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Given Y = 2 xx 10^(11)N//m^(2), rho = 10...

Given `Y = 2 xx 10^(11)N//m^(2), rho = 10^(4) kg//m^(3)`
Find out elongation in rod.

Text Solution

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mass of shaded portion
`m' = (m)/(l)(l-x)` , [where `m =` total mass `= rho A l`]
`T = m'[(l-x)/(2) + x] rArr T = (m)/(l)(l - x)omega^(2)((l-x)/(2)), T = (momega^(2))/(2l)(l^(2) - x^(2))`
`m'omega^(2){(l-x)/(2)+x}`

this tension will be maximum at `A ((mw^(2)l)/(2))` and minimum at `'B'` (zero), elongation in element of width `'dx' = (Tdx)/(AY)` Total elongation
`delta=int(Tdx)/(AY)=overset(l)underset(0)(int)(momega^(2)(l^(2)-x^(2)))/(2lAY)dx`
`delta=(momega^(2))/(2lAY)[l^(2)xx-(x^(3))/(3)]_(0)^(l)=(momega^(2)xx2l^(3))/(2lAYxx3)=(momega^(2)l^(2))/(3AY)=(rhoAlomega^(2)l^(2))/(3AY)`
`delta=(rhoomega^(2)l^(3))/(3AY)=(10^(4)xx(400)xx(1.5)^(3))/(3xx2xx10^(11))=9xx10^(-3)m=9mm`
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