Question

Use I=∫r2 dm to calculate I of a slender uniform rod of length L, about an axis at one end perpendicular to the rod. note: a "slender rod" often refers to a rod of neglible cross sectional area, so that the volume is the Length, and the mass density X Length.

Answer 1

Answer:

The moment of inertia of a slender uniform rod of length L about an axis at one end perpendicular to the rod is $I = \frac{1}{3}\cdot m \cdot L^{2}$.

Explanation:

Let be an uniform rod of length L whose origin is located at one one end and axis is perpendicular to the rod, such that:

$\lambda = \frac{dm}{dr}$

Where:

$\lambda$ - Linear density, measured in kilograms per meter.

$m$ - Mass of the rod, measured in kilograms.

$r$ - Distance of a point of the rod with respect to origin.

Mass differential can translated as:

$dm = \lambda \cdot dr$

The equation of the moment of inertia is represented by the integral below:

$I = \int\limits^{L}_{0} {r^{2}} \, dm$

$I = \lambda \int\limits^{L}_{0} {r^{2}} \, dr$

$I = \lambda \cdot \left(\frac{1}{3}\cdot L^{3} \right)$

$I = \frac{1}{3}\cdot m \cdot L^{2}$ (as $m = \lambda \cdot L$)

The moment of inertia of a slender uniform rod of length L about an axis at one end perpendicular to the rod is $I = \frac{1}{3}\cdot m \cdot L^{2}$.