What is the geometrical interpretation of this quantity $$\frac{\sin(\alpha)}{\alpha}$$ such that $\alpha\in(0,\pi/2)$
For example, this is the abscissa of the centroid of the arcs of the unit circle centered at the origin of angle $2\alpha$ and symmetric with respect to the axis of abscissa.
Is there any other geometrical interpretation of this quantity or can we characterize this centroid by another method
On
When we consider one half of the arc only:
$$ \bar{x}= \dfrac{\int x ds}{\int ds} = \dfrac{\int x ds}{\int ds} $$ with $ s = R \theta,\,$ the centroid $x$ location is
$$ \dfrac{\int_0^\alpha R \cos \theta R d\theta}{\int _0^\alpha R d\theta} =R\dfrac{\sin \alpha}{\alpha}$$
similarly for $ y= R \cos \theta $
$$ \bar{y}=R\dfrac{\;1-\cos \alpha}{\alpha}$$
Centroid locus for a half one side extending arc is shown.
So $\alpha $ need not be bounded.
$$ \alpha\rightarrow \infty, \bar{r} \rightarrow 0$$
Based on the previous hints, in a circle of radius $R$, take lines from the center, forming an angle $2\alpha$. The lines intersect the circle at $A$ and $B$. Then the length of the arc between them is $2R\alpha$, and the length of the chord is $2R\sin\alpha$. So $$\frac{\sin\alpha}{\alpha}=\frac{|\overparen{AB}|}{|AB|}$$ is the ratio of the cord length to the arc length for two points on the circle, forming an angle $2\alpha$ from the origin. ** As requested** I've added an image. The ratio is the length of the blue line to the length of the red arc.