4.5:
Cartesian Form for Vector Formulation
The moment of force is defined as the cross-product of position and force vectors.
The cross-product for the moment can be expressed in the determinant form, using unit vectors and the Cartesian form of position and force vectors.
The determinant is expanded to determine the Cartesian vector formulation of the moment of force.
Consider that a force of 10 N is applied to a revolving door at a point given by the position vector in the cartesian components form. Here, the moment of force in the vector formulation needs to be determined.
In this case, the force vector is perpendicular to the y–z plane , which results in the y and z components of the force vector being zero. A determinant is constructed using the position vector and the force vector.
Finally, by expanding the determinant, the revolving door's moment of the force about the z-axis is determined.
Here, the moment is in the positive z-direction, implying the door rotates counterclockwise.
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.
The determinant can then be expanded to determine the Cartesian vector formulation of the moment of the force.
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.
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.
Suggested Reading
- Hibbeler, R.C. (2016). Engineering Mechanics ‒ Statics and Dynamics. Hoboken, New Jersey: Pearson Prentice Hall. pp 129.