2. (30 points). To hang a New Year's decoration, Anya leans a ladder against a smooth wall so that its base is at the maximum possible distance from the wall. The coefficient of friction between the ladder and the floor is $\mu=2 / 3$, Anya's mass $M=70 \mathrm{kg}$, and the mass of the ladder $\mathrm{m}=20$ kg. The length of the ladder L, as well as the ceiling height H, are both 5 m.

Will Anya be able to climb the ladder? If so, to what height?
1.2 Two weights with masses \( m_{1}=180 \) g and \( m_{2}=120 \) g hang from the ends of a weightless, inextensible string, which is passed over a fixed pulley. Initially, the weights were at the same level. The weights were released and came into motion without initial velocity. What will be the vertical distance between the weights after \(\Delta t=1.5\) s from the start of the motion? Air resistance and friction in the pulley can be neglected.

![](https://cdn.mathpix.com/cropped/2024_05_06_6ec43fa2f035f1ec876bg-02.jpg?height=540&width=368&top_left_y=1746&top_left_x=136)

Let's write Newton's second law for the first and second weights \( m_{1,2}=T_{1,2}-m_{1,2} g \).

Considering the inextensibility and weightlessness of the string \(\left(a_{1}=-a_{2}\right)\) and \(T_{1}=T_{2}=T\), we obtain the expression for the magnitude of the acceleration of each weight

\[
a=\frac{m_{1}-m_{2}}{m_{1}+m_{2}} g
\]

The magnitude of their relative acceleration is \(a_{\text{rel}}=2a\). Then the distance between the weights after time \(\Delta t\) will be

\[
\Delta h=\frac{a_{\text{rel}} \Delta t^{2}}{2}=\left(\frac{m_{1}-m_{2}}{m_{1}+m_{2}}\right) g \Delta t^{2}
\]

Substituting the numerical values, we get \(\Delta h=\frac{180-120}{180+120} \cdot 10 \frac{M}{s^{2}} \cdot(1.5)^{2} s^{2}=4.5\) m.