A solid copper cylinder of length `l=65cm` is placed on a horizontal surface and subjected to a vertical compressive force `F=1000N` directed downward and distributed uniformly over the end face. What will be the resulting change of the volume of the cylinder in cubic millimeters?

Explanation

Volume of a solid cylinder
`V=pir^2l`
So, `(DeltaV)/(V)=(pi2rDeltarl)/(pir^2l)+(pir^2Deltal)/(pir^2l)=(2Deltar)/(r)+(Deltal)/(l)` (1)
But longitudinal strain `Deltal//l` and accompanying lateral strain `Deltar//r` are related as
`(Deltar)/(r)=-mu(Deltal)/(l)` (2)
Using (2) in (1), we get:
`(DeltaV)/(V)=(Deltal)/(l)(1-2mu)` (3)
But `(Deltal)/(l)=(-F//pir^2)/(E)`
(Because the increment in the length of cylinder `Deltal` is negative)
So, `(DeltaV)/(V)=(-F)/(pir^2E)(1-2mu)`
Thus, `DeltaV=(-Fl)/(E)(1-2mu)`
Negative sign means that the volume of the cylinder has decreased.