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Obtain relation between coefficient of volume expansion (alpha_(V)) and coefficient of linear expansion (alpha_(l)). |
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Answer» Solution :Suppose there is a cube of side of length .l.. When its TEMPERATURE is increased by `DeltaT`, it expands EQUALLY in all DIMENSIONS. Hence from `V=l^(3)` `DeltaV=(l+Deltal)^(3)-l^(3)` `=l^(3)+3l^(2)Deltal+3l(Deltal)^(2)+(Deltal)^(3)-l^(3)` But `(Deltal)^(2)` and `(Deltal)^(3)` are much more less than l, hence by neglecting them `DeltaV=3l^(2)Deltal`. . .(1) But, from LINEAR expansion. `Deltal=alpha_(l)lDeltaT`. . .(2) `:.` By using value of equation (1) in equation (2), `DeltaV=3l^(2)(alpha_(l)lDeltaT)` `=3l^(3)alpha_(l)DeltaT` `DeltaV=3Valpha_(l)DeltaT""`. . .(3) `""(becausel^(3)=" VOLUME of cube")` By comparing equation (3) with general equation of volume expansion `DeltaV=alpha_(V)VDeltaT`. `alpha_(V)=3alpha_(l)` which is relation between coefficient of volume and linear expansion. |
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