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

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Presentation transcript:

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 rotation and 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

ES2501: Statics/Unit 9-2: Moment of Force (3D cases) Cross-Product of two vectors: 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

ES2501: Statics/Unit 9-3: Moment of Force (3D cases) Cross-Product of two vectors: 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

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

ES2501: Statics/Unit 9-5: Moment of Force (3D cases) Moment of a force about a point (Vector Form): 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 !!

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

ES2501: Statics/Unit 9-7: Moment of Force (3D cases) Moment of a Couple 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