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I've recently done a problem where I've been given adiagram with some forces applied to a beam trolley, and where I needed to determine the coordinates of the point on the $y$-axis through which the resultant force $\vec{R}$ passes.

The problem itself isn't what's interesting in this case, so I won't put out all details and show a whole solution. Rather I've a question regarding this method that'll present to you. So here's my procedure of calculating this coordinate:

  1. Calculate the resultant force $\vec{R}$

  2. Calculate the moment $\vec{M_0}$ about the origin $O$

  3. Solving the equation $\vec{M_0} = (y \hat{j}) \times \vec{R}$ for $y$

So, my first question is, why does the last step work? I know that the resultant force $\vec{R}$ can be moved along it's line of action without affecting the moment about the origin. So, are we essentially moving our resultant vector to a starting point that lies on the $y$ - axis, to then calculate for what $y$ the force system becomes equivalent to our original one?

I'd be glad if you could present any visual representation that could help me understand this.

Lastly, I wonder why it's even important to care about the $y$-intercept in the first place? Are there any applications of this?

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