Why is the $y$ centroid of a semicircle and that of a semicircular arc different? Using Pappus' second theorem on a semicircle of radius $r$,
$\bar{y}=\frac{V}{2\pi A}=\frac{\frac{4}{3}\pi r^3}{2\pi (\frac{\pi r^2}{2})}=\frac{4r}{3\pi}$
Using Pappus' first theorem on a semicircular arc,
$\bar{y}=\frac{S}{2\pi s}=\frac{4\pi r^2}{2\pi \frac{2\pi r}{2}}=\frac{2r}{\pi}$
This is confirmed by Wikipedia. That page also claims that the area of the arc is $\pi r$. What does that even mean?
$\endgroup$ 11 Answer
$\begingroup$To answer your first question - why the centroids of a half-disk and half-arc are different, imagine their physical manifestations: both are objects having the same mass. The former is a half-disk of material that is evenly distributed throughout. The latter is also a half-disk, but it has all of its mass concentrated at the edge. Clearly, their centers of mass (i.e., centroids) will be different.
To answer your second question, they use the term "area" very loosely to mean that density times area = mass. For the arc, density must be a linear density rather than an areal density.
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