 2 r A / B (IV) Here, α f is the angular acceleration of the arm A B . Conclusion: Refer Figure (1). Write the position vector ( r A / B ) directed from B to A . r A / B = ( − 15 cos 30 ° ) i + ( 15 sin 30 ° ) j = − 12.99 i + 7.5 j Write the position vector ( r C / A ) directed from A to C . r C / A = − 8 i − 0 j = − 8 i The angular velocity vector of the arm A B is ω f = 1 k (counter clockwise) Calculate the velocity at A of the arm A B . Substitute 1 k for ω f and − 12.99 i + 7.5 j for r A / B in Equation (III). v A = 1 k × ( − 12.99 i + 7.5 j ) = | i j k 0 0 1 − 12.99 7.5 0 | = i ( 0 − 7.5 ) − j ( 0 + 12.99 ) + k ( 0 − 0 ) = ( − 7.5 i − 12.99 j ) ft/s Here, the arm has constant angular velocity, hence the angular acceleration of the arm A B is α f = 0 . Calculate the velocity at B of the arm A B

Engineering Mechanics: Dynamics (14th Edition)

14th Edition

ISBN: 9780133915389

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 16.8, Problem 144P

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The velocity and acceleration of the passenger at C at the instant shown.

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Chapter 16.8, Problem 143P

Chapter 16.8, Problem 145P

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The velocity and acceleration of the passenger at C at the instant shown.

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