there is no change in volume of a wire due to change in its length on stretching. what is the Poisson Ratio of the material. Answer this please.
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Poission's Ratio is defned as the ratio of lateral strain to the longitudinal strain.σ
where
Lateral strain: ratio of change in radius or diameter to original radius
Longitudinal strain : Ratio of change in length to original length
σ=lateral strain/longitudinal strain = -(dr/r)/dL/L
here negative sign indicates that the radius of the bar dereases when it is stretched.
Volume of wire =πr²l
but r=D/2
V=π(D/2)² l
=πD² l/4 -------- equation 1
differentiating equation 1
dV=πl/4 .2D.dD + πD²x 1/4 dl
As volume remains constant we get:
0=πl/2D.dD+π D²/4d
-π1/2DdD=πD²/4dl
σ=-(dD/D )l/dl=2/4 =0.5
∴Poisson Ratio of the material is 0.5
where
Lateral strain: ratio of change in radius or diameter to original radius
Longitudinal strain : Ratio of change in length to original length
σ=lateral strain/longitudinal strain = -(dr/r)/dL/L
here negative sign indicates that the radius of the bar dereases when it is stretched.
Volume of wire =πr²l
but r=D/2
V=π(D/2)² l
=πD² l/4 -------- equation 1
differentiating equation 1
dV=πl/4 .2D.dD + πD²x 1/4 dl
As volume remains constant we get:
0=πl/2D.dD+π D²/4d
-π1/2DdD=πD²/4dl
σ=-(dD/D )l/dl=2/4 =0.5
∴Poisson Ratio of the material is 0.5
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