A rod AB of length l is pivoted at an end A and freely rotated in a horizontal plane at an angular speed omega about a vertical axis passing through A. If coefficient of linear expansion of material of rod is alpha, find the percentagechangein its angular velocity if temperature of system is increased by DeltaT.

A rod AB of length l is pivoted at an end A and freely rotated in a ho

1.	A rod AB of length l is pivoted at an end A and freely rotated in a horizontal plane at an angular speed omega about a vertical axis passing through A. If coefficient of linear expansion of material of rod is alpha, find the percentagechangein its angular velocity if temperature of system is increased by DeltaT.
Answer» Solution :If TEMPERATURE of surroundingincreases by `DeltaT`, the new lengthof rod becomes `l.=l(1+alphaDeltaT)` Due to change in LENGTH, moment of inertia of rod also changes and its is about and end A and is given as `I._(A)= (Ml^(2))/(3)` As no EXTERNALFORCE of torque is actingon rod, its angular momentumremainsconstant during HEATING. Thus we have `I_(A)omega=I_(A).omega.` [where `omega.` is the final anuglarvelocity of rod after heating] (or) `(Ml^(2))/(3)omega=(Ml^(2)(1+alpha DeltaT)^(2))/(3)omega.` `rArr omega=(1+2alpha DeltaT)omega^(1) or omega. = omega(1-2alphaDeltaT)` [using binomialexpansion for small `alpha`] Thus percentagechange in angularvelocity of rod due to heating can be given as `Delta omega=(omega-omega.)/(omega) xx 100%=2alpha DeltaT xx 100%`