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| <h2>Solutions of Savchenko Physics Textbook</h2> | | <h2>Solutions of Savchenko Physics Textbook</h2> |
| <p class="author"> | | <p class="author"> |
| Aliaksandr Melnichenka <br/> | | Aliaksandr Melnichenka <br/> |
| October 2023 | | October 2023 |
| $1.1.12^*.$ Two rods intersect at an angle $2 \alpha$ and move with equal velocities $v$ perpendicular to themselves. What is the velocity of the intersection point of the rods? | | $1.1.12^*.$ Two rods intersect at an angle $2 \alpha$ and move with equal velocities $v$ perpendicular to themselves. What is the velocity of the intersection point of the rods? |
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| <img src="kin29.png" | | <img src="kin29.png" |
| loading="lazy" width="230" /> | | loading="lazy" width="230" /> |
| <figcaption> | | <figcaption> |
| For problem $1.1.12^*$ | | For problem $1.1.12^*$ |
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| </figure> | | </figure> |
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| <p> | | <p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="animation.gif" | | <img src="animation.gif" |
| loading="lazy" alt="1.1.12" width="80%" /> | | loading="lazy" alt="1.1.12" width="80%" /> |
| <figcaption> | | <figcaption> |
| Animation of rod movement | | Animation of rod movement |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| As shown in the animation, the intersection point will remain on the bisection of the angle $2\alpha$ between them as they move | | As shown in the animation, the intersection point will remain on the bisection of the angle $2\alpha$ between them as they move |
| </p> | | </p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="sol.png" | | <img src="sol.png" |
| loading="lazy" alt="1.1.12" width="300" /> | | loading="lazy" alt="1.1.12" width="300" /> |
| <figcaption> | | <figcaption> |
| Movement in a small amount of time | | Movement in a small amount of time |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| <p> | | <p> |
| Consider the change of the intersection point on the horizontal axis during the time interval $dt$ | | Consider the change of the intersection point on the horizontal axis during the time interval $dt$ |
| </p> | | </p> |
| <p> | | <p> |
| From the geometry of a right triangle, the horizontal coordinate (aka hypotenuse) has changed to $dx = v dt / \sin\,\alpha$ | | From the geometry of a right triangle, the horizontal coordinate (aka hypotenuse) has changed to $dx = v dt / \sin\,\alpha$ |
| </p> | | </p> |
| <p> | | <p> |
| From where the speed: | | From where the speed: |
| </p> | | </p> |
| <p style="text-align: center;"> | | <p style="text-align: center;"> |
| $$\fbox{$u = \frac{dx}{dt} = \frac{v}{\sin\,\alpha}$}$$ | | $$\fbox{$u = \frac{dx}{dt} = \frac{v}{\sin\,\alpha}$}$$ |
| </p> | | </p> |
| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$u = v/ \sin α$$ | | $$u = v/ \sin α$$ |
| </p> | | </p> |
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