Moment of a Force and Vector Formulation of Resultants

Course Assignment: Engineering Mechanics – ECC 1202.
Lecture Number and Title: Lecture 4 – Force System Resultants.
Institutional Affiliation: University of Guyana, Faculty of Engineering & Technology, Department of Civil Engineering.

Quiz Topic: Equilibrium of a suspended cylinder.
Problem Statement: A $10\,kg$ cylinder is supported by two cables, CA and CB.
Given Conditions:
- The tension in cable CB is specified as being exactly twice the tension in cable CA.
- The mass of the cylinder is $10\,kg$ .
Required Determinations:
- The angle, $\theta$ , necessary to maintain the equilibrium of the $10\,kg$ cylinder.
- The specific tension values for wire CA and wire CB.
Allocated Time: 10 minutes.

Definition of Tendency to Rotate: If a body at rest is subjected to a force, it will tend to rotate about a point that is not located on the line of action of that force.
Terminology: This rotation tendency is technically referred to as a Torque. However, in the context of engineering mechanics, it is more commonly called a Moment of a force or simply a Moment.
Vector Nature: The moment $M_o$ about a point $O$ (or about an axis passing through $O$ that is perpendicular to the plane containing the force) is a vector quantity.
- It possesses both a specific magnitude and a specific direction.
- It acts in a plane horizontal to point $O$ and the force $F$ .

Formula for Magnitude: The magnitude of the moment is calculated using the product of the force and the moment arm:
- $M_o = Fd$
Moment Arm ( $d$ ): The distance $d$ is the perpendicular distance from the axis (point $O$ ) to the line of action of the force $F$ .
Standard Units:
- SI Units: Newton-meters ( $N \cdot m$ ).
- U.S. Customary Units: Pound-feet ( $lb \cdot ft$ ).
Directional Definition: The direction of the moment is defined by its moment axis. This axis is perpendicular to the plane formed by the force $F$ and its corresponding moment arm $d$ .

Planar Constraints: In two-dimensional problems, all forces are assumed to lie within the $x-y$ plane.
The Resultant Moment ( $M_{Ro}$ ): The resultant moment about point $O$ (which corresponds to the $z-axis$ ) is determined by calculating the algebraic sum of the moments caused by all individual forces within the system.
Sign Conventions:
- Counterclockwise (CCW): Moments tending toward a counterclockwise rotation are generally considered positive ( $+$ , $\curvearrowleft$ ) as they are directed along the positive $z-axis$ .
- Clockwise (CW): Moments tending toward a clockwise rotation are considered negative ( $-$ , $\curvearrowright$ ).
Mathematical Representation:
- $\curvearrowleft +, M_{Ro} = \sum F d$
- Example expansion: $M_{RO} = F_1 d_1 - F_2 d_2 + F_3 d_3$
Procedural Establishment of the Moment Arm: To establish the moment arm ( $d$ ) for any given force, the line of action of each force should be extended as a dashed line.
Example Application: Calculation of the resultant moment of four distinct forces acting on a rod about a specific point $O$ .

The Cross Product: Vector multiplication of two vectors $\mathbf{A}$ and $\mathbf{B}$ yields a third vector $\mathbf{C}$ .
- Expression: $\mathbf{C} = \mathbf{A} \times \mathbf{B}$
Geometric Definition: The magnitude of the resultant vector is defined with respect to the angle $\theta$ ( $0^{\circ} \le \theta \le 180^{\circ}$ ) between the two vectors:
- $C = AB \sin(\theta)$
Cartesian Unit Vector Identalities: Using the right-hand rule, the cross products of Cartesian unit vectors $i, j, k$ are as follows:
- $i \times j = k$
- $i \times k = -j$
- $i \times i = 0$
- $j \times k = i$
- $j \times i = -k$
- $j \times j = 0$
- $k \times i = j$
- $k \times j = -i$
- $k \times k = 0$

General Vector Multiplication: The cross product of $\mathbf{A}$ and $\mathbf{B}$ in Cartesian components is:
- $\mathbf{A} \times \mathbf{B} = (A_x i + A_y j + A_z k) \times (B_x i + B_y j + B_z k)$
- Full Expansion: $A_x B_x (i \times i) + A_x B_y (i \times j) + A_x B_z (i \times k) + A_y B_x (j \times i) + A_y B_y (j \times j) + A_y B_z (j \times k) + A_z B_x (k \times i) + A_z B_y (k \times j) + A_z B_z (k \times k)$
- Simplified Result: $\mathbf{A} \times \mathbf{B} = (A_y B_z - A_z B_y)i - (A_x B_z - A_z B_y)j + (A_x B_y - A_y B_x)k$
Matrix Determinant Form: The cross product can be written as a determinant:
- $\mathbf{A} \times \mathbf{B} = \begin{vmatrix} i & j & k \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$
Moment Representation: A moment can be represented as the cross product of the position vector $\mathbf{r}$ and the force vector $\mathbf{F}$ :
- $\mathbf{M_o} = \mathbf{r} \times \mathbf{F} = \begin{vmatrix} i & j & k \\ r_x & r_y & r_z \\ F_x & F_y & F_z \end{vmatrix}$
Example Application: Determining the moment produced by force $\mathbf{F}$ about point $O$ and expressing the final result as a Cartesian Vector.

Historical Context: The Principle of Moments was initially named after the French mathematician Pierre Varignon.
Definition: The principle states that the moment of a force about a point is exactly equal to the sum of the moments of the individual components of that force about the same point.
Practical Application: This allows for easier calculation of moments by resolving a force into $\mathbf{F_x}$ and $\mathbf{F_y}$ components and summing their individual moments.
Examples:
- Calculation of the moment of a force about point $O$ for a generalized image.
- Calculation of the moment produced by a force $\mathbf{F}$ acting at the end of an angle bracket about point $O$ .