Rigid Bodies Equivalent Force Systems

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Rigid Bodies Equivalent Force Systems

• Transmissibility Principle
• Force F is a sliding vector.

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Vector (Cross) Product

• Multiply 2 vectors P?Q resulting in a vector.
• Define magnitude of product.
• Define direction of product.

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V P?Q

• Magnitude V PQ sinq , 0 lt q lt p
• Direction perpendicular to plane of P and Q
• Sense determined by
• right hand rule

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Geometrical Meaning

• hQ sinq
• ? V PQ sinq Ph
• area of parallelogram formed by P and
  Q

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Properties

• Colinear P?Q 0
• Non-commutative P?Q -Q?P
• Distributive P?(QR) 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 (Pxi Pyj Pzk)?(Qxi Qyj Qzk)
• (PyQz-PzQy)i
• (PzQx-PxQz)j
• (PxQy-PyQx)k

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Component Form by Determinants

• P?Q
• (PyQz-PzQy)i
• (PzQx-PxQz)j
• (PxQy-PyQx)k

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Position Vector

• Position vector from point O to M
• r xi yj zk

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Moment of a Force about a Point

• The moment of a force about point O is defined
  by MO r?F (M rF sinq)
• Units of moment are N m or, lb in

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Moments in the Plane

• MO r?F M rF sinq dF

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Varignons Theorem

• Moment of the resultant
• Resultant of the moments
• r ? (F1 F2 F3 ) r?F1 r?F2 r?F2

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Components of Moment

• r xi yj zk F Fxi Fyj Fzk
• MO r?F
• (yFz-zFy)i (zFx-xFz)j (xFy-yFx)k
• In the plane MO r?F (xFy-yFx)k

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DISMISSED !