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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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| <p class="author"> | | <p class="author"> |
| 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="../../#1.3">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.3">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <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 $D$? | | $1.3.2.$ 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> | | </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 $t$? | | 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> | | </p> |
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| <center> | | <center> |
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| <img src="statement.png" | | <img src="statement.png" |
| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| For problem 1.3.2 | | For problem 1.3.2 |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="animation.gif" | | <img src="animation.gif" |
| loading="lazy" width="350" /> | | loading="lazy" width="350" /> |
| <figcaption> | | <figcaption> |
| Animation of the movement of the balls on the spokes | | Animation of the movement of the balls on the spokes |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </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) 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. |
| $$a = g \cdot\cos{\varphi }$$ | | $$a = g \cdot\cos{\varphi }$$ |
| 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 | | 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 |
| $$r = D \cdot\cos{\varphi }$$ | | $$r = D \cdot\cos{\varphi }$$ |
| Let's write further the equation of accelerated motion of the ball and from it find the time of motion | | Let's write further the equation of accelerated motion of the ball and from it find the time of motion |
| $${r=\frac{at^{2}}{2},\quad D\cos\varphi=\frac{g\cos\varphi}{2}t^{2},\quad t=\sqrt{\frac{2D}{g}} .}$$ | | $${r=\frac{at^{2}}{2},\quad D\cos\varphi=\frac{g\cos\varphi}{2}t^{2},\quad t=\sqrt{\frac{2D}{g}} .}$$ |
| $$\fbox{ $t=\sqrt{\frac{2D}{g}}$ } \; (1)$$ | | $$\fbox{ $t=\sqrt{\frac{2D}{g}}$ } \; (1)$$ |
| 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 | | 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> | | </p> |
| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$\text{a. }t = \sqrt{2D/g}$$ | | $$\text{a. }t = \sqrt{2D/g}$$ |
| $$\text{b. On a circle of radius }gt^{2}/2\text{ with top point }A.$$ | | $$\text{b. On a circle of radius }gt^{2}/2\text{ with top point }A.$$ |
