moment of inertia

Question

What is the moment of inertia of a uniform rod of mass M and length L about an axis perpendicular to it through its center?

RankGuru · Accepted Answer

1. **Definition**: Moment of inertia ($I$) represents a body's resistance to angular acceleration about a specific axis.
2. **Derivation Logic**: Imagine the rod on the X-axis from $-L/2$ to $+L/2$. Mass of an element $dx$ is $dm = (M/L)dx$.
3. **Integral**: $I = \int r^2 dm = \int_{-L/2}^{L/2} x^2 \frac{M}{L} dx$.
4. **Result**: $\frac{M}{L} [\frac{x^3}{3}]_{-L/2}^{L/2} = \frac{M}{L} (\frac{L^3}{24} - (-\frac{L^3}{24})) = \frac{ML^2}{12}$.
5. **Note**: If the axis passes through one end, the limit changes to $0$ to $L$, giving $ML^2/3$.

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