Statics

For cross-products, the end result is a vector. It's perpendicular to the plane created by the source vectors.
For cross-products, the order of the vectors is important. AxB ≠ BxA.
If you reverse the order of the operation, then you are reversing the sign of the answer.

Practice until you have the ability to compute cross products quickly. Ideally, use your calculator for this operation. If your calculator doesn't have this functionality pre-built in, program it in yourself.

Stay tuned: we will discover applications for the cross-product in Lesson 14.

🟦 15.3 Moment of a force in 3D space: equation and example

In this flipbook you will learn how to calculate a moment of a force in 3D space.

The equation for the moment of a force in vector notation is:

This equation is pronounced "MoooRoooaaaaaxxxxxFFFFF." Like "more rocks fffff."

It's also common to simply say "M equals r cross F."

🟦 15.4 Moment of a force: a simple 3D problem

Let's say that you want to turn this crank by applying a force.

Unfortunately, the crank (of undetermined purpose) is in a cramped space in your basement utility room. There are a bunch of HVAC ducts in the way. Instead of being able to push in the y-direction, your push force is at a weird angle, as shown.

You compute the moment about point O like this:

How much of the moment is interesting to us? Since we are trying to turn the crank, the only part of the moment that we'd pay attention to is the moment about axis b-b. How much moment (tendency to rotate) is about axis b-b? Simply project the moment vector to axis b-b to get -380 inch-pounds. (That's just the y-component of the moment.)

🟦 15.5 Moment of a force: more advanced 3D problem

In this example problem, a 520 N force has been applied to the end of bar OA.

Point O can be considered to be fixed, so that the bar is in static equilibrium.

Our goal is to compute the moment at O caused by the force F.

Attribution: I believe that this problem originated from the Hibbeler Statics textbook, although I'm not 100% certain.

First, explore the interactive. It's one of the best ones I have made to date, so inspect it carefully.

Then, work it independently.

❏ Interactive: moment of a force

Here is a written solution.

Here is an alternate written solution. It's the same answer, but a different approach.

🟦 15.6 Moment summations about an axis in 3D space

Let's consider a force, F, acting on a yellow-colored geode.

For some reason, you want to compute the tendency to rotate (the moment) about line CE.

The first step is to compute the moment at C using the cross-product of the force and position vector.

For the second step, we need to do a vector projection operation.

❏ Moment about an axis

We have already learned to use the dot product to project a vector to an axis. This produces a scalar. A positive scalar means it points double arrow from O to A. A negative scalar points from A to O.

To solve for that moment in vector notation, simply multiply the dot product back by the unit vector.

Remember:

If you just do the dot product, the result is a scalar moment.
If you multiply the dot product back by the unit vector, then you can write the projected moment in vector form.

❏ Projecting a moment to an axis

The cool thing about that equation is that line OA doesn't have to be parallel to x, y, or z. It can be any random line in 3D space. That fact will be key to our problem-solving process in 3D space. We sum moments strategically about different axes to solve for unknowns.

Don't waste time projecting to the x-, y-, or z- axes, though. Those moments are the three components of the moment vector itself!

Are you a math geek? You can also project a moment to an axis in one big equation, called the scalar triple product! Dig in if you're interested.

🟦 15.7 Computing 3D centroids

This sculpture is comprised of three identical cuboids.

Each cuboid measures 1" x 1" x 10" and weighs three pounds.

We want to determine the sculpture's center of weight. This is because the sculptor wants to suspend the sculpture from a single cable, and the center of mass isn't computed correctly, the sculpture will tilt.

Thinking through the E.o.E., we quickly deduce that the upward force in the cable must equal 3# x 3 cuboids = 9 pounds.

Next, we project the three planar views of the sculpture, with the 9# force located at (xbar, ybar, zbar) from the origin.

You'll complete this problem on your own; the projections are given to you to help you think through it.

❏ Center of a 3D sculpture

❏ Three 2D FBD projections of the sculpture

Note: since (xbar, ybar) does not lie on the body, it is not possible to support the sculpture in this position with a single cable. You would need to use multiple support points to achieve the sculptor's vision.

🟦 15.8 Putting it all together: 3D Statics problems

Fall 2025 students - I'll be doing these examples in class. One will be a 3D truss ("space truss") example.

🟦 15.9 Summary: the Statics Toolbox for 3D problems

The infographic below provides a summary of our Statics tools for 2D and 3D problem-solving.

The concepts are the same. The tools are a bit different.

➜ Practice Problems

Problem 1:

A sculpture has been created by gluing three cuboids together. Each measures 1" x 1" x 10" and weighs 3 pounds.

Find the center of weight of the sculpture. Report your answer as (xbar, ybar, zbar) relative to the given origin.

Answer:

(8, 2, 6) in.

Problem 2:

This sculpture looks the same as before, but it's a little bit different.

It still was created by gluing three cuboids together. Each measures 1" x 1" x 10." But each one is a different material.

Component 1 is the lightest. It has a density of 0.1 pounds per cubic inch. Component 2 has a density of 0.2 pounds per cubic inch. And component 3 is the heaviest, with a density of 0.3 pounds per cubic inch.

Find the center of weight of the sculpture. Report your answer as (xbar, ybar, zbar) relative to the given origin.

Answer:

(8.75, 2.75, 7.83) in.

Problem 3:

A heavy plate supports two loads and is supported as shown.

Include the self-weight of the plate in your analysis.

Solve for all reactions.

Answer:

Problem 4:

A pin-pin flagpole (member 3) is held in place by three cables (commonly called "guy wires" in this type of application).

Cable 1 is tensioned to 100# as shown.

Solve for the force in Cable 2.

Note: there are two members labeled Cable 2. They are identical and symmetric.

Here is a link to the 3D CAD model (in case the image isn't clear).

Answer: 87.2# (T)

Problem 5:

A box (W = 30 kips) is hung by three cables from three pin supports. The three supports are all at the same elevation and form an equilateral triangle (12' by 12' by 12'). Node D lies 10 feet below the ceiling.

How much tension force is carried in each of the three cables? Here is an interactive model that may help you think through the geometry.