Moment About an Axis

Moment About an Axis Today’s Objectives: Students will be able to determine the moment of a force about an axis using a. scalar analysis, and, b. vector analysis. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Applications With the force P, a person is creating a moment MA using this flex-handle socket wrench. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Scalar Analysis (1 of 2) Recall that the moment of a scalar force about any point O is MO = F dO where dO is the perpendicular (or shortest) distance distance from the point to the force’s line of action. This concept can be extended to find the moment of a force about an axis. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Scalar Analysis (2 of 2) In the figure above, the moment about the y-axis would be My = Fz ( dx ) = F (r cos θ ) . However, unless the force can easily be broken into components and the “dx” quickly, such calculations are not always trivial and vector analysis may be much easier (and less likely to produce errors). Copyright © 2022 Pearson Education, Inc. All Rights Reserved Vector Analysis (1 of 2) Our goal is to find the moment of F (the tendency to rotate the body) about the a-axis. First compute the moment of F about any arbitrary point O that lies on the a-axis using the cross product. MO = r  F Now, find the component of M0 along the a-axis using the dot product. Ma = u a  Mo Copyright © 2022 Pearson Education, Inc. All Rights Reserved Vector Analysis (2 of 2) Ma can also be obtained as ua x uay uaz Ma = ua  ( r  F ) = rx Fx ry Fy rz Fz The above equation is also called the triple scalar product. In the this equation, ua represents the unit vector along the a-axis, r is the position vector from any point on the a-axis to any point A on the line of action of the force, and F is the force vector. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example (1 of 2) Given: A force is applied to the tool as shown. Find: The magnitude of the moment of this force about the x axis of the value. Plan: 1. Use Mz = u • ( r  F ) . 2. First, find F in Cartesian vector form. 3. Note that u = 1 i in this case. 4. The vector r is the position vector from O to A. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Example (2 of 2) Solution: u =1 i rOA = 0 i + 0.3 j + 0.25 k  m F = 200 ( cos 120 i + cos 60 j + cos 45 k ) N = −100 i + 100 j + 141.4 k  N Now find Mz = u  ( rOA  F ) 1 0 0 Mz = 0 0.3 0.25 = 10.3 (141.4 ) − 0.25 (100 )N  m −100 100 141.4 Mz = 17.4N  m CCW Copyright © 2022 Pearson Education, Inc. All Rights Reserved Group Problem Solving (1 of 3) Given: The force of F = 30 N acts on the bracket.  = 60°,  = 60°,  = 45°. Find: The moment of F about the a-a axis. Plan: 1. Find ua and rOA 2. Find F in Cartesian vector form. 3. Use Ma = ua  ( rOA  F ) Copyright © 2022 Pearson Education, Inc. All Rights Reserved Group Problem Solving (2 of 3) Solution: ua = j rOA = −0.1 i + 0.15 k  m F = 30 cos 60 i + cos 60 j + cos 45 k  N F = 15 i + 15 j + 21.21k  N Copyright © 2022 Pearson Education, Inc. All Rights Reserved Group Problem Solving (3 of 3) Now find the triple product, Ma = ua  ( rOA  F ) 0 1 0 Ma = −0.1 0 0.15 N  m 15 15 21.21 Ma = −1−0.1( 21.21) − 0.15 (15 ) = 4.37 N  m Copyright © 2022 Pearson Education, Inc. All Rights Reserved Reading Quiz (1 of 2) 1. When determining the moment of a force about a specified axis, the axis must be along ____________. A. the x axis B. the y axis C. the z axis D. any line in 3-D space E. any line in the x-y plane Copyright © 2022 Pearson Education, Inc. All Rights Reserved Reading Quiz (2 of 2) 2. The triple scalar product u  ( r  F ) results in A. a scalar quantity ( + or − ). B. a vector quantity. C. zero. D. a unit vector. E. an imaginary number. Copyright © 2022 Pearson Education, Inc. All Rights Reserved Attention Quiz 1. For finding the moment of the force F about the x-axis, the position vector in the triple scalar product should be ________. A. rAC B. rBA C. rAB D. rBC 2. If r =1i + 2 j  m and F = 10 i + 20 j +30k  N, then the moment of F about the y-axis is ______ N  m. A. 10 B. −30 C. −40 D. None of the above. Copyright © 2022 Pearson Education, Inc. All Rights Reserved