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| <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> | | <h3 id="back-link"><a href="../../#1.5">$\leftarrow$Back</a></h3> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.5.19.$ A spherical buoy of radius $R$ is attached to the bottom of the reservoir. The water level in the reservoir rises at a speed $u$. What is the speed of movement of the boundary of the flooded part of the buoy on its surface at the moment when the water level is $h$ above the center of the buoy? | | $1.5.19.$ A spherical buoy of radius $R$ is attached to the bottom of the reservoir. The water level in the reservoir rises at a speed $u$. What is the speed of movement of the boundary of the flooded part of the buoy on its surface at the moment when the water level is $h$ above the center of the buoy? |
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| <figcaption> | | <figcaption> |
| For problem $1.5.19$ | | For problem $1.5.19$ |
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| <p> | | <p> |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
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| <img src="https://savchenkosolutions.com/1/1.5.19/draw.png" | | <img src="https://savchenkosolutions.com/1/1.5.19/draw.png" |
| loading="lazy" width="180" /> | | loading="lazy" width="180" /> |
| <figcaption> | | <figcaption> |
| Changing the water level in the buoy | | Changing the water level in the buoy |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
| | | |
| <p>By time $t$ the water level will be $h$</p> | | <p>By time $t$ the water level will be $h$</p> |
| <p>In a short period of time $dt$ the water layer will rise by </p> | | <p>In a short period of time $dt$ the water layer will rise by </p> |
| <p> | | <p> |
| $$ dh = u \, dt $$ | | $$ dh = u \, dt $$ |
| </p> | | </p> |
| <p>In turn, the boundary of the flooded part will also rise by </p> | | <p>In turn, the boundary of the flooded part will also rise by </p> |
| <p> | | <p> |
| $$ dh = v\, dt \cos \alpha $$ | | $$ dh = v\, dt \cos \alpha $$ |
| </p> | | </p> |
| <p>Let's equate both equations</p> | | <p>Let's equate both equations</p> |
| <p> | | <p> |
| $$ v \cos \alpha = u $$ | | $$ v \cos \alpha = u $$ |
| </p> | | </p> |
| <p> | | <p> |
| $$ v = \frac{u}{\cos \alpha} $$ | | $$ v = \frac{u}{\cos \alpha} $$ |
| </p> | | </p> |
| <p> | | <p> |
| $$ \fbox{$v = u \frac{R}{\sqrt{R^2 - h^2}}$} $$ | | $$ \fbox{$v = u \frac{R}{\sqrt{R^2 - h^2}}$} $$ |
| </p> | | </p> |
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| | | |
| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$v = u \frac{R}{\sqrt{R^2 - h^2}}$$ | | $$v = u \frac{R}{\sqrt{R^2 - h^2}}$$ |
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