M2-4 - Module Objective By the end of this Module you...

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CIV100 – Mechanics Module 2: Force and Moment in 3D by: Jinyue Zhang 2011/9/25 1 Module Objective By the end of this Module you should be able to: Understand the concept of vector in 3D. Determine the addition and product of two or more 3D forces. Understand the concept of couple and moment in 3D. Reduce/Simplify a force system Reduce simple distributed loading Draw free body diagram in 3D Understand the concept of Equilibrium in 3D 2011/9/25 2 Today’s Objective Represent a 3D vector in Cartesian coordinate system Study the relationship between a 3D resultant force and its components Understand the concept of unit vector in 3D Add/Subtract Cartesian vectors (forces) 2011/9/25 3 3D in Our Real Life
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2011/9/25 4 Cartesian Coordinate System Cartesian Unit Vector Cartesian coordinate system is a Right-Handed Coordinate System Cartesian Unit Vectors x y z zx y x z y x 2011/9/25 5 Rectangular Components of a Vector Resolve a vector into 3 components along x, y and z axes. F' F F z + = x y z F Fz F’ Fx Fy y x F F F' + = z y x F F F F + + = 2011/9/25 6 Unit Vector What is a unit vector? – Indicate a line of action and sense – Has a magnitude of 1 How do we use unit vector? A u A A = Vector A Magnitude of vector A Unit vector of A (indicate the direction and sense) U A 2011/9/25 7 Cartesian Vector Representation In 2D, we have k j i A z y x + + = j i F F F F y x F + = + = In 3D, similarly To Remember k j i A A A A A z y x + + = + + =
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2011/9/25 8 Magnitude of the Resultant In 2D, we have 2 2 y x R F + = In 3D, similarly 2 2 2 2 2 ' z A + + = + = 2 2 ' + = To Remember 2 2 2 + + = 2011/9/25 9 Direction of a Cartesian Vector k j i k j i k j i u A A + + = + + = + + = = γ β α cos cos cos cos cos (cos ) k j i k j i k j i A u u A A A cos cos cos ) ( 1 + + = + + = + + = = = = cos = cos = cos To Remember 1 cos cos cos 2 2 2 = + + Coordinate Direction Angles 2011/9/25 10 Add Two or More 3D Vectors R = A + B In General )k )j )i B A R )k )j )i B A R k j i B k j i A B + + = = + + + + + = + = + + = + + = ( ( ( ( ( ( To Remember k j i F F + + = = k j i F F + + = = 2011/9/25 11 Example Express the force F shown as a Cartesian vector. (hint: using i,j,k)
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2011/9/25 12 Example Express the force F shown as a Cartesian vector. (hint: using i,j,k) Pay attention to significant figures! 2011/9/25 13 Example Determine the magnitude and the coordinate direction angles of the resultant force acting on the ring. 2011/9/25 14 Example Determine the magnitude and the coordinate direction angles of the resultant force acting on the ring. Pay attention on significant digits. Pay attention to significant figures! 2011/9/25 15 Review Questions The symbols α , β , and γ designate the __________ of a 3-D Cartesian vector. A) unit vectors B) coordinate direction angles C) x, y and z components D) magnitudes of x, y and z components If you know just U A , you can determine the ________ of A uniquely.
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M2-4 - Module Objective By the end of this Module you...

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