Centre of Mass of a Semicircular Disc

Result

The CM of a uniform solid semicircular disc of radius $R$ lies at:

$y_{cm} = \frac{4R}{3\pi} \approx 0.424R$

Derivation

Divide the semicircular disc into thin semicircular rings (strips parallel to the diameter).

A ring at radius $r$ with thickness $dr$:

• Area: $\pi r \, dr$ (half circumference × thickness)
• Mass: $dm = \sigma \cdot \pi r \, dr$ where $\sigma = \frac{M}{\pi R^2/2} = \frac{2M}{\pi R^2}$

The CM of this thin semicircular ring is at $y = \frac{2r}{\pi}$ (from the semicircular ring result).

$y_{cm} = \frac{1}{M}\int_0^R \frac{2r}{\pi} \cdot \sigma \pi r \, dr = \frac{\sigma}{M}\int_0^R 2r^2 \, dr$

$= \frac{2M/(\pi R^2)}{M} \cdot \frac{2R^3}{3} = \frac{2}{\pi R^2} \cdot \frac{2R^3}{3} = \frac{4R}{3\pi}$

$\boxed{y_{cm} = \frac{4R}{3\pi}}$

Comparison with semicircular ring

The disc's CM is closer to the diameter. This makes sense — the disc has mass distributed throughout its area, including near the diameter where $y$ is small. The ring has all its mass at the rim where $y$ is large.