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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.

static equilibrium, vector cross product Mo formula, determinant matrix diagram, physics education

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

3D Moment equation diagram, showcasing static equilibrium principle: \( M_O = (r_y F_z - r_z F_y)i - (r_x F_z - r_z F_x)j + (r_x F_y - r_y F_x)k \).

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.

Moment of inertia matrix representation with vectors i, j, k; static equilibrium equation.

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.

Tags

Cartesian FormVector FormulationMoment Of ForcePosition VectorForce VectorCross productUnit VectorsDeterminantRevolving DoorN mCounterclockwise Rotation

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