Knowing when to decompose weight vector vs. normal vector

In summary: The normal force is perpendicular to the velocity vector so it makes sense to choose the x-axis down the incline.The normal force is perpendicular to the velocity vector so it makes sense to choose the x-axis down the incline.
  • #1
TRB8985
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15
Homework Statement
Knowing whether there is a "correct" way to approach breaking down vectors by decomposing the normal force compared to the weight.
Relevant Equations
F = ma ; W = mg
Good afternoon everyone,

I have a question on Newton's 2nd Law regarding objects on a generic incline. Take for example, a car on a banked curve:

SY6No.png
Here in the picture I've provided, you can see that the normal force has been decomposed into the x and y components via sine and cosine of the angle multiplied by N.

My question is.. is there a particular reason that seems to be the norm? When I was an undergraduate, I seemed to always be under the impression that the normal was strictly perpendicular to a surface, and instead, it was the weight vector that was decomposed with the coordinate system parallel to the tilted surface, like this:

1.png
My initial thought was that these were equivalent ways of approaching the same idea. Just wanted to be sure.
 
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  • #2
First of all, the acceleration of the system will be the same regardless how you draw your ##x## and ##y## axes in terms of which you write your vectors. Nature doesn't know the alphabet. The rule of thumb is that if you have two-dimensional situations, as in the one above, you choose axes so that the acceleration is along one of the principal axes.

This is recommended but not cast in stone. It makes Newton's second law equations easier to handle because the acceleration is zero in the other direction. In the example of the car going around the curve, the acceleration is horizontal so the ##x##-axis is along the horizontal. In the example of the block sliding down, the acceleration is along the incline, so it makes sense to choose the ##x##-axis down the incline.

As an exercise, you might wish to (try and) solve either problem using different axes and see how much more involved it gets mathematically. Nevertheless, if you do everything right, the answer will be the same.
 
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  • #3
Great, glad to hear it.

That actually sounds like a good idea for a YouTube video.. Thanks, Kuruman!
 
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  • #4
Good luck.
 
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  • #5
kuruman said:
In the example of the car going around the curve, the acceleration is horizontal so the x-axis is along the horizontal
The centripetal (normal) acceleration is along the x-axis, the car might as well have tangential acceleration along the z-axis (perpendicular to the plane of the page). But yes in general we choose the axis so that the acceleration (normal or tangential) is along one of the axis.
 
  • #6
Delta2 said:
The centripetal (normal) acceleration
Just to clarify, "normal" there is in relation to the velocity vector, not in relation to the plane.
 
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  • #7
haruspex said:
Just to clarify, "normal" there is in relation to the velocity vector, not in relation to the plane.
Yes I mean normal to the velocity vector ##\mathbf{v}## or normal to the trajectory infinitesimal element ##d\mathbf{l}=\mathbf{v}dt##.
 

Related to Knowing when to decompose weight vector vs. normal vector

1. What is the difference between a weight vector and a normal vector?

A weight vector is a vector that represents the importance or significance of each feature in a machine learning model. It is used to calculate the final prediction of the model. On the other hand, a normal vector is a vector that is perpendicular to a surface and is used to determine the direction of a plane or line.

2. When should I decompose a weight vector instead of a normal vector?

You should decompose a weight vector when you want to understand the contribution of each feature in your model's prediction. This can help you identify which features are most important and potentially improve the performance of your model. You should decompose a normal vector when you want to understand the orientation or direction of a surface or line.

3. How do I decompose a weight vector?

To decompose a weight vector, you can use a technique called feature importance, which involves analyzing the weights of each feature in your model and ranking them based on their importance. Another method is to use a technique called partial dependence plots, which shows the relationship between a specific feature and the model's prediction while keeping other features constant.

4. Can I decompose both a weight vector and a normal vector at the same time?

Yes, it is possible to decompose both a weight vector and a normal vector at the same time. However, the purpose and methods for decomposing these vectors may be different, as explained in the previous questions.

5. Why is it important to know when to decompose a weight vector vs. a normal vector?

Understanding when to decompose a weight vector vs. a normal vector can help you gain insights into your model and improve its performance. It can also help you interpret the results of your model and make more informed decisions. Additionally, knowing when to decompose these vectors can save you time and effort by focusing on the most relevant information for your specific goals.

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