Moment of Force a - College of Engineering | University of ...



Moment of Force about a PointDefinition: Mo=r×F, where r is a position vector from point ? to any point on the line of action of force F (sliding vector).Physical meaning: Measure of rotational tendency of F with respect to ?.4032250381000Units: S.I. – N * m, U.S.C.S. – lb * ftQuestion: Where the moment points out? Varignon’s theorem – moments are distributiveThe moment about a given point O of the resultant of several concurrent forces is equal to the sum of the moments of the various forces about the same point O.r×F1+F2+…=r×F1+r×F2+…31267405395002D MomentsMoment of plane force about a point in the same plane always points out-of-plane. It can, therefore, be considered a scalar.Since for any r and θ, r*sinθ=d:Mo=r×F=r*F*sinθ=d*FSign convention: counter clock-wise is positive (out of the paper plane), clock-wise is negative (into the paper plane).Computation of 2D Moments1st method: Use scalar definition Mo=d*F, by finding d and F=F 2nd method: Resolve F into components and use Varignon’s theorem.Note 1: components don’t have to be rectangular (but should add-up to be equal to the original vector). Note 2: signs/directions of those moments from components can be opposite!3rd method: Use vector definition Mo=r×F and perform multiplication algebraically.Mo=r×F=r*F*sinθ=d*FNote: In 2D r?and?F have zero z-components, while Mo has only z-component.171450419100000239395000Example 1Example Example 3Example 4Example 5Example 6Example 73D MomentsUnlike 2D moments, in 3D all vectors in Mo=r×F can have all three components simultaneously non-putation of 3D moments done almost exclusively by vector algebra: cross product of r=rxi+ryj+rzk and F=Fxi+Fyj+Fzk. Occasionally the second method (Varignon’s theorem) is used.Similarly to 2D, the distance from O to line can be computed using moment formulas:d*F=r×FMoment (3D), Example 1Example 2Example 3:Moment about an axisIn some cases rotation a body can be restricted to occur about an axis. In this case a new useful mechanical quantity can be derived – moment about an axis.To compute this moment:Compute a moment Mo about any point on the axis,Compute a projection of Mo onto the axis. Example from the above figure:Mo=20?0.5=10N?m My=10?35=6N?mAlternatively, find the distance from the line of action to the axis: ?=?. :?My=20?0.3=6N?mVector definition:To find the moment of F about an axis ???:Select an arbitrary convenient point ? on the axis ???Compute moment of F about the point ?Mo=rOA×Fwhere ? is an arbitrary convenient point on line of action of FCompute projection of Mo onto ???: Maa=Mo?uaa=uaa?rOA×F=uxuyuzrxryrzFxFyFz4) Define the vector Maa Maa=Maa?uaa=(uaa?rOA×F)?uaaNote: from the above definitions, we can obtain a physical meaning of determinant:-235585889000319405034290000 3486150152019000-1016003873500Components of moment vector: Mx, My, Mz are moments of force F about the corresponding axes passing through O!Moment about an axis, Example 1Example 2:4885211336508A00A1817392555154Example 3: ................
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