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| + | <meta name="description" content="A bead can move along a ring of radius R, pushed by a spoke that rotates uniformly with angular velocity \omega in the plane of the ring. The axis of rotation of the spoke is located on the ring. Determine the acceleration of the bead."> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="A bead can move along a ring of radius R, pushed by a spoke that rotates uniformly with angular velocity \omega in the plane of the ring. The axis of rotation of the spoke is located on the ring. Determine the acceleration of the bead."> | ||
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| + | <meta property="og:description" content="A bead can move along a ring of radius R, pushed by a spoke that rotates uniformly with angular velocity \omega in the plane of the ring. The axis of rotation of the spoke is located on the ring. Determine the acceleration of the bead."> | ||
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| + | <title>A bead can move along a ring of radius R, pushed by a spoke that rotates uniformly with angular velocity \omega in the plane of the ring. The axis of rotation of the spoke is located on the ring. Determine the acceleration of the bead.</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="../../#1.5">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $1.5.12.$ A bead can move along a ring of radius $R$, pushed by a spoke that rotates uniformly with angular velocity $\omega$ in the plane of the ring. The axis of rotation of the spoke is located on the ring. Determine the acceleration of the bead. | ||
| + | |||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.5.12/statement.png" | ||
| + | loading="lazy" width="150" /> | ||
| + | <figcaption> | ||
| + | For problem $1.5.12$ | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | </p> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | |||
| + | </p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="https://savchenkosolutions.com/1/1.5.12/draw.png" | ||
| + | loading="lazy" width="350" /> | ||
| + | <figcaption> | ||
| + | Displaying velocity vectors | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | <p> | ||
| + | |||
| + | <p>The bead will move along the spoke with the velocity $\upsilon_{\parallel}$, and since the spoke rotates, the bead will have a velocity component perpendicular to the spoke: $\upsilon_{\perp}$</p> | ||
| + | <p>Then the absolute velocity $\vec{v}$ in Non-inertial reference frame will be equal to the vector sum of $\vec{v}_{rel}$ and $\vec{v}_{cl}$</p> | ||
| + | $$ \upsilon_{\perp}=\omega r=\omega\cdot 2R\cos\alpha $$ | ||
| + | <p>On the other hand,</p> | ||
| + | <p>Then the absolute velocity $\vec{v}$ in Non-inertial reference frame will be equal to the vector sum of $\vec{v}_{rel}$ and $\vec{v}_{cl}$</p> | ||
| + | $$ \upsilon_{\perp}=\upsilon \cos\alpha $$ | ||
| + | <p>Since the speed of a bead moving in a circle is directed perpendicular to the radius.</p> | ||
| + | <p>Then</p> | ||
| + | $$ \omega\cdot 2R\cos\alpha=\upsilon \cos\alpha $$ | ||
| + | |||
| + | $$ \upsilon=2R\omega=\text{const} $$ | ||
| + | <p>Since the speed does not change in magnitude, the bead has only normal acceleration</p> | ||
| + | $$ a_n=\frac{\upsilon^2}{R}=\frac{4\omega^2R^2}{R}=4\omega^2R $$ | ||
| + | |||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$a_n=4\omega^2R$$ | ||
| + | </p> | ||
| + | |||
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| + | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | ||
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| + | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> alex@savchenkosolutions.com <br></small> | ||
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| @@ -0,0 +1,113 @@ | |||
| <!DOCTYPE html> | |||
| <html lang="en"> | |||
| <head> | |||
| <meta charset="utf-8"> | |||
| <meta name="viewport" content="width=device-width, initial-scale=1.0"> | |||
| <meta http-equiv="content-language" content="en"> | |||
| <meta name="keywords" content="Savchenko Problems in Physics, Savchenko solutions, physics problems, physics olympiad preparation, IPhO, Jaan Kalda"> | |||
| <meta name="description" content="A bead can move along a ring of radius R, pushed by a spoke that rotates uniformly with angular velocity \omega in the plane of the ring. The axis of rotation of the spoke is located on the ring. Determine the acceleration of the bead."> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="A bead can move along a ring of radius R, pushed by a spoke that rotates uniformly with angular velocity \omega in the plane of the ring. The axis of rotation of the spoke is located on the ring. Determine the acceleration of the bead."> | |||
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| <title>A bead can move along a ring of radius R, pushed by a spoke that rotates uniformly with angular velocity \omega in the plane of the ring. The axis of rotation of the spoke is located on the ring. Determine the acceleration of the bead.</title> | |||
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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="../../#1.5">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $1.5.12.$ A bead can move along a ring of radius $R$, pushed by a spoke that rotates uniformly with angular velocity $\omega$ in the plane of the ring. The axis of rotation of the spoke is located on the ring. Determine the acceleration of the bead. | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.5.12/statement.png" | |||
| loading="lazy" width="150" /> | |||
| <figcaption> | |||
| For problem $1.5.12$ | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| </p> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="https://savchenkosolutions.com/1/1.5.12/draw.png" | |||
| loading="lazy" width="350" /> | |||
| <figcaption> | |||
| Displaying velocity vectors | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <p> | |||
| <p>The bead will move along the spoke with the velocity $\upsilon_{\parallel}$, and since the spoke rotates, the bead will have a velocity component perpendicular to the spoke: $\upsilon_{\perp}$</p> | |||
| <p>Then the absolute velocity $\vec{v}$ in Non-inertial reference frame will be equal to the vector sum of $\vec{v}_{rel}$ and $\vec{v}_{cl}$</p> | |||
| $$ \upsilon_{\perp}=\omega r=\omega\cdot 2R\cos\alpha $$ | |||
| <p>On the other hand,</p> | |||
| <p>Then the absolute velocity $\vec{v}$ in Non-inertial reference frame will be equal to the vector sum of $\vec{v}_{rel}$ and $\vec{v}_{cl}$</p> | |||
| $$ \upsilon_{\perp}=\upsilon \cos\alpha $$ | |||
| <p>Since the speed of a bead moving in a circle is directed perpendicular to the radius.</p> | |||
| <p>Then</p> | |||
| $$ \omega\cdot 2R\cos\alpha=\upsilon \cos\alpha $$ | |||
| $$ \upsilon=2R\omega=\text{const} $$ | |||
| <p>Since the speed does not change in magnitude, the bead has only normal acceleration</p> | |||
| $$ a_n=\frac{\upsilon^2}{R}=\frac{4\omega^2R^2}{R}=4\omega^2R $$ | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$a_n=4\omega^2R$$ | |||
| </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> alex@savchenkosolutions.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||