I Original direction of force versus vector components

What happens to a mechanical force's real original direction i.e. when we divide it into components of basis vectors, which in turn change as per problem at hand (like gravity components at inclined plane ), how we arrive at correct physics by taking two/three arbitrary directions of our choice and forgetting about all other directions and of-course the original direction?

Basically does the object not feel the force in the original direction in which the force was applied to it?
 

Doc Al

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Basically does the object not feel the force in the original direction in which the force was applied to it?
Dividing a force into components just makes things easier to analyze. You still need to consider the entire force, thus all of the components. (In some cases, you know that certain components are 'canceled out' by other forces. That makes things easier in determining the net force.)
 
Dividing a force into components just makes things easier to analyze. You still need to consider the entire force, thus all of the components. (In some cases, you know that certain components are 'canceled out' by other forces. That makes things easier in determining the net force.)
I am comfortable within purview of pure Mathematics that we can divide components of vectors into basis vectors and also I can understand the theorem that any vector is uniquely represented as linear combination of basis vectors. Those Basis vectors we are free to choose, but once we choose, it has unique components in those basis vectors directions.

What I am struggling to understand is physical justification apart from ease of analysis.

Is it empirical observations that we can divide force into components or is there a deeper Mathematics involved that I can further study...?
 

robphy

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What I am struggling to understand is physical justification apart from ease of analysis.

Is it empirical observations that we can divide force into components or is there a deeper Mathematics involved that I can further study...?
Are you asking how we know "force" is a vector?
[appears to satisfy the parallelogram rule of addition, etc...]
 

Doc Al

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Is it empirical observations that we can divide force into components or is there a deeper Mathematics involved that I can further study...?
The mathematics is simply that of vectors. If, as robphy suspects (and I agree), you are asking how do we know that force is a vector, then that's where empirical experience comes in. It works.
 
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Everyday intuitive experience is enough. If you compare a single resultant force with its components you'll see two equivalent force systems. Both viewpoints are interchangeable as far as net practical effect.
 
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Newton's Principia, Book I, Laws of Motion, Third Law, Corollary I is where he proves the vector sum. He justifies it by his second law, which states: "The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed." The idea is that since the action of the force is rectilinear, the action of two forces simultaneously will give the same result as if the forces acted consecutively, which after all is what vector addition amounts to (think of the "tip to tail" method of vector addition). The mere fact that they are acting at the same time does not affect the motion, since they are rectilinear and therefore do not interfere with each other. This gives us the freedom to examine them one at a time, i.e. to decompose the net force into vectors.
 
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The term vector can correctly be defined several ways. A vector can be thought of as an imaginary object used, for example, to mathematically model a force. Additionally, a scientist's definition (must transform under a proper rotation) may be more restrictive than a mathematician's.
 

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