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| <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> | | <span><img src="../../img/book.png"></span><span>Savchenko Solutions</span> |
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| <p class="author"> | | <p class="author"> |
| Solutions of Savchenko Problems in Physics <br> | | Solutions of Savchenko Problems in Physics <br> |
| <i><b>knowledge must be free</b></i> | | <i><b>knowledge must be free</b></i> |
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| <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#2.1">$\leftarrow$Back</a></h3> |
| | | |
| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $2.1.45^*.$ A coin is placed on a plane whose slope tangent is equal to the coefficient of friction. In the horizontal direction along the plane, the coin was given a velocity $v$. Find the steady speed of the coin. | | $2.1.45^*.$ A coin is placed on a plane whose slope tangent is equal to the coefficient of friction. In the horizontal direction along the plane, the coin was given a velocity $v$. Find the steady speed of the coin. |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.45/sol.jpg" | | <img src="https://savchenkosolutions.com/2/2.1.45/sol.jpg" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| Forces acting on the body | | Forces acting on the body |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| Projecting $m\vec{g}$ on $OY$: | | Projecting $m\vec{g}$ on $OY$: |
| </p> | | </p> |
| <br> | | <br> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.45/sol1.jpg" | | <img src="https://savchenkosolutions.com/2/2.1.45/sol1.jpg" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| Projecting on $OY$ | | Projecting on $OY$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| <br> | | <br> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.45/sol3.jpg" | | <img src="https://savchenkosolutions.com/2/2.1.45/sol3.jpg" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| Drawing in the $xOy$ plane | | Drawing in the $xOy$ plane |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| At the final moment $(\varphi=0^{\circ})$: | | At the final moment $(\varphi=0^{\circ})$: |
| </p> | | </p> |
| <br> | | <br> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.45/sol4.jpg" | | <img src="https://savchenkosolutions.com/2/2.1.45/sol4.jpg" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| Forces acting on the body at the final moment | | Forces acting on the body at the final moment |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| Let's consider an arbitrary moment: | | Let's consider an arbitrary moment: |
| </p> | | </p> |
| <br> | | <br> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/2/2.1.45/sol2.jpg" | | <img src="https://savchenkosolutions.com/2/2.1.45/sol2.jpg" |
| loading="lazy" width="250" /> | | loading="lazy" width="250" /> |
| <figcaption> | | <figcaption> |
| Forces acting on a body at an arbitrary moment | | Forces acting on a body at an arbitrary moment |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| Note that $\varphi$ changes from $0^{\circ}$ to $90^{\circ}$: | | Note that $\varphi$ changes from $0^{\circ}$ to $90^{\circ}$: |
| </p> | | </p> |
| <p> | | <p> |
| Let's write Newton's second law: | | Let's write Newton's second law: |
| $$ \left\{\begin{matrix} | | $$ \left\{\begin{matrix} |
| \frac{du_x}{dt} = \frac{F \sin \varphi }{m}& \\ | | \frac{du_x}{dt} = \frac{F \sin \varphi }{m}& \\ |
| \frac{du_y}{dt} = \frac{F (1- \cos \varphi) }{m} & | | \frac{du_y}{dt} = \frac{F (1- \cos \varphi) }{m} & |
| \frac{d\varphi}{dt} = \frac{F \sin \varphi }{mu}& (a)\\ | | \frac{d\varphi}{dt} = \frac{F \sin \varphi }{mu}& (a)\\ |
| \frac{du}{dt} = \frac{F (1- \cos \varphi) }{m} & (b) | | \frac{du}{dt} = \frac{F (1- \cos \varphi) }{m} & (b) |
| \end{matrix}\right. $$ | | \end{matrix}\right. $$ |
| We divide $(b)$ by $(a)$: | | We divide $(b)$ by $(a)$: |
| $$ \frac{du}{d\varphi} = u\frac{1- \cos \varphi}{\sin \varphi} $$ | | $$ \frac{du}{d\varphi} = u\frac{1- \cos \varphi}{\sin \varphi} $$ |
| | | |
| $$ \frac{du}{u} = \frac{1- \cos \varphi}{\sin \varphi} d\varphi $$ | | $$ \frac{du}{u} = \frac{1- \cos \varphi}{\sin \varphi} d\varphi $$ |
| | | |
| $$ \frac{du}{u} = tg (\frac{\varphi}{2}) d\varphi $$ | | $$ \frac{du}{u} = tg (\frac{\varphi}{2}) d\varphi $$ |
| We integrate both parts of the equation: | | We integrate both parts of the equation: |
| <!-- $\int_{v_0}^{v}\frac{du}{u} = \int_{90^{\circ}}^{0^{\circ}}tg (\frac{\varphi}{2}) d\varphi \; (c)$ --> | | <!-- $\int_{v_0}^{v}\frac{du}{u} = \int_{90^{\circ}}^{0^{\circ}}tg (\frac{\varphi}{2}) d\varphi \; (c)$ --> |
| $$ \int \frac{du}{u} = \int tg (\frac{\varphi}{2}) d\varphi \; (c) $$ | | $$ \int \frac{du}{u} = \int tg (\frac{\varphi}{2}) d\varphi \; (c) $$ |
| | | |
| $$ \int\frac{du}{u}=ln|u| \; (d) $$ | | $$ \int\frac{du}{u}=ln|u| \; (d) $$ |
| | | |
| $$ \int tg (\frac{\varphi}{2}) d\varphi=-2 ln(\cos(\frac{x}{2})) \; (e) $$ | | $$ \int tg (\frac{\varphi}{2}) d\varphi=-2 ln(\cos(\frac{x}{2})) \; (e) $$ |
| Substitute $(e)$ and $(d)$ into $(c)$: | | Substitute $(e)$ and $(d)$ into $(c)$: |
| <div class="scroll-wrapper"> | | <div class="scroll-wrapper"> |
| $$ ln(v)+C=ln(\sin(\varphi)) -(ln(\sin(\frac{\varphi}{2})-\cos(\frac{\varphi}{2})))\; (e) $$ | | $$ ln(v)+C=ln(\sin(\varphi)) -(ln(\sin(\frac{\varphi}{2})-\cos(\frac{\varphi}{2})))\; (e) $$ |
| </div> | | </div> |
| $$ \boxed{v_x=\frac{v}{2}} $$ | | $$ \boxed{v_x=\frac{v}{2}} $$ |
| </p> | | </p> |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$ v_x=\frac{v}{2} $$ | | $$ v_x=\frac{v}{2} $$ |
| </p> | | </p> |
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