8. Two material points $(P, p)$ and $(Q, q)$ lie on the same side of a certain line $M N$ (Fig. 16). Their distances from the line $M N$ are $a$ and $b$ respectively. Find the distance $z$ of their center of gravity $Z$ from the line $M N$. (U. 121, P. 144.)

Let there be a certain axis $s$ on the plane. The static moment of a material point

![](https://cdn.mathpix.com/cropped/2024_05_21_27f86caa754616304e40g-026.jpg?height=275&width=478&top_left_y=1045&top_left_x=618)

Fig. 16. $A \equiv(A, m)$ relative to this axis is the product of the mass $m$ and its distance $y$ from the axis $s$, i.e., $m \cdot y$,

$$
\operatorname{Mom}_{s} A=m \cdot y
$$

If there is a system of $n$ material points $A_{1}, A_{2}, A_{3}, \ldots, A_{n}$ on the plane, located on the same side of a certain axis $s$, then the moment of this system relative to the axis is the sum of the moments of the material points of the system, i.e.,

$$
\operatorname{Mom}_{s} A_{1}+\text { Mom }_{s} A_{2}+\ldots+\text { Mom }_{s} A_{n}
$$

or, in other words, the sum

$$
m_{1} y_{1}+m_{2} y_{2}+\ldots+m_{n} y_{n}
$$

where $y_{1}, y_{2}, \ldots, y_{n}$ are the distances of the points $A_{1}, A_{2}, \ldots, A_{n}$ from the axis (Fig. 17), and $m_{1}, m_{2}, \ldots, m_{n}$ are the masses of these material points.

![](https://cdn.mathpix.com/cropped/2024_05_21_27f86caa754616304e40g-027.jpg?height=338&width=562&top_left_y=442&top_left_x=349)

Fig. 17.
8. Let for definiteness $a<b$. Further, let $F$ (see Fig. 16) lie on $Z_{1} Z, E$ on $Q_{1} Q, P F\|Z E\| M N$. Since $\triangle P F Z \sim \triangle Z E Q$, we have

$$
\frac{F Z}{E Q}=\frac{P Z}{Q Z}=\frac{q}{p}
$$

Thus,

$$
\frac{z-a}{b-z}=\frac{q}{p}
$$

From this, we find

$$
\begin{aligned}
(p+q) \cdot z & =p \cdot a+q \cdot b \\
z & =\frac{p a+q b}{p+q}
\end{aligned}
$$

Our reasoning does not apply when $P Q \perp M N$ (Fig. 128). Let $M N \times P Q \equiv S$. Then, by problem 1, $z=S Z=\frac{p a+q b}{p+q}$.

![](https://cdn.mathpix.com/cropped/2024_05_21_27f86caa754616304e40g-145.jpg?height=354&width=463&top_left_y=1159&top_left_x=139)

Fig. 128.

The previous reasoning also does not apply if $a=b$. But then it is clear that $z=a$, so formula (2) remains valid.

Thus, the last formula is applicable in all cases provided by the problem condition.