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| <title>A molecular projector creates a ribbon-shaped narrow beam of silver's atoms that arrive to internal surface of a cylinder of radius R = 30 cm and form a spot. The device rotates with angular velocity \omega = 100\pi s^{-1}. Find the beam's velocity, if the spot deviates an angle \varphi=0.314 rad respect to initial position. </title> | | <title>A molecular projector creates a ribbon-shaped narrow beam of silver's atoms that arrive to internal surface of a cylinder of radius R = 30 cm and form a spot. The device rotates with angular velocity \omega = 100\pi s^{-1}. Find the beam's velocity, if the spot deviates an angle \varphi=0.314 rad respect to initial position. </title> |
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| <h3 id="back-link"><a href="../../#5.2">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#5.2">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
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| <p> | | <p> |
| $5.2.9$ | | $5.2.9$ |
| A molecular projector creates a ribbon-shaped narrow beam of silver's atoms that arrive to internal surface of a cylinder of radius $R = 30$ cm and form a spot. The device rotates with angular velocity $\omega = 100\pi$ s$^{-1}$. Find the beam's velocity, if the spot deviates an angle $\varphi=0.314$ rad respect to initial position. | | A molecular projector creates a ribbon-shaped narrow beam of silver's atoms that arrive to internal surface of a cylinder of radius $R = 30$ cm and form a spot. The device rotates with angular velocity $\omega = 100\pi$ s$^{-1}$. Find the beam's velocity, if the spot deviates an angle $\varphi=0.314$ rad respect to initial position. |
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| loading="lazy" alt="5.2.9" width="200" /> | | loading="lazy" alt="5.2.9" width="200" /> |
| <figcaption> | | <figcaption> |
| For problem 5.2.9 | | For problem 5.2.9 |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| When first atom of the beam arrives to surface after a time $t$, the cylinder has sweep the angle $\varphi$. So | | When first atom of the beam arrives to surface after a time $t$, the cylinder has sweep the angle $\varphi$. So |
| $$\varphi=\omega t \;(1)$$ | | $$\varphi=\omega t \;(1)$$ |
| The atoms are displaced from center to surface during same time $t$, considering constant velocity $v$ | | The atoms are displaced from center to surface during same time $t$, considering constant velocity $v$ |
| $$R=vt \;(2)$$ | | $$R=vt \;(2)$$ |
| Dividing $(2)$ by $(1)$ and separating $v$ | | Dividing $(2)$ by $(1)$ and separating $v$ |
| $$\boxed{v = \frac{\omega R}{\varphi}=300 \;{\rm{m/s}}}$$ | | $$\boxed{v = \frac{\omega R}{\varphi}=300 \;{\rm{m/s}}}$$ |
| </p> | | </p> |
| <p style="text-align: right; font-style: italic; font-size: 14;"> | | <p style="text-align: right; font-style: italic; font-size: 14;"> |
| BSc. Luis Daniel Fernández Quintana<br> | | BSc. Luis Daniel Fernández Quintana<br> |
| Physics Department (FCNE)<br> | | Physics Department (FCNE)<br> |
| Universidad de Oriente, Cuba<br> | | Universidad de Oriente, Cuba<br> |
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