# Force Resultant is equal to the sum of the components -- why?

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• Bassel AbdulSabour
In summary, the resultant's X and Y components of two forces are equal to the sum of the X and Y components of the two forces because of the addition laws of a vector space and the Principle of Superposition in Classical Mechanics.
Bassel AbdulSabour
Why are the resultant's X and Y components of two forces equal the sum of the X and Y components of the two forces?

Bassel AbdulSabour said:
Why are the resultant's X and Y components of two forces equal the sum of the X and Y components of the two forces?
Why not? What else would they be equal to?

Bassel AbdulSabour said:
Why are the resultant's X and Y components of two forces equal the sum of the X and Y components of the two forces?
Because these are the addition laws of a vector space: ##\begin{bmatrix}u_x\\u_y\end{bmatrix}+\begin{bmatrix}v_x\\v_y\end{bmatrix}=\begin{bmatrix}(u+v)_x\\(u+v)_y\end{bmatrix}\,.## This coincides with the geometric vector addition of arrows. But whatever you take as a definition, they should yield the same result.

berkeman

If one more person was added to the right end of the rope, which side would win?

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Pulling or pushing ?

Bassel AbdulSabour said:
Why are the resultant's X and Y components of two forces equal the sum of the X and Y components of the two forces?
It follows from one of the axioms of Classical mech.: the Principle of Superposition

## 1. What is the definition of force resultant?

The force resultant is the net or overall force acting on an object, taking into account both magnitude and direction.

## 2. Why is the force resultant equal to the sum of the components?

This is based on the principle of vector addition, where forces acting on an object can be broken down into their individual components. The sum of these components in each direction results in the total force acting on the object.

## 3. How does the direction of the force resultant relate to the components?

The direction of the force resultant is determined by the direction of the individual components. If the components are acting in the same direction, the force resultant will also act in that direction. If the components are acting in opposite directions, the force resultant will act somewhere in between.

## 4. Is the force resultant always equal to the sum of the components?

Yes, as long as the components are acting on the same object and in the same direction. If the components are acting on different objects or in different directions, the force resultant will not necessarily be equal to the sum of the components.

## 5. Can the force resultant be greater than the sum of the components?

No, the force resultant can never be greater than the sum of the components. This is because the force resultant takes into account both magnitude and direction, and the sum of the components already includes the magnitude of each individual force.

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