Final answer:
The moment of a couple is calculated using the cross-product of position vectors and force vectors of the two forces in the couple, and is expressed in Cartesian vector form using the components μx, μy, and μz.
Step-by-step explanation:
The moment of a couple in physics refers to a pair of forces, equal in magnitude but opposite in direction, whose lines of action do not coincide. This creates a turning effect or rotational motion without an overall force in any direction. To express the moment (μ) of a couple acting on an assembly in Cartesian vector form, we use the cross-product of the position vector (š) with the force vector (F).
Calculating the Moment of a Couple
The position vector from an origin to a point P can be represented as š(t) = x(t)î + y(t)ķ + z(t)k. If we have two forces represented by their components, say F1 = F1xî + F1yķ + F1zk and F2 = F2xî + F2yķ + F2zk, and their respective position vectors r1 and r2, the moment created by the couple can be found by evaluating the cross-products r1 ✕ F1 and r2 ✕ F2 and then adding the results vectorially.
Therefore, the moment of the couple (μ) is given by:
μ = r1 ✕ F1 + r2 ✕ F2
The cross-product components can be calculated using the determinants of the matrix with unit vectors in the first row, position vector components in the second row, and force vector components in the third row. The resulting moment of the couple in Cartesian vector form will have the components μx, μy, and μz which can be expressed in the î, ķ, k unit vector notation.