moment of inertia - 9

[logistic_guy]

Find the moment of inertia of a long uniform rod of length $\displaystyle l$ and mass $\displaystyle M$ when the rotation is through its end.

 

[khansaheb]

Find the moment of inertia of a long uniform rod of length $\displaystyle l$ and mass $\displaystyle M$ when the rotation is through its end.

Please show us what you have tried and exactly where you are stuck.

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[logistic_guy]

$\displaystyle I_{\text{end}} = \int R^2 \ dm = \int x^2 \ dm = \int_{0}^{l} x^2 \lambda \ dx$

This is the easiest integral so far.



$\displaystyle I_{\text{end}} = \lambda \frac{x^3}{3}\bigg|_{0}^{l} = \frac{M}{l}\frac{l^3}{3} = \textcolor{blue}{\frac{1}{3}Ml^2}$