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| <meta name="description" content="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? "> | | <meta name="description" content="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? "> |
| <meta name="author" content="Aliaksandr Melnichenka"> | | <meta name="author" content="Aliaksandr Melnichenka"> |
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | | <meta name="date" content="2023-10" scheme="YYYY-MM"> |
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| <title>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? </title> | | <title>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? </title> |
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| <h3 id="back-link"><a href="../../#2.8">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.8">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| | | |
| <p> | | <p> |
| $2.8.34$ | | $2.8.34$ |
| 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? | | 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> | | </p> |
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| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" alt="2.8.34" width="300" /> | | loading="lazy" alt="2.8.34" width="300" /> |
| <figcaption> | | <figcaption> |
| For problem 2.8.34 | | For problem 2.8.34 |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| See the following figure... | | See the following figure... |
| </p> | | </p> |
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| <center> | | <center> |
| <figure> | | <figure> |
| <img src="draw.png" | | <img src="draw.png" |
| loading="lazy" alt="2.8.34" width="350" /> | | loading="lazy" alt="2.8.34" width="350" /> |
| <figcaption> | | <figcaption> |
| Force analysis | | Force analysis |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <p> | | <p> |
| Let's consider the Newton Second Law for both axis. Firstly, for $x$-axis, | | Let's consider the Newton Second Law for both axis. Firstly, for $x$-axis, |
| $$N_1+f_{r2} = mg$$ | | $$N_1+f_{r2} = mg$$ |
| $$N_1+\mu_2 N_2 = mg \;(1)$$ | | $$N_1+\mu_2 N_2 = mg \;(1)$$ |
| | | |
| For $y$-axis, | | For $y$-axis, |
| $$N_2 = f_{r1}$$ | | $$N_2 = f_{r1}$$ |
| $$N_2 = \mu_1 N_1 \;(2)$$ | | $$N_2 = \mu_1 N_1 \;(2)$$ |
| | | |
| Substituting $(2)$ into $(1)$: | | Substituting $(2)$ into $(1)$: |
| $$N_1 (1+\mu_1 \mu_2) = mg \;(3)$$ | | $$N_1 (1+\mu_1 \mu_2) = mg \;(3)$$ |
| | | |
| Applying Newton Second Law for the rotation about the pivot point, | | Applying Newton Second Law for the rotation about the pivot point, |
| $$\sum \vec{\tau} = \vec{0}$$ | | $$\sum \vec{\tau} = \vec{0}$$ |
| $$-mg\frac{l}{2}\sin{\frac{\pi}{2}-\alpha}+N_1 l\sin{\frac{\pi}{2}+\alpha}-f_{r1} l \sin{\pi-\alpha} = 0$$ | | $$-mg\frac{l}{2}\sin{\frac{\pi}{2}-\alpha}+N_1 l\sin{\frac{\pi}{2}+\alpha}-f_{r1} l \sin{\pi-\alpha} = 0$$ |
| | | |
| where $l$ is the staircase's length. Now, for avoiding that staircase slides over ground $\sum \tau \leq 0$, then | | where $l$ is the staircase's length. Now, for avoiding that staircase slides over ground $\sum \tau \leq 0$, then |
| $$-mg\frac{l}{2}\sin{\frac{\pi}{2}-\alpha}+N_1 l\sin{\frac{\pi}{2}+\alpha}-f_{r1} l \sin{\pi-\alpha} \leq 0$$ | | $$-mg\frac{l}{2}\sin{\frac{\pi}{2}-\alpha}+N_1 l\sin{\frac{\pi}{2}+\alpha}-f_{r1} l \sin{\pi-\alpha} \leq 0$$ |
| $$-mg\frac{l}{2}\sin{\frac{\pi}{2}-\alpha}+N_1 l\sin{\frac{\pi}{2}+\alpha}-\mu_1 N_1 l \sin{\pi-\alpha} \leq 0$$ | | $$-mg\frac{l}{2}\sin{\frac{\pi}{2}-\alpha}+N_1 l\sin{\frac{\pi}{2}+\alpha}-\mu_1 N_1 l \sin{\pi-\alpha} \leq 0$$ |
| | | |
| Developing the above expression, | | Developing the above expression, |
| $$mg \geq 2N_1(1-\mu_1 \tan{\alpha}) \;(4)$$ | | $$mg \geq 2N_1(1-\mu_1 \tan{\alpha}) \;(4)$$ |
| | | |
| Putting $(3)$ into $(4)$, and separating $\tan{\alpha}$ term, | | Putting $(3)$ into $(4)$, and separating $\tan{\alpha}$ term, |
| $$\boxed{\tan{\alpha} \geq \frac{1-\mu_1 \mu_2}{2 \mu_1}}$$ | | $$\boxed{\tan{\alpha} \geq \frac{1-\mu_1 \mu_2}{2 \mu_1}}$$ |
| </p> | | </p> |
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| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| BSc. Luis Daniel Fernández Quintana<br> | | BSc. Luis Daniel Fernández Quintana<br> |
| Physics Department (FCNE)<br> | | Physics Department (FCNE)<br> |
| Universidad de Oriente, Cuba<br> | | Universidad de Oriente, Cuba<br> |
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