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[I have created a GeoGebra link to the system I describe below. It helps a lot in understanding.]

Assume I have a 2D body whose border is a curve. This body is pivoting about a point $(0,0)$. A constant vertical downward force is applied on such body on the point that is always at the same $x$-coordinate. What is the curve profile, such that the force is always perpendicular to the tangent at every point of application, even when the body rotates?

In the GeoGebra model I have used an ellipse, but obviously this doesn't work. I am only really interested for the angle values between $\pm\frac\pi6$ rad.

Thank you vey much for your time.

J. M. ain't a mathematician
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1 Answers1

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The curve you seek is precisely the involute of a circle, as the animation on the Wikipedia article should make clear:

enter image description here

Image by Wikimedia Commons user "Van helsing", reproduced under CC-BY-2.5.

The perpendicular line at any point on the curve remains at a constant distance to the origin. If you simultaneously rotated the curve in the opposite direction to keep the perpendicular vertical, it would look exactly like your Geogebra sketch.