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A thin uniform rod of length L and radius r rotates with an angular velocity in a horizontal plane about a vertical axis passing through one of its ends. The density of the material is rho and Youngs modulus is Y. 1) Find tension in the rod at distance Y from the axis. 2) Find elongation of the rod

Answer» <html><body><p></p>Solution :1) Consider and element dx at a <a href="https://interviewquestions.tuteehub.com/tag/distance-116" style="font-weight:bold;" target="_blank" title="Click to know more about DISTANCE">DISTANCE</a> from the axis of rotation. <br/>2) The centripetal force on this portion is <br/> `dt= dmr <a href="https://interviewquestions.tuteehub.com/tag/omega-585625" style="font-weight:bold;" target="_blank" title="Click to know more about OMEGA">OMEGA</a>^(2) dT=(rho A dr) <a href="https://interviewquestions.tuteehub.com/tag/r-611811" style="font-weight:bold;" target="_blank" title="Click to know more about R">R</a> omega^(2)` <br/> This force is provided by the tension in the rod <br/> `int dT= overset(L) underset(r) int rho A dr omega^(2), T rho A omega^(2)overset(L) underset(r) intrdr, T= rho A omega^(2)[ (r^(2))/(2)]_(r)^(L)` <br/> `rArr T=(rhoA omega^(2))/(2)[ L^(2)-r^(2)] T=(1)/(2) rho A omega^(2)(L^(2)-r^(2))` <br/> (3) strain `=("stress")/(Y)` If dy is the elongation in element of length dx<br/> `(dy)/(dx)=(T)/(<a href="https://interviewquestions.tuteehub.com/tag/ay-386628" style="font-weight:bold;" target="_blank" title="Click to know more about AY">AY</a>) rArr dy=(Tdx)/(AY) rArr dy=((1)/(2) rho A omega^(2)(L^(2)-r^(2)) dr)/(AY)` <br/> `int dy=(rho omega^(2))/(2y) overset(L) underset(<a href="https://interviewquestions.tuteehub.com/tag/0-251616" style="font-weight:bold;" target="_blank" title="Click to know more about 0">0</a>) int (L^(2)-r^(2))dr rArr DeltaL=(rho omega^(2))/(2y)[ L^(2)r-(r^(3))/(3)]_(0)^(L)` <br/> `DeltaL=(rho omega^(2))/(2Y)[L^(2)-(L^(3))/(3)] rArr DeltaL=( rho omega^(2))/(2y)[(2L)/(3)], DeltaL-=( rho omega^(2)L^(3))/(3Y)`</body></html>


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