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Vector Moments Definition and Calculations in Statics Lecture

Learn about moment of force, rigid bodies, and equivalent force-couple systems in this statics lecture. Understand vector moment definition, moment about an axis, and moment of a couple. Solve Chapter 3 problems by hand calculations with examples provided. Check your work using MathCAD. Homework due Friday, September 26, Quiz on Wednesday, September 24. Supplement your understanding with detailed explanations and practical tips.

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Vector Moments Definition and Calculations in Statics Lecture

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  1. ME221 StaticsLECTURE # 10Sections 3.4 & 3.6 Lecture 10

  2. Homework #4 • Chapter 3 problems: • 1, 4, 8, 11, 17, 25, 26, 28, 35 & 40 • To be solved using hand calculations • May check work using MathCAD, etc. • Due Friday, September 26 Quiz #3 • Wednesday, September 24 Lecture 10

  3. Last Lecture • Cross Product • Rigid Bodies • Moment of a Force about a Point Today • Moment about an Axis • Moment of Couple • Equivalent Force Couple System Lecture 10

  4. F O rA/O A Vector Moment Definition The moment about point O of a force acting at point A is: MO = rA/O x F Compute the cross product with whichever method you prefer. Lecture 10

  5. 200 N 60 o O.4 0.2 60 o 0.285 x d MA =200N *0.247m= 49.4 Nm Example Method # 1 tan 60°=0.2m/x A x=0.115m sin 60°=d/0.285m d = 0.247 m Lecture 10

  6. 200 sin 60 200 N 60 o O.4 0.2 A + M Method # 2 200 cos60 =200N (sin 60)(0.4m)- 200N (cos 60)(0.2m) = 49.4 Nm Lecture 10

  7. 200 N 60 o O.4 0.2 r A F=200N cos 60 i + 200N sin 60 j r =0.4 i + 0.2 j ^ ^ ^ i j k MA= Method # 3 0.4 0.2 0 =200 (sin 60)(0.4) - 200 (cos 60)(0.2) 200cos60 200sin60 0 = 49.4 Nm Lecture 10

  8. 200 N 60 o O.4 0.2 A F=200N cos 60 i + 200N sin 60 j r =0.285 i MA= Method # 4 r =0.285 i i j k 0.285 0 0 = 49.4 Nm 200cos60 200sin60 0 Lecture 10

  9. y ^ |Mn| =MA·n ^ n ^ =n·(rB/A x F ) rAB=rB/A O x B A z Moment of a Force about an Axis F Same as the projection of MA along n nx ny nz rB/Ax r B/A y r B/Az |Mn|= F x F y F z Lecture 10

  10. ^ n y F A ^ Mn=|Mn|n rAB=rB/A B O Mp = MA - Mn x z ^ ^ =n x [(r B/Ax F) x n] Resolve the vector MA into two vectors one parallel and one perpendicular to n. MA Mp Mn Lecture 10

  11. B Let F1 = -F2 y A Mo=rA x F2+ rB x F1 =(rB - rA ) x F1 =rAB x F1= C O x z Moment of a Couple F1 d rAB=rB/A F2 rB rA The Moment of two equal and opposite forces is called a couple |C|=|F1| d Lecture 10

  12. Moment of a Couple (continued) • The two equal and opposite forces form a couple (no net force, pure moment) • The moment depends only on the relative positions of the two forces and not on their position with respect to the origin of coordinates Lecture 10

  13. Moment of a Couple (continued) • Since the moment is independent of the origin, it can be treated as a free vector, meaning that it is the same at any point in space • The two parallel forces define a plane, and the moment of the couple is perpendicular to that plane Lecture 10

  14. Example Lecture 10

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