Vector Products and Moments of Force in Statics

Moment of a Force About a Point

The moment of a force $\mathbf{F}$ about a specific point $O$ measures the tendency of that force to cause the rigid body to rotate about that point. Mathematically, the moment $\mathbf{M}_O$ is defined using the cross product of a position vector $\mathbf{r}$ and the force vector $\mathbf{F}$:

$$\mathbf{M}_O = \mathbf{r} \times \mathbf{F}$$

In this equation, $\mathbf{r}$ is the vector directed from the point $O$ to any point on the line of action of the force $\mathbf{F}$.

Magnitude of the Moment

The magnitude of the moment can be simplified if you know the perpendicular distance from the point $O$ to the line of action of the force. This distance is known as the moment arm ($d$):

$$M_O = r F \sin \theta = F d$$

Exam Tip: When calculating the magnitude, always look for the shortest distance from the pivot point to the force's line of action. This "perpendicular distance" is often the fastest way to solve 2D problems without full vector expansion.

The Principle of Transmissibility

A fundamental concept in statics is the Principle of Transmissibility, which states that the conditions of equilibrium or motion are not affected by transmitting a force along its line of action.

Essentially, if you move the point of application of a force $\mathbf{F}$ to a different location, as long as the force maintains the same magnitude, direction, and line of action, the external effect on the rigid body remains identical.

Warning: While the principle holds for external behavior (motion and equilibrium), it does not apply to internal forces or deformations. Two different loading points might produce the same external motion but will cause different internal stresses within the material.

Moment of a Force in 2D Problems

In two-dimensional (coplanar) problems, all forces and the point of rotation lie within a single plane (typically the $xy$-plane). Because the moment must be perpendicular to the plane containing the force and the position vector, the moment vector $\mathbf{M}_O$ will point along the $z$-axis.

By convention:

• Counterclockwise (CCW) rotation: The moment is considered positive.
• Clockwise (CW) rotation: The moment is considered negative.

Varignon's Theorem

Calculating moments can often be simplified using Varignon's Theorem. This theorem states that the moment of a resultant of several concurrent forces about a point $O$ is equal to the sum of the moments of the individual forces about that same point:

$$\mathbf{r} \times (\mathbf{F}_1 + \mathbf{F}_2 + \dots) = \mathbf{r} \times \mathbf{F}_1 + \mathbf{r} \times \mathbf{F}_2 + \dots$$

This allows you to resolve a complex force into its $x$ and $y$ components and sum their individual moments to find the total moment.

Moment of a Force in Space (3D)

In three-dimensional space, the moment is a full Cartesian vector. To find the moment of a force $\mathbf{F}$ applied at point $A$ about an origin $O$, we use the components of the position vector $\mathbf{r} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$ and the force $\mathbf{F} = F_x\mathbf{i} + F_y\mathbf{j} + F_z\mathbf{k}$:

$$\begin{aligned} M_x &= y F_z - z F_y \\ M_y &= z F_x - x F_z \\ M_z &= x F_y - y F_x \end{aligned}$$

Moment About an Arbitrary Point

If you need to find the moment about a point $B$ that is not the origin, you must use the relative position vector $\mathbf{r}_{A/B}$, which is the vector from $B$ to the point of application $A$:

$$\mathbf{r}_{A/B} = \mathbf{r}_A - \mathbf{r}_B = (x_A - x_B)\mathbf{i} + (y_A - y_B)\mathbf{j} + (z_A - z_B)\mathbf{k}$$

The moment is then calculated using the determinant form of the cross product:

$$\mathbf{M}_B = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ x_{A/B} & y_{A/B} & z_{A/B} \\ F_x & F_y & F_z \end{vmatrix}$$

Summary of Equivalent Systems

Two systems of forces are considered equivalent (or equipollent) if they produce the same total force and the same total moment about a given point. This means they impart the same translation and the same rotation to a rigid body.

Any system of forces can be reduced to a single force $\mathbf{R}$ and a single couple $\mathbf{M}_O$ acting at a specific point. In specific cases, such as concurrent, coplanar, or parallel forces, this can be further reduced to a single resultant force.