Question

The moment of inertia of a thin uniform rod of
mass M and length L about an axis
perpendicular to the rod, through its centre is
L. The moment of inertia of the rod about an axis
perpendicular to rod through its end point is​

Answer 1

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Answer 2

Answer:

Moment of Inertia, I = ML²/3

Explanation:

Given:-

       Rod type = uniform

               Mass = M

            Length = L

 Angle about which it rotates on axis = 90°

Since, its not a point mass object but a rod whose mass m is spread uniformly over the length L. So, in order to find the moment of inertia, we take small change in distance and mass. As:

       After distance x, we have: dx & dm

[Refer to the attached image to visualize the scenario]

Using differentiation & integration, we find the value of moment of inertia now.

We know that, by formula:

    Density, λ = M/L _______(1)

i.e density equals mass per unit length.

We take small changes in distance and mass as:

        dI = dmx²   [since, I = mx²]

        dI = x² [M/L dx]

       ∫dI = M/L ∫x²dx

          I = M/L [x³/3]  

      lim ⇒ 0 to L.

          I = M/L × L³/ 3

         I = ML²/3

Hence, the moment of inertia of the rod about an axis perpendicular to rod through its end point is​ ML²/3.

Hope it helps! ;-))