Moment of Inertia Calculator — Theory
1. What is moment of inertia?
Moment of inertia measures how difficult it is to rotate an object about an axis. It depends not only
on the total mass, but also on how far the mass is from the rotation axis.
\[
I=\int r^2\,dm.
\]
The factor \(r^2\) means that mass farther from the axis contributes much more to the moment of inertia.
2. Moments of inertia for a lamina
A lamina is a flat two-dimensional plate in the \(xy\)-plane. If its surface density is
\(\rho(x,y)\), then
\[
I_x=\iint_\Omega y^2\rho(x,y)\,dA,
\]
\[
I_y=\iint_\Omega x^2\rho(x,y)\,dA.
\]
Since the lamina lies in the plane \(z=0\), the perpendicular-axis theorem gives
\[
I_z=I_x+I_y.
\]
3. Moments of inertia for a solid
For a solid with volume density \(\rho(x,y,z)\), the three coordinate-axis moments are
\[
I_x=\iiint_\Omega (y^2+z^2)\rho(x,y,z)\,dV,
\]
\[
I_y=\iiint_\Omega (x^2+z^2)\rho(x,y,z)\,dV,
\]
\[
I_z=\iiint_\Omega (x^2+y^2)\rho(x,y,z)\,dV.
\]
Each formula uses the squared distance from the chosen axis.
4. Mass and centroid
For a lamina, the total mass is
\[
M=\iint_\Omega \rho(x,y)\,dA.
\]
For a solid, the total mass is
\[
M=\iiint_\Omega \rho(x,y,z)\,dV.
\]
The centroid or center of mass is found from first moments:
\[
\bar x=\frac{1}{M}\int x\,dm,
\qquad
\bar y=\frac{1}{M}\int y\,dm,
\qquad
\bar z=\frac{1}{M}\int z\,dm.
\]
5. Parallel-axis theorem
If an axis is shifted parallel to a centroidal axis by distance \(d\), then
\[
I_{\text{parallel}}=I_{\text{centroid}}+Md^2.
\]
For example, if \(I_x\) is about the origin and the centroid is
\((\bar x,\bar y,\bar z)\), then
\[
I_{x,c}=I_x-M(\bar y^2+\bar z^2).
\]
6. Common uniform-density formulas
7. Worked example: rectangular plate
For a uniform rectangular plate of width \(W\), height \(H\), and mass \(M\), centered at the origin,
the moment about the \(x\)-axis is
\[
I_x=\iint_\Omega y^2\rho\,dA.
\]
Since \(\rho=M/(WH)\),
\[
I_x=
\frac{M}{WH}
\int_{-W/2}^{W/2}
\int_{-H/2}^{H/2}
y^2\,dy\,dx.
\]
The inner integral is
\[
\int_{-H/2}^{H/2}y^2\,dy=\frac{H^3}{12}.
\]
Therefore,
\[
I_x=\frac{1}{12}MH^2.
\]
8. Variable density
If density is not constant, the same formulas still apply, but \(\rho\) must stay inside the integral.
For example,
\[
I_y=\iint_\Omega x^2\rho(x,y)\,dA.
\]
Variable density can shift the centroid and change the moments of inertia. That is why the calculator
computes mass, centroid, and centroidal moments separately.
9. Rotational physics connection
Moment of inertia appears in rotational kinetic energy:
\[
K_{\text{rot}}=\frac{1}{2}I\omega^2.
\]
It also appears in Newton’s second law for rotation:
\[
\tau=I\alpha.
\]
Larger \(I\) means the object is harder to angularly accelerate.
10. Common mistakes
- Using the wrong distance: \(I_x\) uses distance from the \(x\)-axis, not the \(x\)-coordinate.
- Confusing lamina and solid formulas: laminas use \(dA\), solids use \(dV\).
- Ignoring density: variable density can shift the centroid and change \(I\).
- Forgetting the parallel-axis theorem: centroidal moments and origin moments are not always the same.
- Using \(I_z=I_x+I_y\) for solids: that theorem applies to flat laminas, not general 3D solids.
