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I have a point $P$ situated on the circumference of a circle with radius $r=2$ (as shown here) and the circle rotates along the x-axis by $180^°$ to the right.

If the circle rotates by that amount, it should travel $2\pi$ units (half its circumference) to the right, and as a result of the rolling motion, the point $P$ should move up by $4$ (its diameter) units.

I tried to make this an animation in desmos, but instead of the circle traveling $2\pi$ units, it seems to travel only $\pi$ units after rotating $180^°$

And if I do try to move it $2\pi$ units, the point $P$ instead of moving up comes back down.

I've been stuck here for so long, I really can't seem to figure out what I did wrong, is it the animation or am I getting the maths wrong? I believe that the latter is more unlikely.

It would be really helpful if someone looked at the animation here and told me what's going on.

Noor
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  • Hi, Your calculations are correct, You have calculated the distance travelled by the point $P$ due to the rotation. But, The position of the point $P$ depends on the angle of rotation. I suggest you to look up the concept of Radians. – vishalnaakar25 Apr 18 '25 at 16:04

1 Answers1

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The coordinate of the center of the circle is $2a$ not $a$. So when you rotate $\pi$ radians you move $r\pi=2\pi$ along $x$ axis. So modify the $x$ for the circle from $x-a$ to $x-2a$, and the $x$ for the two points from $a\pm2\sin a$ to $2a\pm\sin a$.

Andrei
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  • Thanks! This was exactly the problem, I get why $x-a$ became $x-2a$ and why $a\pm 2 \sin a$ became $2a\pm 2\sin a$, but why is the x coordinate of the center $2a$ and not $a$, is it because the radius $r=2$? I'm still in high school, your patience is much appreciated. – Noor Apr 18 '25 at 16:19
  • Correct. Try to do the same thing with a different radius – Andrei Apr 18 '25 at 16:23