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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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| 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="../../#1.5">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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. | | $1.5.12.$ 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. |
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| For problem $1.5.12$ | | For problem $1.5.12$ |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
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| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| Displaying velocity vectors | | Displaying velocity vectors |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
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| <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>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> | | <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> |
| $$ \upsilon_{\perp}=\omega r=\omega\cdot 2R\cos\alpha$$ | | $$ \upsilon_{\perp}=\omega r=\omega\cdot 2R\cos\alpha$$ |
| <p>On the other hand,</p> | | <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> | | <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> |
| $$ \upsilon_{\perp}=\upsilon\cos\alpha$$ | | $$ \upsilon_{\perp}=\upsilon\cos\alpha$$ |
| <p>Since the speed of a bead moving in a circle is directed perpendicular to the radius.</p> | | <p>Since the speed of a bead moving in a circle is directed perpendicular to the radius.</p> |
| <p>Then</p> | | <p>Then</p> |
| $$ \omega\cdot 2R\cos\alpha =\upsilon\cos\alpha$$ | | $$ \omega\cdot 2R\cos\alpha =\upsilon\cos\alpha$$ |
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| $$ \upsilon =2R\omega =\text{const} $$ | | $$ \upsilon =2R\omega =\text{const} $$ |
| <p>Since the speed does not change in magnitude, the bead has only normal acceleration</p> | | <p>Since the speed does not change in magnitude, the bead has only normal acceleration</p> |
| $$ a_n=\frac{\upsilon^2}{R}=\frac{4\omega^2R^2}{R}=4\omega^2R $$ | | $$ a_n=\frac{\upsilon^2}{R}=\frac{4\omega^2R^2}{R}=4\omega^2R $$ |
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| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$a_n=4\omega^2R$$ | | $$a_n=4\omega^2R$$ |
