Question

A uniform rod of length 2L' has mass per unit length ‘m'. Find the moment of inertia of the rod about an axis passing through its centre and perpendicular to its length.​

Answer 1

The moment of inertia is 2mL³/3

Explanation:

If we take a small mass dm at a distance x from the given axis and if the length of the mass is dx then

$dm=mdx$

The moment of inertia of this mass will be

$dI=dmx^2=mx^2dx$

Thus, the moment of inertia of the half rod of length L

$I=\int_0^{L}mx^2dx$

$I=m\frac{x^3}{3}$

$\implies I=m\frac{L^3}{3}$

The other half will also have this moment of inertia

Therefore, the moment of inertia of the rod

$=2I$

$=2\times m\frac{L^3}{3}$

$=\frac{2}{3}mL^3$

Hope this answer is helpful.

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