Question

Use equation I=∫r2dm to calculate the moment of inertia of a slender, uniform rod with mass M and length L about an axis at one end, perpendicular to the rod. Express your answer in terms of the variables M and L.

Answer 1

Answer:

The moment of inertia is    $I= \frac{1}{3} ML^2$

Explanation:

From the question we a given an equation

                        $I = \int\limits{r^2} \, dm$

Now let look at a small length da of this uniform rod that a meters form the x-axis

 The mass of this small section can be mathematically evaluated as

                       $dM = da *\frac{M}{L}$

For this small portion for the rod the moment of inertia is

                   $dI = (dM)a^2$

Substituting for dM

                  $dI = da [\frac{M}{L} ] a^2$

to get I we integrate both sides

                 $I = \int\limits^L_0 {\frac{M}{L} * a^2} \, da$

                   $= [\frac{M}{L} ][\frac{a^3}{3} ]\left L} \atop {0}} \right.$

                   $I= \frac{1}{3} ML^2$