Question

find the moment of inertia of a uniform rod (rotated about one end) of mass M and length L starting from the definition of moment of inertia

PLEASE INCLUDE ANY RELEVANT CALCULUS CONCEPTS

Answer 1

Moment of inertia of a point object about an axis is given by [ where M = mass of the object, r = perpendicular distance of the object from the axis]

refer the diagram :

Cosnider a small element (can be assumed as a point object) of length "dx" at a distance 'x' from the axis of rotation.

mass of this element will 'dm'.

As the rod is of uniform mass distribution i.e; mass per unit length is same throughout.

then,  

As the element is of very small in size, the moment of inertia if the element about the axis is

integrating the right hand side of the above equation from 0 to L , we get moment of inertia of the complete rod about the axis.

[answer]