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| <title>The reel of tape is played back during time t at the film drawing speed v. The initial radius of the reel (with film) is R, and the final radius (without film) is r. What is the thickness of the film?</title> | | <title>The reel of tape is played back during time t at the film drawing speed v. The initial radius of the reel (with film) is R, and the final radius (without film) is r. What is the thickness of the film?</title> |
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| <h3> Statement </h3> | | <h3> Statement </h3> |
| <p> | | <p> |
| $1.5.20.$ The reel of tape is played back during time $t$ at the film drawing speed $v$. The initial radius of the reel (with film) is $R$, and the final radius (without film) is $r$. What is the thickness of the film? | | $1.5.20.$ The reel of tape is played back during time $t$ at the film drawing speed $v$. The initial radius of the reel (with film) is $R$, and the final radius (without film) is $r$. What is the thickness of the film? |
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| <h3>Solution</h3> | | <h3>Solution</h3> |
| <p> | | <p> |
| <center> | | <center> |
| <figure> | | <figure> |
| <img src="https://savchenkosolutions.com/1/1.5.20/draw.png" | | <img src="https://savchenkosolutions.com/1/1.5.20/draw.png" |
| loading="lazy" width="150" /> | | loading="lazy" width="150" /> |
| <figcaption> | | <figcaption> |
| Film reel | | Film reel |
| </figcaption> | | </figcaption> |
| </figure> | | </figure> |
| </center> | | </center> |
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| <p>Film wound on a reel takes up an area</p> | | <p>Film wound on a reel takes up an area</p> |
| <p> | | <p> |
| $$ S = \pi (R^2 - r^2) $$ | | $$ S = \pi (R^2 - r^2) $$ |
| </p> | | </p> |
| <p>If the film is completely unwound at a speed of $v$ in time $t$, then the length of the film </p> | | <p>If the film is completely unwound at a speed of $v$ in time $t$, then the length of the film </p> |
| <p> | | <p> |
| $$ l = vt $$ | | $$ l = vt $$ |
| </p> | | </p> |
| <p>In cross-section, the film is a rectangle whose area is $S$ and length is $l$. Then the width of this rectangle (it is also the thickness of the film)</p> | | <p>In cross-section, the film is a rectangle whose area is $S$ and length is $l$. Then the width of this rectangle (it is also the thickness of the film)</p> |
| <p> | | <p> |
| $$ \fbox{$d = \frac{S}{l} = \frac{\pi (R^2 - r^2)}{vt}$} $$ | | $$ \fbox{$d = \frac{S}{l} = \frac{\pi (R^2 - r^2)}{vt}$} $$ |
| </p> | | </p> |
| </p> | | </p> |
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| <h4>Answer</h4> | | <h4>Answer</h4> |
| <p> | | <p> |
| $$d = \frac{\pi (R^2 - r^2)}{vt}$$ | | $$d = \frac{\pi (R^2 - r^2)}{vt}$$ |
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