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4.5: Cartesian Form for Vector Formulation

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Cartesian Form for Vector Formulation
 
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4.5: Cartesian Form for Vector Formulation

The Cartesian form for vector formulation is a process to calculate  the moment of force using the position and force vectors. The moment of force is defined as the cross-product of these vectors, making it a vector quantity. The Cartesian form of the position and force vectors involves unit vectors, which can be used to express the cross-product in determinant form.

Equation 1

The determinant can then be expanded to determine the Cartesian vector formulation of the moment of the force.

Equation 2

For example, suppose a force of 10 N is applied to a revolving door along the x-direction at a point given by the position vector r = 0i - 4j + 0k in the Cartesian component form. In this case, the force vector is perpendicular to the y-z plane along which the door stands, which means that the y and z components of the force vector are zero. To calculate the moment of force, a determinant is constructed using the position vector and the force vector, F = 10i + 0j + 0k.

Equation 3

Finally, by expanding the determinant, we can determine that the revolving door's moment of force equals 40 N.m about the z-axis. The positive moment implies that the door rotates counterclockwise.

The Cartesian form for vector formulation allows us to calculate the moment of force using position and force vectors in a straightforward manner.


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Keywords: Cartesian Form Vector Formulation Moment Of Force Cross-product Unit Vectors Determinant Cartesian Vector Formulation Revolving Door Force Vector Position Vector Counterclockwise Rotation

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