Let’s talk about couple moments. Imagine you’re the captain of a ship and in front of you is
the ships wheel. To turn it, you need both hands and you apply a force here and here. Notice that
while one force faces up, another faces down. By applying these 2 forces, we can turn this wheel,
in this case, counter clockwise. This moment we create, is called a couple moment. In 2D problems,
all we need is the distance from one force to the other. Even if we move this force to this point,
or this point, it won’t make a difference, because what determines the couple moment is the
perpendicular distance between them. This is also why a couple moment is called a free vector. So in
scalar, we can find the moment by multiplying the force by the perpendicular distance.
When it comes to a vector formulation, we take the cross product between a position vector and
the force. Imagine a pipe system like this and we have a couple force applied at these locations.
When we find the couple moment, the position vector can be directed from any point on the
line of action of one force to any point on the line of action of the other force.
So for example, here, we can use a position vector from A to B to find the couple moment.
One thing you have to keep in mind is that since we chose a position vector going from A to B,
we have to ignore the force at A. That is because at point A, force 1 can’t create a moment about
point A. Instead, we find the couple moment by assuming that the pipe turns about point A. If you
had a position vector from B to A, then we would ignore the force at B instead. This will become
much clearer when we look at the last 2 examples. So without further ado, let’s get to started.
Let’s take a look at this question where we need to find the couple moment produced by
the 2 forces. The distance between the forces is 0.3 m, so all we need to do is multiply
that by the force. We will assume clockwise movement to be positive. And that’s our answer.
Let’s look at this question where we need to figure out the magnitude of force F so that
the resultant couple moment is 400 newton meters clockwise. So what we need to do is add up all
the couple moments created by each of the forces and equal it to 400. First, let’s figure out the
length of the triangle. Using trigonometry, you will find that each side of the triangle
is 0.653 m. So let’s write our equation, we will assume clockwise rotation to be positive. We see
that both the 250 N force, and the unknown force will create a clockwise rotation while the 600 N
force will create a counter clockwise rotation. The distance between the 250 N force is 1 m, and
the distances between each of the other forces are 0.653 m. Let’s solve for F and that’s our answer.
Let’s take a look at this problem where we have 2 wrenches applying a force and we need to figure
out the couple moment created. So a moment can be found using this equation, which means what
we need is a position vector from points A to B, since that’s the distance between the two forces.
We also need to express the 125 N force in cartesian form.
Since our position vector goes from points A to B, the moment of the force at point A is 0, so we
ignore it. The force at B is straight upwards, so it’ll just have a positive k component.
Now that we have a position vector and a force, we can take the cross product.
Let’s take a look at one last example where we have to express the resultant moment created by
4 forces effecting the pipes. So the way to solve a problem like this is to figure out the couple
moment created by each of the pairs of forces. Let’s look at the upwards forces first. To find
the couple moment, we need a position vector from A to B or B to A, I am going to pick A to B. To
get that, we first need to figure out the location of point B. Using trigonometry, the y component
is 0.4cos30 degrees and the zed component is 0.4sin30 degrees. The x component is already
given, which is 0.35m. Let’s write down the locations of point A and C as well. Note that the
difference between points B and C is only in the x direction. Now, a position vector from A to B.
Since our position vector starts from A, we ignore the force at A, so we consider the other force,
which is straight upwards in the zed axis. So we can now take the cross product.
Let’s look at the other couple force now. For this, we can use the same position vector as
before, because the line of action of force C crosses the position vector, but I will use
a position vector from C to A to show that its also possible to get the answer this way as well.
You can also use a position vector from A to C. Since we chose a position vector going from C to
A, we ignore the force at C and consider the force at A. Now we take the cross product.
The last step is to find the resultant moment, so all we do is add these together. Remember,
we add I components to I components, j components to j components and k components to k components.
That should cover the types of problems you will face when it comes to couple moments. I hope this
video helped, and if it did, please consider sharing it with your friends and classmates.
They too might find it helpful. Thanks for watching and best of luck with your studies.