eli5: moments about a specified axis? conceptual understanding.

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Yes, I know the formula is U dotted with (r x F), but I have no conceptual understanding. Any video or text I find, just tells me how to do it…but I have no idea what I am doing. How does the dot product help with this, I just expect something… cross-product related only.

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How does dotting the unit vector of a specified axis with the moment vector we calculated give us the moment around said specified axis. Do I just have a dumb understanding of the dot-product? Can someone kindly explain like I am 1.

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M is the magnitude of the moment, a dimensionless scalar. U is the unit vector, which defines the direction around which the moment acts. It’s a dot product because you might calculate a moment force that rotates around an axis that doesn’t align with the any of the x,y, or z axes. By using the dot product, you can calculate the total magnitude of M when it doesn’t align with an axis.

Imagine you’re trying to figure out a good way to represent a rotation. How do you do it?

At first it probably makes sense to use some kind of vector that shows the direction and magnitude of the rotation. The magnitude is no problem, you just figure out how fast some point is moving at any given time. Except, wait, a bit near the rim of the object moves faster than a bit near the axis. Hmm, how can you pick a point that represents how fast something is rotating?

You can’t use any physical bit of the object because they’re all moving at different speeds based on how far they are from the axis. Instead, look at the invariant. What is *not* changing? The angle per second. Okay, so you can simply measure how many degrees the object rotates through per second and that’s the magnitude. Good.

But we’re only halfway there…what about direction? Every bit of the thing is also changing the direction it’s moving every moment, too.

Again, look at the invariant. What’s *not* changing direction if something is spinning around an axis? The axis. So to signify direction, just represent the “direction of spin” as a vector along the axis.

An axis points in two directions. But that object can also spin in two direction, clockwise and counterclockwise. So, you can use the right hand rule as an arbitrarily chosen convention. If the object is rotating in the direction of the curled fingers of a right hand, the direction of the vector points along the thumb.

Now you have a convention for representing both magnitude (angular speed) and direction (according to right hand rule) to represent an angular velocity.

If you think about how a force works in a translational context, you can use the same thinking to translate that into a rotational context to understand what a moment is. In translational problem if you apply a force to a block, you get how all the mechanics work, but in a rotational context, the force is not enough information to understand how that translates into angular acceleration. The reason is that if you apply the same force close to an axis of rotation vs. far from it, the lever arm matters. The longer the arm a force acts over, the more influence it will exert on rotational acceleration.

You can see this if you think about a force that goes straight toward the axis of rotation. In that case, if all of the force of a kick is directed toward the center of the ball, none of that force will cause a rotation. The ball will just get pushed forward, no rotation.

If the force is directed off-center, some component of the force goes into translation, and some component goes into rotation. The more off-center the force, the more of that force that’s directed into rotation instead of translation. At the extreme, in an ideal scenario, a grazing kick could act over the whole radius arm of the ball and go completely into rotation, leaving the ball spinning in place.

If you think about how the dot and cross product work together in the equation, you’ll see that the cross is responsible for sorting out how to create the rotational vector, and the dot works out how much force is directed into rotating (whereas the remainder goes into translating).