Moment about an arbitrary axis

[member 392791]

Hello,

I am having difficulty in understanding how the moment about an arbitrary axis is found as the scalar product of the moment about the origin with the unit vector of the arbitrary axis. Can anyone elucidate this?

 

[Doc Al]

At the risk of stating the obvious (and thus not being particularly helpful) the moment about the origin is a vector, and taking the scalar product of a vector with a unit vector is how one finds the component of the vector along the axis of the unit vector.

 

[member 392791]

Here are the pages in the textbook that I am referring to, which may help you understand my question more clearly since I am using it as my reference. The second page is in this thread (it appears you can't link the same file in multiple threads)

The topic is figure 3.27 and the confusion is how equation 3.42 is the way it is


EDIT: This is where the 2nd page is linked
https://www.physicsforums.com/showthread.php?t=710626

 

[Doc Al]

The topic is figure 3.27 and the confusion is how equation 3.42 is the way it is

I see the figure but not the equation. Where again is the second page? (Your link just referred to this thread.)

Edit: I see that you edited your post to link to the second page.

Equation 3.42 just defines the moment along an axis as the component of the moment about a point along the axis.

 

[member 392791]

I read that, but to me it still isn't something that I have visualized, and thus I can't really rationalize it, therefore I will have to memorize it, but I want to understand what that means. Why is it that I just dot a unit vector with the moment to get the moment around the entire axis?? Is there another way of explaining this so that I can understand it and feel competent with it?

 

[member 392791]

I think I don't understand why its a dot product of lambda. Also, it's strange to me that you can find the moment at a certain point, then pick any point along an arbitrary axis to find the moment around that axis

 

[Doc Al]

Why is it that I just dot a unit vector with the moment to get the moment around the entire axis??

When you say "entire" axis it sounds like you think that the moment about the axis is somehow greater than the moment about a point. Just the opposite. The moment about an axis is just a component of the total moment about a point.

I think I don't understand why its a dot product of lambda.

Lambda is just a unit vector; taking the dot product with a unit vector is how you get a component of a vector in some direction. Imagine you have a vector $\vec{F}$ in the x-y plane. To get the component of $\vec{F}$ in the x-direction, you'd compute $\vec{F}\cdot\hat{x} = F \cos\theta$, where $\theta$ is the angle the vector makes with the x-axis. That should be familiar to you.

Also, it's strange to me that you can find the moment at a certain point, then pick any point along an arbitrary axis to find the moment around that axis

Yes, you need to convince yourself of this fact. That you can calculate the moment along an axis using any point along the axis as your origin. Try it with some simple examples until it clicks. Obviously the moment changes as you pick a different point along the axis, but the component of the moment parallel to the axis remains the same.