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| ho. Determine the force exerted on the nail if the rope sections on opposite sides of the nail form an angle lpha. At what speed will the rope move away from the nail?"> | | ho. Determine the force exerted on the nail if the rope sections on opposite sides of the nail form an angle lpha. At what speed will the rope move away from the nail?"> |
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| ho. Determine the force exerted on the nail if the rope sections on opposite sides of the nail form an angle lpha. At what speed will the rope move away from the nail?"> | | ho. Determine the force exerted on the nail if the rope sections on opposite sides of the nail form an angle lpha. At what speed will the rope move away from the nail?"> |
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| ho. Determine the force exerted on the nail if the rope sections on opposite sides of the nail form an angle lpha. At what speed will the rope move away from the nail?"> | | ho. Determine the force exerted on the nail if the rope sections on opposite sides of the nail form an angle lpha. At what speed will the rope move away from the nail?"> |
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| ho. Determine the force exerted on the nail if the rope sections on opposite sides of the nail form an angle lpha. At what speed will the rope move away from the nail?</title> | | ho. Determine the force exerted on the nail if the rope sections on opposite sides of the nail form an angle lpha. At what speed will the rope move away from the nail?</title> |
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| <h2>Solutions of Savchenko Problems in Physics</h2> | | <h2>Solutions of Savchenko Problems in Physics</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| $2.2.44^*.$ A rope thrown over a smooth nail is dragged at a speed of $v$ through the gap. The friction force in the slot $F$, the mass of the rope length unit $\rho$. Determine the force exerted on the nail if the rope sections on opposite sides of the nail form an angle $\alpha$. At what speed will the rope move away from the nail? | | $2.2.44^*.$ A rope thrown over a smooth nail is dragged at a speed of $v$ through the gap. The friction force in the slot $F$, the mass of the rope length unit $\rho$. Determine the force exerted on the nail if the rope sections on opposite sides of the nail form an angle $\alpha$. At what speed will the rope move away from the nail? |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" width="150" /> | | loading="lazy" width="150" /> |
| <figcaption> | | <figcaption> |
| For problem $2.2.44^*$ | | For problem $2.2.44^*$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| </p> | | </p> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="draw1.jpg" | | <img src="draw1.jpg" |
| loading="lazy" width="180" /> | | loading="lazy" width="180" /> |
| <figcaption> | | <figcaption> |
| Forces acting on the system | | Forces acting on the system |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| $$ | | $$ |
| \rho = \frac{m}{l} = \frac{\Delta m}{\Delta l} | | \rho = \frac{m}{l} = \frac{\Delta m}{\Delta l} |
| $$ | | $$ |
| | | |
| $$\cos\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1+\cos\alpha}{2}}$$ | | $$\cos\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1+\cos\alpha}{2}}$$ |
| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="draw2.jpg" | | <img src="draw2.jpg" |
| loading="lazy" width="130" /> | | loading="lazy" width="130" /> |
| <figcaption> | | <figcaption> |
| For problem $2.2.44^*$ | | For problem $2.2.44^*$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| For the thread to be stretched at a constant speed $\vec{v}$, there must be a force on the first half that is modulo the frictional force $\vec{F}$: | | For the thread to be stretched at a constant speed $\vec{v}$, there must be a force on the first half that is modulo the frictional force $\vec{F}$: |
| | | |
| $$F'^2=F^2+F^2-2F^2\cos(\pi-\alpha)$$ | | $$F'^2=F^2+F^2-2F^2\cos(\pi-\alpha)$$ |
| $$F'=F\sqrt{2(1+\cos\alpha)}$$ | | $$F'=F\sqrt{2(1+\cos\alpha)}$$ |
| $$F'=2F\cos\frac{\alpha}{2}$$ | | $$F'=2F\cos\frac{\alpha}{2}$$ |
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| | | |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="draw3.jpg" | | <img src="draw3.jpg" |
| loading="lazy" width="130" /> | | loading="lazy" width="130" /> |
| <figcaption> | | <figcaption> |
| For problem $2.2.44^*$ | | For problem $2.2.44^*$ |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| | | |
| Similiarly, | | Similiarly, |
| | | |
| $$p'^2=p^2+p^2-2p^2\cos(\pi-\alpha)$$ | | $$p'^2=p^2+p^2-2p^2\cos(\pi-\alpha)$$ |
| $$p'=p\sqrt{2(1+\cos\alpha)}$$ | | $$p'=p\sqrt{2(1+\cos\alpha)}$$ |
| $$p'=2p\cos\frac{\alpha}{2}$$ | | $$p'=2p\cos\frac{\alpha}{2}$$ |
| | | |
| By definition of momentum | | By definition of momentum |
| | | |
| $$p=\Delta m v$$ | | $$p=\Delta m v$$ |
| | | |
| From where, considering that $p'$ and $F'$ acts against $Ox$ | | From where, considering that $p'$ and $F'$ acts against $Ox$ |
| | | |
| $$p'=-2\Delta m v\cos\frac{\alpha}{2}$$ | | $$p'=-2\Delta m v\cos\frac{\alpha}{2}$$ |
| $$F'=2F\cos\frac{\alpha}{2}$$ | | $$F'=2F\cos\frac{\alpha}{2}$$ |
| Let's write Newton's second law | | Let's write Newton's second law |
| $$\sum \vec{F}=0$$ | | $$\sum \vec{F}=0$$ |
| $$N-F'-F''=0$$ | | $$N-F'-F''=0$$ |
| $$N-2F\cos\frac{\alpha}{2}+2\Delta m v\cos\frac{\alpha}{2}=0$$ | | $$N-2F\cos\frac{\alpha}{2}+2\Delta m v\cos\frac{\alpha}{2}=0$$ |
| $$F''=\frac{p'}{\Delta t}$$ | | $$F''=\frac{p'}{\Delta t}$$ |
| From where we obtain the force $\vec{N}$ | | From where we obtain the force $\vec{N}$ |
| | | |
| $$N=2\cos\frac{\alpha}{2}(\frac{-\Delta m v}{\Delta t}+F)$$ | | $$N=2\cos\frac{\alpha}{2}(\frac{-\Delta m v}{\Delta t}+F)$$ |
| By definition of density | | By definition of density |
| $$\Delta m = \rho \Delta x$$ | | $$\Delta m = \rho \Delta x$$ |
| By definition of speed | | By definition of speed |
| $$ v = \frac{\Delta x}{\Delta t}$$ | | $$ v = \frac{\Delta x}{\Delta t}$$ |
| $$ N = 2F\cos\frac{\alpha}{2} - 2\rho v^2\cos\frac{\alpha}{2}$$ | | $$ N = 2F\cos\frac{\alpha}{2} - 2\rho v^2\cos\frac{\alpha}{2}$$ |
| $$ \boxed{N = 2\cos\frac{\alpha}{2} (F -\rho v^2)}$$ | | $$ \boxed{N = 2\cos\frac{\alpha}{2} (F -\rho v^2)}$$ |
| When the rope comes off | | When the rope comes off |
| $$N\leq0$$ | | $$N\leq0$$ |
| $$F-\rho v^2\leq0$$ | | $$F-\rho v^2\leq0$$ |
| $${F\leq \rho v^2}$$ | | $${F\leq \rho v^2}$$ |
| From where we finally get | | From where we finally get |
| $$\boxed{v \geq \sqrt{\frac{F}{\rho}}}$$ | | $$\boxed{v \geq \sqrt{\frac{F}{\rho}}}$$ |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$N=2(F-\rho v^2)\cos\frac\alpha2;\text{ при }v\geqslant\sqrt{F/\rho}.$$ | | $$N=2(F-\rho v^2)\cos\frac\alpha2;\text{ при }v\geqslant\sqrt{F/\rho}.$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| Almaskhan Arsen<br> | | Almaskhan Arsen<br> |
| </p> | | </p> |
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