8. (10 points) There are two light springs of equal length but with different stiffnesses. The springs were placed vertically one on top of the other. A load of mass $m=100$ kg was placed on top. As a result, the construction was compressed by $x_{1}=12.5$ cm. Then the springs were placed side by side and the same load was placed on top again. In this case, the construction was compressed by $x_{2}=2$ cm. Find the stiffnesses of the springs. The acceleration due to gravity $g=10 \mathrm{~m} / \mathrm{c}^{2}$.
Answer: $k_{1}=400 \frac{\mathrm{H}}{\mathrm{M}}$ and $k_{2}=100 \frac{\mathrm{H}}{\mathrm{M}}$

Solution. In the first situation, the stiffness of the obtained structure is $k_{0}=\frac{k_{1} k_{2}}{k_{1}+k_{2}}$. (2 points)

The equilibrium condition in this case is: $k_{0} x_{1}=m g$.

As a result: $\frac{k_{1} k_{2}}{k_{1}+k_{2}}=\frac{m g}{x_{1}}=80 \frac{\mathrm{H}}{\mathrm{M}}$.

In the second situation, the stiffness of the obtained structure is $k_{\kappa}=k_{1}+k_{2}$. (2 points)

The equilibrium condition in this case is: $k_{\kappa} x_{2}=m g$.

As a result: $k_{1}+k_{2}=\frac{m g}{x_{2}}=500 \frac{\mathrm{H}}{\mathrm{M}}$.

Solving this system of equations, we get: $k_{1}=400 \frac{\mathrm{H}}{\mathrm{M}}$ and $k_{2}=100 \frac{\mathrm{H}}{\mathrm{M}}$.

## Multidisciplinary Engineering Olympiad "Star"

 in natural sciences 9th grade

Final stage

$2018-2019$

Variant 2

Tasks, answers, and evaluation criteria