Find the centroid of a homogeneous semicircle of radius a
Find the centroid of a homogeneous semicircle of radius $a$.
Explanation of Method
Placing the diameter on the $x$-axis makes the region symmetric about the $y$-axis, so only the static moment about the diameter needs to be computed — done cleanly in polar coordinates.
- Use symmetry to get $\bar x=0$.
- Compute the area $A$.
- Compute the static moment $M_x$ in polar coordinates.
- Divide to get $\bar y$.
Step 1 · Setup and symmetry
With $D:x^2+y^2\le a^2,\,y\ge0$, the region is symmetric about the $y$-axis, so $\bar x=0$.
Step 2 · Area
$$ A = \frac{\pi a^2}2. $$
Step 3 · Static moment $M_x$
In polar coordinates $x=r\cos\theta,\,y=r\sin\theta$, $dA=r\,dr\,d\theta$, with $0\le r\le a$, $0\le\theta\le\pi$:
$$ M_x = \int_0^\pi\int_0^ar\sin\theta\cdot r\,dr\,d\theta = \int_0^\pi\sin\theta\,d\theta\cdot\int_0^ar^2\,dr = 2\cdot\frac{a^3}3 = \frac{2a^3}3. $$
Step 4 · Centroid coordinate $\bar y$
$$ \bar y = \frac{M_x}A = \frac{2a^3/3}{\pi a^2/2} = \frac{4a}{3\pi}. $$
Additional Notes
- This derivation matches the source exactly. This is a standard reference result used to cross-check several other centroid problems in this collection (e.g. the sector and segment centroids reduce to it in the appropriate limits).
- Verified numerically (e.g. $a=2.3$): direct double integration gives $\bar y\approx0.97615$, matching $\dfrac{4a}{3\pi}$ exactly.
Result
The centroid of the homogeneous semicircle lies on the axis of symmetry at: