Added Luis's English solution of 2.8.34 and minor change in 2.8.4

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+ <header style="text-align:center;">
+ <h2>Solutions of Savchenko Physics Textbook</h2>
+ <p class="author">
+ Aliaksandr Melnichenka <br/>
+ October 2023
+ </p>
+ </header>
+
+ <h3 id="back-link"><a href="../">$\leftarrow$Back</a></h3>
+
+ <h3> Statement </h3>
+
+ <p>
+ $2.8.34$
+ The staircase rests on a vertical wall and floor. At what values of the angle between the ladder and the floor can it stand if the coefficients of friction of the ladder on the floor and on the wall are equal to $\mu_1$ and $\mu_2$, respectively?
+ </p>
+ <center>
+ <figure>
+ <img src="statement.png"
+ loading="lazy" alt="2.8.34" width="300" />
+ <figcaption>
+ For problem 2.8.34
+ </figcaption>
+ </figure>
+ </center>
+
+ <h3>Solution</h3>
+ <p>
+ See the following figure...
+ </p>
+
+ <center>
+ <figure>
+ <img src="draw.png"
+ loading="lazy" alt="2.8.34" width="350" />
+ <figcaption>
+ Force analysis
+ </figcaption>
+ </figure>
+ </center>
+
+ <p>
+ Let's consider the Newton Second Law for both axis. Firstly, for $x$-axis,
+ $$N_1+f_r_2 = mg$$
+ $$N_1+\mu_2 N_2 = mg \;(1)$$
+
+ For $y$-axis,
+ $$N_2 = f_r_1$$
+ $$N_2 = \mu_1 N_1 \;(2)$$
+
+ Substituting $(2)$ into $(2)$:
+ $$N_1 (1+\mu_1 \mu_2) = mg \;(3)$$
+
+ Applying Newton Second Law for the rotation about the pivot point,
+ $$\sum \vec{\tau} = \vec{0}$$
+ $$-mg\frac{l}{2}\sin{\frac{\pi}{2}-\alpha}+N_1 l\sin{\frac{\pi}{2}+\alpha}-f_r_1 l \sin{\pi-\alpha} = 0$$
+
+ where $l$ is the staircase's length. Now, for avoiding that staircase slides over ground $\sum \tau \leq 0$, then
+ $$-mg\frac{l}{2}\sin{\frac{\pi}{2}-\alpha}+N_1 l\sin{\frac{\pi}{2}+\alpha}-f_r_1 l \sin{\pi-\alpha} \leq 0$$
+ $$-mg\frac{l}{2}\sin{\frac{\pi}{2}-\alpha}+N_1 l\sin{\frac{\pi}{2}+\alpha}-\mu_1 N_1 l \sin{\pi-\alpha} \leq 0$$
+
+ Developing the above expression,
+ $$mg \geq 2N_1(1-\mu_1 \tan{\alpha}) \;(4)$$
+
+ Putting $(3)$ into $(4)$, and separating $\tan{\alpha}$ term,
+ $$\boxed{\tan{\alpha} \geq \frac{1-\mu_1 \mu_2}{2 \mu_1}}$$
+ </p>
+
+ <p style="text-align: right; font-style: italic; font-size: 14;">
+ BSc. Luis Daniel Fernández Quintana<br>
+ Physics Department (FCNE)<br>
+ Universidad de Oriente, Cuba<br>
+ </p>
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