Explanation Calculation: Given, the rigid body is rotating with a constant angular speed of 3 2 rad / s and the rotation appears to be clockwise when viewed from A towards the origin. The line through origin O and the point A ( 1 , 1 , 1 ) is the axis of rotation of a rigid body. Let, a = i + j + k be the vector from O to A . b = i + 3 j + 2 k be the vector from O to B . The vector v is orthogonal to a and b . Therefore, v is parallel to b × a as the rotation is clockwise. The cross product of b × a is calculated as follows, b × a = | i j k 1 3 2 1 1 1 | = i [ ( 3 × 1 ) − ( 2 × 1 ) ] − j [ ( 1 × 1 ) − ( 2 × 1 ) ] + k [ ( 1 × 1 ) − ( 3 × 1 ) ] = i [ 3 − 2 ] − j [ 1 − 2 ] + k [ 1 − 3 ] = i + j − 2 k The vector projection of b onto a is the vector, proj a b = ( a ⋅ b a ⋅ a ) a = [ ( 1 × 1 ) + ( 1 × 3 ) + ( 1 × 2 ) ( 1 × 1 ) + ( 1 × 1 ) + ( 1 × 1 ) ] ( i + j + k ) = ( 1 + 3 + 2 1 + 1 + 1 ) ( i + j + k ) = 2 i + 2 j + 2 k Therefore, ( 2 , 2 , 2 ) is the center of the circular path ( 1 , 3 , 2 ) takes the radius as follows, radius = ( 2 − 1 ) 2 + ( 2 − 3 ) 2 + ( 2 − 2 ) 2 = ( 1 ) 2 + ( − 1 ) 2 + ( 0 ) 2 = 2 Therefore, the arc length per second covered by the point is 3 2 2 units / sec is equal to the | v | (velocity is constant)

University Calculus: Early Transcendentals Plus MyLab Math -- Access Card Package (3rd Edition) (Integrated Review Courses in MyMathLab and MyStatLab)

3rd Edition

ISBN: 9780321999573

Author: Joel R. Hass, Maurice D. Weir, George B. Thomas Jr.

Publisher: PEARSON

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Chapter 11, Problem 4AAE

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Find the velocity v of the point of the body that is at the position B(1,3,2).

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Find the velocity v of the point of the body that is at the position B ( 1 , 3 , 2 ) .

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Find the velocity v of the point of the body that is at the position B(1,3,2).

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