7. Center Of Mass Calculator — Theory
This calculator finds the centroid (uniform density) or center of mass (variable density) of a planar lamina/region.
It supports regions defined by two curves \(y_B(x)\le y\le y_T(x)\) over \([a,b]\).
1) Mass, moments, and centroid
\[
M=\iint_R \rho(x,y)\,dA,\qquad
M_y=\iint_R x\,\rho(x,y)\,dA,\qquad
M_x=\iint_R y\,\rho(x,y)\,dA
\]
\[
\bar x=\frac{M_y}{M},\qquad
\bar y=\frac{M_x}{M}
\]
If \(\rho\equiv 1\), then \(M\) is just the area \(A\), and \((\bar x,\bar y)\) is the geometric centroid.
2) Region between curves (single-integral shortcuts)
Suppose the region is
\[
R=\{(x,y)\mid a\le x\le b,\; y_B(x)\le y\le y_T(x)\}.
\]
Then \(dA=(y_T(x)-y_B(x))\,dx\) when using vertical strips.
Uniform density \(\rho=1\)
\[
A=\int_a^b\left(y_T-y_B\right)\,dx
\]
\[
M_y=\int_a^b x\left(y_T-y_B\right)\,dx,\qquad
M_x=\int_a^b \frac12\left(y_T^2-y_B^2\right)\,dx
\]
\[
\bar x=\frac{M_y}{A},\qquad \bar y=\frac{M_x}{A}
\]
Density depending only on \(x\): \(\rho=\rho(x)\)
\[
M=\int_a^b \rho(x)\left(y_T-y_B\right)\,dx
\]
\[
M_y=\int_a^b x\,\rho(x)\left(y_T-y_B\right)\,dx,\qquad
M_x=\int_a^b \rho(x)\,\frac12\left(y_T^2-y_B^2\right)\,dx
\]
These are the exact formulas used by the calculator in “Uniform” and “\(\rho(x)\)” modes.
For \(\rho(x,y)\), the calculator evaluates the double integral numerically using an inner Simpson integral in \(y\).
4) Worked example: semicircle of radius \(r\)
Consider the upper semicircle \(y=\sqrt{r^2-x^2}\) on \([-r,r]\) with uniform density.
By symmetry about the y-axis, \(\bar x=0\).
\[
A=\frac{\pi r^2}{2}
\]
Using the strip formula for \(M_x\) (moment about the x-axis):
\[
M_x=\int_{-r}^{r}\frac12\,y^2\,dx
=\int_{-r}^{r}\frac12\left(r^2-x^2\right)\,dx
=\frac{2r^3}{3}
\]
\[
\bar y=\frac{M_x}{A}
=\frac{\frac{2r^3}{3}}{\frac{\pi r^2}{2}}
=\frac{4r}{3\pi}
\]
For \(r=1\): \(\bar y=\dfrac{4}{3\pi}\approx 0.42441\). This matches the calculator’s semicircle preset.
5) Engineering tip
For beams/plates, variable density \(\rho(x)\) is a common model (e.g., material grading along the length).
Use presets as a starting point, then adjust \(\rho\) to match your design.
