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ES2501: Statics/Unit 9-1: Moment of Force (3D cases)

ES2501: Statics/Unit 9-1: Moment of Force (3D cases). General:. - To extend the definition of the moment to 3D problems a vector definition based on the cross-product of two vectors must be used. The moment of a force is a vector whose direction represents the

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ES2501: Statics/Unit 9-1: Moment of Force (3D cases)

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  1. ES2501: Statics/Unit 9-1: Moment of Force (3D cases) General: • - To extend the definition of the moment to 3D problems a vector • definition based on the cross-product of two vectors must be used. • The moment of a force is a vector whose direction represents the • axis of rotationand whose magnitude represents the moment of • the force about the axis. • Two definitions are equivalent but the vector definition is more • convenient for 3D problems. • - Need knowledge of the cross-product of two vectors

  2. Cross-Product of two vectors: ES2501: Statics/Unit 9-2: Moment of Force (3D cases) Definition: ------ new vector Note: The result is a Vector Applications: Moment of a force Angular momentum Area of the parallelogram spanned by the two vectors

  3. Cross-Product of two vectors: ES2501: Statics/Unit 9-3: Moment of Force (3D cases) Properties: ( Non-commutative) ( associative with multiplication by a scalar)) ( distributive) ( condition for parallel lines) ( cross-product of coordinate unit vectors) Equivalence?? Expression in terms of Cartesian Components

  4. Cross-Product of two vectors(con’d): ES2501: Statics/Unit 9-4: Moment of Force (3D cases) Ezample: Check: The same results!

  5. Moment of a force about a point (Vector Form): ES2501: Statics/Unit 9-5: Moment of Force (3D cases) Moment of about point O Position vector from O to ANY point A on the line of action of the forceabout point O General Comments: • Moment of a force about a point is represented by a VECTOR determined • by the cross product of a position vector from the point to any point on • the line action of the force and the force itself. • Direction: use the right-hand rule • Magnitude: • For a system of forces: • -A can be ANY point on the line action of the force In consistent with 2D definition! Additive the same d !!

  6. Calculation of Moment in term Cartesian Components: ES2501: Statics/Unit 9-6: Moment of Force (3D cases) Example: Fine the total moment of all three forces about point O easier systematic

  7. Moment of a Couple ES2501: Statics/Unit 9-7: Moment of Force (3D cases) Couple ----- a pair of forces with the same magnitude and opposite direction Moments of a Couple: Distance between lines of action of the couple of force A Position vector from B to A A --- Any point on the line of action of B --- Any point on the line of action of Moments of a Couple B O’ O Moments of a couple about ANY point is the same

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