Explanation Given: The angular velocity, ω 1 = 6 rad/s . The angular velocity, ω 2 = 4 rad/s . The angular acceleration, ω ˙ 1 = 0 rad/s 2 . The angular acceleration, ω ˙ 2 = 0 rad/s 2 . The position of origin O , r O = 0 . The position of P with respect to O , r P / O = 4.243 j → + 4.243 k → ft . The conveyor belt running at constant rate, r ˙ = 5 ft . The angle of conveyor belt, θ = 45 ° . The velocity at point O , v O = 0 . The acceleration at point O , a O = 0 . Write the expression position of the point measured from the fixed point. ( r ˙ P / O ) x y z = r ˙ cos θ j → + r ˙ sin θ k → (I). Write the expression of the velocity of P with respect to O by rotating x , y , z frame of reference. ( v P / O ) x y z = ( r ˙ P / O ) x y z + ω 2 × r P / O (II) Write the expression of the acceleration of P with respect to O by rotating x , y , z frame of reference. ( a P / O ) x y z = [ ( r ¨ P / O ) x y z + ω 2 × ( r ˙ P / O ) x y z ] + ω ˙ 2 × r P / O + ω 2 × v P / O (III) Here, x , y , z for the translating-rotating frame of reference, x ′ , y ′ , z ′ for the second translating-rotating frame of reference, ( r ˙ B / A ) for second time derivative position of the point measured from the fixed point, and ( r ¨ B / A ) for third time derivative position of the point measured from the fixed point. Write the expression of the velocity. v P = v O + ω 1 × r P / O + ( v P / O ) x y z (IV) Write the expression of the acceleration. a P = a O + ω ˙ 1 × r P / O + ω 1 × ( ω 1 × r P / O ) + 2 ω 1 × ( v P / O ) x y z + ( a P / O ) x y z (V) Conclusion: Substitute 5 ft for r ˙ , and 45 ° for θ in Equation (I). ( r ˙ P / O ) x y z = 5 cos 45 ° j → + 5 sin 45 ° k → Substitute 5 cos 45 ° j → + 5 sin 45 ° k → for ( r ˙ P / O ) x y z , 4 rad/s for ω 2 , and 4.243 j → + 4.243 k → for r P / O in Equation (II). ( v P / O ) x y z = ( 5 cos 45 ° j → + 5 sin 45 ° k → ) + 4 i → × ( 4.243 j → + 4.243 k → ) = − 13.44 j → + 20.51 k → ft/s Substitute 4.243 j → + 4.243 k → for r P / O , − 13

Engineering Mechanics: Dynamics Study (Book and Pearson eText)

14th Edition

ISBN: 9780134116990

Author: Russell C. Hibbeler

Publisher: PEARSON

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Chapter 20.4, Problem 50P

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The acceleration of package P .

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Chapter 20.4, Problem 49P

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Engineering Mechanics: Dynamics Study (Book and Pearson eText)

Ch. 20.3 - Prob. 1P Ch. 20.3 - Prob. 2P Ch. 20.3 - Prob. 3P Ch. 20.3 - Prob. 4P Ch. 20.3 - Prob. 5P Ch. 20.3 - Prob. 6P Ch. 20.3 - Prob. 7P Ch. 20.3 - The disk rotates about the shaft S, while the...Ch. 20.3 - The electric fan is mounted on a swivel support...Ch. 20.3 - Prob. 11P

Ch. 20.3 - Prob. 12P Ch. 20.3 - The right circular cone rotates about the z axis...Ch. 20.3 - Prob. 14P Ch. 20.3 - Prob. 15P Ch. 20.3 - Prob. 16P Ch. 20.3 - Prob. 17P Ch. 20.3 - Prob. 18P Ch. 20.3 - Prob. 20P Ch. 20.3 - Prob. 21P Ch. 20.3 - Prob. 22P Ch. 20.3 - Prob. 23P Ch. 20.3 - Prob. 24P Ch. 20.3 - Prob. 25P Ch. 20.3 - Rod AB is attached to collars at its ends by using...Ch. 20.3 - Rod AB is attached to collars at its ends by using...Ch. 20.3 - If the rod is attached with ball-and-socket joints...Ch. 20.3 - Prob. 29P Ch. 20.3 - If collar A has a speed vA = 4 m/s, determine the...Ch. 20.3 - Prob. 31P Ch. 20.3 - If the collar A in Prob. 20-31 has a deceleration...Ch. 20.3 - Prob. 33P Ch. 20.3 - Rod CD is attached to the rotating arms using...Ch. 20.3 - Prob. 35P Ch. 20.3 - Prob. 36P Ch. 20.4 - So1ve Example 20.5 such that the x, y, z axes move...Ch. 20.4 - Prob. 38P Ch. 20.4 - Prob. 39P Ch. 20.4 - At the instant = 60, the construction lift is...Ch. 20.4 - Prob. 41P Ch. 20.4 - Prob. 42P Ch. 20.4 - Prob. 43P Ch. 20.4 - Prob. 44P Ch. 20.4 - Prob. 45P Ch. 20.4 - Prob. 46P Ch. 20.4 - Prob. 47P Ch. 20.4 - At the given instant the rod is turning about the...Ch. 20.4 - Prob. 49P Ch. 20.4 - Prob. 50P Ch. 20.4 - Prob. 51P Ch. 20.4 - Prob. 52P Ch. 20.4 - Prob. 53P Ch. 20.4 - At the instant shown, the arm AB is rotating about...

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The velocity of package P . The acceleration of package P .

Solution Summary: The author determines the velocity of package P and its acceleration.