CH3-Force System Resultants: Moment of a Force & Moment About an Axis

Moment of a Force about a Specific Axis

• Scalar Analysis:

  • Moment about point O: $M<em>O = F d</em>O$, where $d_O$ is the perpendicular distance.

  • Can be complex for 3D forces and axes.

• Vector Analysis:

  • Goal: Find moment of force $F$ about an $a$-axis ($M_a$).

  • First, compute moment of $F$ about any point O on the $a$-axis: $M_O = r \times F$.

  • Then, find the component of $M<em>O$ along the $a$-axis using the dot product: $M</em>a = u<em>a \bullet M</em>O$.

  • Alternatively, use the triple scalar product: $M<em>a = u</em>a \bullet (r \times F)$.

    • $u_a$: Unit vector along the $a$-axis.

    • $r$: Position vector from any point on the $a$-axis to any point on the line of action of $F$.

    • $F$: Force vector.

Moment of a Couple

• Definition: Two parallel forces of equal magnitude and opposite directions, separated by a perpendicular distance $d$.

• Effect: Produces only rotation (resultant force is zero).

• Couple Moment: The moment produced by a couple.

  • Determined by summing moments of both couple forces about any arbitrary point.

  • It is a free vector: it can act at any point as its effect depends only on the relative position between the forces, not the origin.

• Scalar Formulation:

  • Magnitude: $M = Fd$, where $F$ is the magnitude of one force and $d$ is the perpendicular distance.

  • Direction: Determined by the right-hand rule, perpendicular to the plane containing the forces.

• Vector Formulation:

  • $M = r \times F$, where $r$ is a position vector from any point on the line of action of $-F$ to any point on the line of action of $F$.

• Equivalent Couples: Two couples are equivalent if they produce moments with the same magnitude and direction.

• Resultant Couple Moment: Since couple moments are free vectors, their resultant is found by vector addition: $M_R = \sum M$.