moment of inertia (rod)

Question

Moment of inertia of a uniform thin rod (mass M, length L) about axis through center perpendicular to length?

RankGuru · Accepted Answer

1. **Definition**: Moment of inertia ($I$) measures an object's resistance to angular acceleration.
2. **Integration for Rod**: For a rod along the x-axis from $-L/2$ to $+L/2$:
   - $I = \int r^2 \, dm = \int_{-L/2}^{L/2} x^2 \left(\frac{M}{L} dx\right)$
   - $I = \frac{M}{L} \left[ \frac{x^3}{3} \right]_{-L/2}^{L/2}$
   - $I = \frac{M}{L} \left[ \frac{L^3}{24} - \left(-\frac{L^3}{24}\right) \right] = \frac{M}{L} \cdot \frac{2L^3}{24}$.
3. **Result**: **$I = \frac{1}{12} ML^2$**.
4. **Note**: About one end, the result is $\frac{1}{3} ML^2$ (using Parallel Axis Theorem).

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moment of inertia (rod)

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