Q. 26

A steel rod of length 2l, cross sectional area A and mass M is set rotating in a horizontal plane about an axis passing through the centre. If Y is the Young’s modulus for steel, find the extension in the length of the rod. (Assume the rod is uniform.)

Answer :



Force acting in this element due to rotating is



Where


df=force on that small element under consideration


dm=mass of element under consideration


ω=angular velocity (which will remain same for all particles)


x=distance from the axis of rotation


Now, Taking mass per unit length



We have taken 2L as we considering the length of half part to be L




Now we know a relation between young modulus and tension force, but since, force is a function of x hence we can only write the equation for a small element of length dx



Here we have represented extension in length as dr, since our original length is dx



Now we need to multiply it by 2 as we were considering only half length of rod,



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