Added 1.2.20-1.2.21; 1.3.1-1.3.2
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| + | <meta name="description" content="a. From the top point of the circle, a ball begins to slide along a smooth chute at an angle \varphi to the vertical. How long will it take for it to reach the circle, if its diameter is D?b. From point A, small beads begin to slide along the spokes with different slopes at the same time without friction. What curve will the beads be on at time t?"> | ||
| + | <meta name="author" content="Aliaksandr Melnichenka"> | ||
| + | <meta name="date" content="2023-10" scheme="YYYY-MM"> | ||
| + | <meta property="og:title" content="a. From the top point of the circle, a ball begins to slide along a smooth chute at an angle \varphi to the vertical. How long will it take for it to reach the circle, if its diameter is D?b. From point A, small beads begin to slide along the spokes with different slopes at the same time without friction. What curve will the beads be on at time t?"> | ||
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| + | <title>a. From the top point of the circle, a ball begins to slide along a smooth chute at an angle \varphi to the vertical. How long will it take for it to reach the circle, if its diameter is D?b. From point A, small beads begin to slide along the spokes with different slopes at the same time without friction. What curve will the beads be on at time t?</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="../#1.3">$\leftarrow$Back</a></h3> | ||
| + | |||
| + | <h3> Statement </h3> | ||
| + | <p> | ||
| + | $1.3.2.$ a. From the top point of the circle, a ball begins to slide along a smooth chute at an angle $\varphi$ to the vertical. How long will it take for it to reach the circle, if its diameter is $D$? | ||
| + | </p><p> | ||
| + | b. From point $A$, small beads begin to slide along the spokes with different slopes at the same time without friction. What curve will the beads be on at time $t$? | ||
| + | </p> | ||
| + | |||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="statement.png" | ||
| + | loading="lazy" width="350" /> | ||
| + | <figcaption> | ||
| + | For problem 1.3.2 | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | |||
| + | <h3>Solution</h3> | ||
| + | <p> | ||
| + | <center> | ||
| + | <figure> | ||
| + | <img src="animation.gif" | ||
| + | loading="lazy" width="350" /> | ||
| + | <figcaption> | ||
| + | Animation of the movement of the balls on the spokes | ||
| + | </figcaption> | ||
| + | </figure> | ||
| + | </center> | ||
| + | a) A ball will move along a smooth chute with acceleration equal to the projection of the free-fall acceleration in the direction of motion, i.e. | ||
| + | $$a = g \cdot cos{\varphi }$$ | ||
| + | The displacement of the ball is the chord of a circle of diameter $D$, the magnitude of which is related to the diameter, by the following relation | ||
| + | $$r = D \cdot cos{\varphi }$$ | ||
| + | Let's write further the equation of accelerated motion of the ball and from it find the time of motion | ||
| + | $${r=\frac{at^{2}}{2},\quad D\cos\varphi=\frac{g\cos\varphi}{2}t^{2},\quad t=\sqrt{\frac{2D}{g}} .}$$ | ||
| + | $$\fbox{ $t=\sqrt{\frac{2D}{g}}$ } \; (1)$$ | ||
| + | b) Note that the expression $(1)$ does not include the value of the angle, so all balls will be dropped simultaneously. They will lie on a circle of radius $r = g t^2/2$, as shown in the animation | ||
| + | </p> | ||
| + | |||
| + | <h4>Answer</h4> | ||
| + | <p> | ||
| + | $$\text{a. }t = \sqrt{2D/g}$$ | ||
| + | $$\text{b. On a circle of radius }gt^{2}/2\text{ with top point }A.$$ | ||
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| + | <small> © <strong>Savchenko Solutions</strong>, 2023-2024 <br></small> | ||
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| + | <p> | ||
| + | <small>All rights belong to the authors. <br> Commercial use of materials - with the written permission of the authors. <br> astrosander01@gmail.com <br></small> | ||
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| <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. From the top point of the circle, a ball begins to slide along a smooth chute at an angle \varphi to the vertical. How long will it take for it to reach the circle, if its diameter is D?b. From point A, small beads begin to slide along the spokes with different slopes at the same time without friction. What curve will the beads be on at time t?"> | |||
| <meta name="author" content="Aliaksandr Melnichenka"> | |||
| <meta name="date" content="2023-10" scheme="YYYY-MM"> | |||
| <meta property="og:title" content="a. From the top point of the circle, a ball begins to slide along a smooth chute at an angle \varphi to the vertical. How long will it take for it to reach the circle, if its diameter is D?b. From point A, small beads begin to slide along the spokes with different slopes at the same time without friction. What curve will the beads be on at time t?"> | |||
| <meta property="og:image" content="img/logo.png"> | |||
| <meta property="og:description" content="a. From the top point of the circle, a ball begins to slide along a smooth chute at an angle \varphi to the vertical. How long will it take for it to reach the circle, if its diameter is D?b. From point A, small beads begin to slide along the spokes with different slopes at the same time without friction. What curve will the beads be on at time t?"> | |||
| <meta name="yandex-verification" content="6cfda41f74038368"> | |||
| <title>a. From the top point of the circle, a ball begins to slide along a smooth chute at an angle \varphi to the vertical. How long will it take for it to reach the circle, if its diameter is D?b. From point A, small beads begin to slide along the spokes with different slopes at the same time without friction. What curve will the beads be on at time t?</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="../#1.3">$\leftarrow$Back</a></h3> | |||
| <h3> Statement </h3> | |||
| <p> | |||
| $1.3.2.$ a. From the top point of the circle, a ball begins to slide along a smooth chute at an angle $\varphi$ to the vertical. How long will it take for it to reach the circle, if its diameter is $D$? | |||
| </p><p> | |||
| b. From point $A$, small beads begin to slide along the spokes with different slopes at the same time without friction. What curve will the beads be on at time $t$? | |||
| </p> | |||
| <center> | |||
| <figure> | |||
| <img src="statement.png" | |||
| loading="lazy" width="350" /> | |||
| <figcaption> | |||
| For problem 1.3.2 | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| <h3>Solution</h3> | |||
| <p> | |||
| <center> | |||
| <figure> | |||
| <img src="animation.gif" | |||
| loading="lazy" width="350" /> | |||
| <figcaption> | |||
| Animation of the movement of the balls on the spokes | |||
| </figcaption> | |||
| </figure> | |||
| </center> | |||
| a) A ball will move along a smooth chute with acceleration equal to the projection of the free-fall acceleration in the direction of motion, i.e. | |||
| $$a = g \cdot cos{\varphi }$$ | |||
| The displacement of the ball is the chord of a circle of diameter $D$, the magnitude of which is related to the diameter, by the following relation | |||
| $$r = D \cdot cos{\varphi }$$ | |||
| Let's write further the equation of accelerated motion of the ball and from it find the time of motion | |||
| $${r=\frac{at^{2}}{2},\quad D\cos\varphi=\frac{g\cos\varphi}{2}t^{2},\quad t=\sqrt{\frac{2D}{g}} .}$$ | |||
| $$\fbox{ $t=\sqrt{\frac{2D}{g}}$ } \; (1)$$ | |||
| b) Note that the expression $(1)$ does not include the value of the angle, so all balls will be dropped simultaneously. They will lie on a circle of radius $r = g t^2/2$, as shown in the animation | |||
| </p> | |||
| <h4>Answer</h4> | |||
| <p> | |||
| $$\text{a. }t = \sqrt{2D/g}$$ | |||
| $$\text{b. On a circle of radius }gt^{2}/2\text{ with top point }A.$$ | |||
| </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> astrosander01@gmail.com <br></small> | |||
| </p> | |||
| </footer> | |||
| </body> | |||
| </html> | |||