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| <title>To save space, the entrance to one of the highest bridges in Japan is arranged in the form of a spiral line that wraps around a cylinder of radius R. The roadbed forms an angle \alpha with the horizontal plane. What is the acceleration of a car moving along it at a constant modulo velocity v?</title> | | <title>To save space, the entrance to one of the highest bridges in Japan is arranged in the form of a spiral line that wraps around a cylinder of radius R. The roadbed forms an angle \alpha with the horizontal plane. What is the acceleration of a car moving along it at a constant modulo velocity v?</title> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.3.29.$ To save space, the entrance to one of the highest bridges in Japan is arranged in the form of a spiral line that wraps around a cylinder of radius $R$. The roadbed forms an angle $\alpha$ with the horizontal plane. What is the acceleration of a car moving along it at a constant modulo velocity $v$? | | $1.3.29.$ To save space, the entrance to one of the highest bridges in Japan is arranged in the form of a spiral line that wraps around a cylinder of radius $R$. The roadbed forms an angle $\alpha$ with the horizontal plane. What is the acceleration of a car moving along it at a constant modulo velocity $v$? |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.3.29/Bridge.jpg" | | <img src="https://savchenkosolutions.com/1/1.3.29/Bridge.jpg" |
| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| Spiral Bridge in Japan | | Spiral Bridge in Japan |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
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| <p> Find the value of acceleration $ a $: </p> | | <p> Find the value of acceleration $ a $: </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ a = \sqrt{a_{\tau}^2 + a_{n}^2} $$ | | $$ a = \sqrt{a_{\tau}^2 + a_{n}^2} $$ |
| </p> | | </p> |
| <p> Let $ a_{\tau} = 0 $, then $ a_{n} = \frac{v^2}{R} $, where $ v $ is the horizontal component of the velocity directed perpendicular to the axis of rotation. </p> | | <p> Let $ a_{\tau} = 0 $, then $ a_{n} = \frac{v^2}{R} $, where $ v $ is the horizontal component of the velocity directed perpendicular to the axis of rotation. </p> |
| <p> Then the required acceleration will be equal to: </p> | | <p> Then the required acceleration will be equal to: </p> |
| <p class="exp"> | | <p class="exp"> |
| $$ a = a_n $$ | | $$ a = a_n $$ |
| </p> | | </p> |
| <p class="exp"> | | <p class="exp"> |
| $$\fbox{$ a = \frac{v^2}{R} \cdot \cos^2 \alpha $}$$ | | $$\fbox{$ a = \frac{v^2}{R} \cdot \cos^2 \alpha $}$$ |
| </p> | | </p> |
| </p> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$a = \frac{v^2}{R} \cdot \cos^2 \alpha $$ | | $$a = \frac{v^2}{R} \cdot \cos^2 \alpha $$ |
| </p> | | </p> |
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