5. Two loads with masses $m$ and $2 m$ are connected by three threads as shown in the figure. The thread $\mathrm{AB}$ is horizontal, and the thread CD forms an angle $\alpha$ with the vertical. Find the tensions in the threads AB and CD. The thread AB is cut. Find the accelerations of the loads immediately after this. The threads are inextensible and weightless.

![](https://cdn.mathpix.com/cropped/2024_05_06_5a5c4a883292a9d82de7g-1.jpg?height=401&width=386&top_left_y=1224&top_left_x=1516)
5. First, let's find the forces acting in all threads in equilibrium. On the body of mass \( m \), the force of gravity \( m \vec{g} \) and three tension forces of the threads \( \vec{T}_{A B}, \vec{T}_{C D} \), and the tension force of the lower thread, equal to \( 2 m \vec{g} \) (see the figure). Therefore, the equilibrium condition for this body in projections on the horizontal and vertical directions gives

![](https://cdn.mathpix.com/cropped/2024_05_06_1c50a76ea7b7bad6fbcag-3.jpg?height=406&width=383&top_left_y=2344&top_left_x=1542)

\[
\begin{aligned}
& T_{A B}=T_{C D} \sin \alpha \\
& 3 m g=T_{C D} \cos \alpha
\end{aligned}
\]

From these formulas, we find

\[
T_{C D}=\frac{3 m g}{\cos \alpha}, \quad T_{A B}=3 m g \operatorname{tg} \alpha
\]

Immediately after cutting the thread \( \mathrm{AB} \), the body of mass \( m \) will start moving under the action of two forces: \( 3 m \vec{g} \), directed vertically downward, and \( \vec{T}_{C D} \), directed along the thread CD. Since the velocity of the body is zero, the vector of its acceleration is perpendicular to the thread CD. And since the tension force in the lower thread does not change immediately after cutting the thread \( \mathrm{AB} \) (for the tension forces to change, the bodies must move), the second law of Newton for the body of mass \( m \) in projections on the direction perpendicular to the thread \( \mathrm{CD} \) gives

\[
m a_{m}=3 m g \sin \alpha
\]

From this, we get

\[
a_{m}=3 g \sin \alpha
\]

Due to the fact that the tension force in the thread connecting the bodies does not change, the acceleration of the body of mass \( 2 m \) is zero

\[
a_{2 m}=0
\]

Evaluation criteria for the problem (maximum score for the problem - 2 points)