1. There is a system of two material points $\boldsymbol{P}$ and $\boldsymbol{Q}$, located on some ray. The masses of these points $p$ and $q$ and the distances $x$ and $y$ of each of these points from the origin of the ray $S$ are known.

What is the distance of the center of gravity of the system from the origin of the ray? (O. 116, U. 121, R. 135.)
1. Let $Z$ (Fig. 123) be the center of gravity of a system of two material points, $S Z=z$. Let for definiteness $x<y$. Since $\frac{P Z}{Q Z}=\frac{q}{p}$, then $\frac{z-x}{y-z}=\frac{q}{p}$, from which

$$
\begin{aligned}
(p+q) z & =p x+q y \\
z & =\frac{p x+q y}{p+q}
\end{aligned}
$$

Let $Z \equiv(Z, p+q)$. Then equation (1) can also be written as:

$$
\operatorname{Mom}_{s} Z=\text { Mom }_{s} \boldsymbol{P}+\operatorname{Mom}_{s} \mathbf{Q} .
$$

In more detail: if a system of two material points is located on some ray, then the static moment of the system relative to the origin of the ray is equal to the moment of the combined material points of this system relative to the same origin of the ray.

![](https://cdn.mathpix.com/cropped/2024_05_21_27f86caa754616304e40g-137.jpg?height=89&width=565&top_left_y=441&top_left_x=352)

Fig. 123.