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The circle's circumference is $c$ meters, and it rotates at a speed of $v$ RPM. The hole in the circle has a breadth of $l$ meters, and the pole (colored black) has a diameter of $d$ meters. The pole is in the center of the hole.

When will the circle's wall hit the pole?

Well, any point on the circle's circumference moves at a speed of $\frac{1}{60}vc \ \text{m}/\text s$. The red arrow is supposed to represent a straight line, going tangentially from the corner of the hole, to the surface of the pole. It has a length of $(l-d) \ \text m/2 $. From this, one might conclude that the wall and the pole will collide after the following amount of seconds:

$$\frac{(l-d) \ \text m/2 }{\frac{1}{60}vc \ \text{m}/\text s} =\frac{(l-d)60 \ \text s}{2vc} = \frac{(l-d)30 \ \text s}{vc} $$

But I reckon this is wrong, because the corner isn't moving in a straight line. Its path is curved, and that changes things. So, when will the wall hit the pole?

user110391
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  • You are correct that the straight-line answer has some inaccuracy (at least in theory). For a more accurate answer you need trigonometry. – David K Dec 24 '22 at 20:22
  • @DavidK But in practical applications dealing with a large circles (like a kiln with a circumference of about 20 meters), the innacuracy is negligible? – user110391 Dec 24 '22 at 20:54
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    There will be some percentage error which depends on how large $l$ and $d$ are compared to $c.$ For example, if $c = 20$ and $l = 1$ then you have an error of about half a percent at most, less if $d$ is much smaller than $l.$ – David K Dec 24 '22 at 21:24
  • @DavidK I see, thank you! – user110391 Dec 24 '22 at 21:57

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