A metal cylinder of length `L` is subjected to a uniform compressive to a uniform compressive force `F` as shown in the figure. The material of the cylinder has Young's modulus `Y` and poisson's ratio `sigma` The change in volume of the cylinder is

Options

A. `(sigmaFL)/(Y)`
B. `((1-sigma)FL)/(Y)`
C. `((1+2sigma)FL)/(Y)`
D. `((1-2sigma)FL)/(Y)`

Answer

D

Explanation

Volume of the cylinder,
`V=pir^2L`
Volumetric strain `=(triangleV)/(V)=(triangle(pir^2L))/(pir^2L)`
`=(triangleV)/(V)=(pir^2triangleL+2pirLtriangler)/(pir^2L)`
`=(triangleL)/(L)+(2triangler)/(r )`
Poisson's ratio, `sigma=(((triangler)/(r )))/(((triangleL)/(L)))`
or `(triangler)/( r)=-(sigmatriangleL)/(L)`
On substituting this value of `(triangler)/(r )` in Eq. (i) we get
`(triangleV)/(V)=(triangleL)/(L)(1-2sigma)` .(ii)
Young's modulus, `Y=(((F)/(pir^2)))/(((triangleL)/(L)))`
Or `(triangleL)/(L)=(F)/(pir^2Y)`
On subsitituting this value of `(triangleL)/(L)` in Eq. (ii) we get
`(triangleV)/(V)=(F)/(pir^2Y)(1-2sigma)`
`(triangleV)/(pir^2L)=(F)/(pir^2Y)(1-2sigma)`
`triagleV=(FL)/(Y)(1-2sigma)`