# Problem № 6 (10 points)

A cylinder with a mass of $M=0.5$ kg was placed on rails inclined at an angle $\alpha=45^{\circ}$ to the horizontal (the side view is shown in the figure). What is the minimum mass $m$ of the load that needs to be attached to the thread wound around the cylinder so that it starts rolling upwards? Slipping is absent.

![](https://cdn.mathpix.com/cropped/2024_05_06_7ac9c551b647ccda756fg-6.jpg?height=320&width=457&top_left_y=1071&top_left_x=822)

Answer: 1.2 kg

#
# Solution and evaluation criteria:

![](https://cdn.mathpix.com/cropped/2024_05_06_7ac9c551b647ccda756fg-6.jpg?height=460&width=460&top_left_y=1643&top_left_x=821)

The moment of forces relative to the point of contact of the cylinder with the plane:

$m g(R - R \sin \alpha) = M g R \sin \alpha$

(5 points)

$m \left(1 - \frac{\sqrt{2}}{2}\right) = M \frac{\sqrt{2}}{2}$

$m = M \frac{\sqrt{2}}{2 - \sqrt{2}} = 1.2 \kappa 2$

(5 points)

#