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| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="A vertical cylindrical vessel of radius R partially filled with liquid rotates with the liquid around its axis. A balloon of radius r is tied to the side wall of the vessel on a thread of length l; during rotation the thread forms an angle \alpha with the wall. Determine the angular velocity of rotation of the vessel."> | ||
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| + | <meta property="og:description" content="A vertical cylindrical vessel of radius R partially filled with liquid rotates with the liquid around its axis. A balloon of radius r is tied to the side wall of the vessel on a thread of length l; during rotation the thread forms an angle \alpha with the wall. Determine the angular velocity of rotation of the vessel."> | ||
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| + | <title>A vertical cylindrical vessel of radius R partially filled with liquid rotates with the liquid around its axis. A balloon of radius r is tied to the side wall of the vessel on a thread of length l; during rotation the thread forms an angle \alpha with the wall. Determine the angular velocity of rotation of the vessel.</title> | ||
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| + | <header style="text-align:center;"> | ||
| + | <h2>Solutions of Savchenko Problems in Physics</h2> | ||
| + | <p class="author"> | ||
| + | Aliaksandr Melnichenka <br/> | ||
| + | October 2023 | ||
| + | </p> | ||
| + | </header> | ||
| + | |||
| + | <h3 id="back-link"><a href="../../#4.2">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $4.2.26.$ A vertical cylindrical vessel of radius R partially filled with liquid rotates with the liquid around its axis. A balloon of radius $r$ is tied to the side wall of the vessel on a thread of length $l$; during rotation the thread forms an angle $\alpha$ with the wall. Determine the angular velocity of rotation of the vessel. | ||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="4.2.26.png" | ||
| + | loading="lazy" width="200" /> | ||
| + | <figcaption> | ||
| + | For problem $4.2.26$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="forces.png" | ||
| + | loading="lazy" width="240" /> | ||
| + | <figcaption> | ||
| + | Forces acting on the balloon | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | Since the ball does not float, the vertical Archimedean force is completely compensated by the vertical component of the thread tension force | ||
| + | $$T\cos\alpha=\frac{4\pi r^3}{3}\rho g$$ | ||
| + | From where we could find the thread tension force | ||
| + | $$T=\frac{4\pi r^3}{3\cos\alpha}\rho g\quad (1)$$ | ||
| + | Alternatively, we could write an equilibrium condition for the <b>Horizontal Archimedean force</b>, which is compensated by the horizontal force of the thread tension | ||
| + | $$T\sin\alpha = \rho V\omega^2x\quad (2)$$ | ||
| + | It can be seen from the figure that the ball's center of mass is at a distance $l+r$ from the point of attachment of the thread: | ||
| + | $$x=R-(l+r)\sin\alpha$$ | ||
| + | After substituting into $(2)$ we obtain | ||
| + | $$\rho \frac{4\pi r^3}{3}\omega^2\left(R-(l+r)\sin\alpha\right) = T\sin\alpha \quad (3)$$ | ||
| + | Substituting $(1)$ into $(3)$ | ||
| + | <div class="scroll-wrapper">$$\rho \frac{4\pi r^3}{3}\omega^2\left(R-(l+r)\sin\alpha\right) = \frac{4\pi r^3}{3\cos\alpha}\rho g\sin\alpha$$</div> | ||
| + | From where we find the angular velocity | ||
| + | $$\boxed{\omega = \sqrt{\frac{g\tan\alpha}{R-(l+r)\sin\alpha}}}$$ | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$\omega=\sqrt{(g\operatorname{tg}\alpha)/[R-(l+r)\operatorname{sin}\alpha]}.$$ | ||
| + | </p> | ||
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| <meta name="description" content="A vertical cylindrical vessel of radius R partially filled with liquid rotates with the liquid around its axis. A balloon of radius r is tied to the side wall of the vessel on a thread of length l; during rotation the thread forms an angle \alpha with the wall. Determine the angular velocity of rotation of the vessel."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="A vertical cylindrical vessel of radius R partially filled with liquid rotates with the liquid around its axis. A balloon of radius r is tied to the side wall of the vessel on a thread of length l; during rotation the thread forms an angle \alpha with the wall. Determine the angular velocity of rotation of the vessel."> | |||
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| </head> | |||
| <body style=""> | |||
| <header style="text-align:center;"> | |||
| <h2>Solutions of Savchenko Problems in Physics</h2> | |||
| <p class="author"> | |||
| Aliaksandr Melnichenka <br/> | |||
| October 2023 | |||
| </p> | |||
| </header> | |||
| <h3 id="back-link"><a href="../../#4.2">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $4.2.26.$ A vertical cylindrical vessel of radius R partially filled with liquid rotates with the liquid around its axis. A balloon of radius $r$ is tied to the side wall of the vessel on a thread of length $l$; during rotation the thread forms an angle $\alpha$ with the wall. Determine the angular velocity of rotation of the vessel. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="4.2.26.png" | |||
| loading="lazy" width="200" /> | |||
| <figcaption> | |||
| For problem $4.2.26$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="forces.png" | |||
| loading="lazy" width="240" /> | |||
| <figcaption> | |||
| Forces acting on the balloon | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| Since the ball does not float, the vertical Archimedean force is completely compensated by the vertical component of the thread tension force | |||
| $$T\cos\alpha=\frac{4\pi r^3}{3}\rho g$$ | |||
| From where we could find the thread tension force | |||
| $$T=\frac{4\pi r^3}{3\cos\alpha}\rho g\quad (1)$$ | |||
| Alternatively, we could write an equilibrium condition for the <b>Horizontal Archimedean force</b>, which is compensated by the horizontal force of the thread tension | |||
| $$T\sin\alpha = \rho V\omega^2x\quad (2)$$ | |||
| It can be seen from the figure that the ball's center of mass is at a distance $l+r$ from the point of attachment of the thread: | |||
| $$x=R-(l+r)\sin\alpha$$ | |||
| After substituting into $(2)$ we obtain | |||
| $$\rho \frac{4\pi r^3}{3}\omega^2\left(R-(l+r)\sin\alpha\right) = T\sin\alpha \quad (3)$$ | |||
| Substituting $(1)$ into $(3)$ | |||
| <div class="scroll-wrapper">$$\rho \frac{4\pi r^3}{3}\omega^2\left(R-(l+r)\sin\alpha\right) = \frac{4\pi r^3}{3\cos\alpha}\rho g\sin\alpha$$</div> | |||
| From where we find the angular velocity | |||
| $$\boxed{\omega = \sqrt{\frac{g\tan\alpha}{R-(l+r)\sin\alpha}}}$$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$\omega=\sqrt{(g\operatorname{tg}\alpha)/[R-(l+r)\operatorname{sin}\alpha]}.$$ | |||
| </p> | |||
| <footer class="row container"> | |||
| <br> | |||
| <p> | |||
| <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | |||
| </p> | |||
| <p> | |||
| <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small> | |||
| </p> | |||
| </footer> | |||
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