ECS221L7

# ECS221L7 - Position vector from point O to M r = xi yj zk x...

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Rigid Bodies; Equivalent Force Systems Transmissibility Principle Force F is a sliding vector. = F F

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Vector (Cross) Product Multiply 2 vectors P × Q resulting in a vector. Define magnitude of product. Define direction of product.
V = P × Q Magnitude: V = PQ sin θ , 0 < θ < π Direction: perpendicular to plane of P and Q Sense: determined by right hand rule θ V P Q

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Geometrical Meaning h=Q sin θ V = PQ sin θ = Ph = area of parallelogram formed by P and Q Q P h θ
Properties Colinear : P × Q = 0 Non-commutative : P × Q = -Q × P Distributive: P × ( Q +R ) = P × Q + P × R Non-associative : (P × Q ) × R P × ( Q × R )

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Component Form i × i = 0 j × j = 0 k × k = 0 i × j = k k × i = j j × k = i P × Q = (P x i + P y j + P z k ) × (Q x i + Q y j + Q z k ) = (P y Q z -P z Q y )i + (P z Q x -P x Q z )j + (P x Q y -P y Q x )k j i k i k j
Component Form by Determinants P × Q = = (P y Q z -P z Q y )i + (P z Q x -P x Q z )j + (P x Q y -P y Q x )k z y x z y x Q Q Q P P P k j i

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Position Vector

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Unformatted text preview: Position vector from point O to M: r = xi + yj + zk x z y r M O Moment of a Force about a Point The moment of a force about point O is defined by: M O = r × F (M = rF sin θ ) Units of moment are N m or, lb in r O F M o Moments in the Plane M O = r × F M = rF sin θ = dF F r M o d θ F r M o d θ Varignon’s Theorem “Moment of the resultant = Resultant of the moments” r × ( F 1 + F 2 + F 3 + …) = r × F 1 + r × F 2 + r × F 2 Components of Moment r = xi + yj + zk F = F x i + F y j + F z k M O = r × F = (yF z-zF y )i + (zF x-xF z )j + (xF y-yF x )k In the plane: M O = r × F = (xF y-yF x )k...
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