# How to find total force on an object?

If there are a multitude of forces acting on an object in different directions, how do we find the TOTAL force?

I know we add up the $x$- and $y$-components of the forces individually, but how do we find the total force? Do we take the $x$- and $y$-components and apply the Pythagorean Theorem to obtain its magnitude?

Force is a vector which itself has components along different axes, but if you just want the magnitude of the vector, then yes, you use the Pythagorean theorem. There's a good page on vector addition here if you want more info.

Quite simply: yes.

When you add up the x and y components, you result in a vector whose x and y components have already been broken down. The total force is equal to the magnitude of that vector.

As a simple example, if you have $10N$ at $0^o$ and $10N$ at $45^o$ you get a resulting vector $\vec{F}$ whose components are $F_x = 17.07N$ and $F_y = 7.07N$. The Force is then $|\vec{F}| = 18.48N$.

Force is a vector, so it has components in both the $x$ and $y$ directions in your specific example. You have to add the components. You find the components by using trig.

$$F_x = F_{x,A}+F_{x,B} + F_{x,C}$$ $$F_y = F_{y,A}+F_{y,B} + F_{y,C}$$

Then the total force is ${\boldsymbol F} = F_x \hat{x} + F_y \hat{y}$

The total force (vector) is the sum of all the forces:$$\vec{F_T}=\sum\limits_i\vec{F_i}$$

Mathematically:$$||\vec{F_T}||=\sqrt{\sum\limits_i||\vec{F_i}||^2}\tag{1}$$

If you have doubt on that you can prove $(1)$ using the mathematical induction.

Yes. Add up the like components and then use the Pythagorean Theorem.