Irdov Problem 1.314

The work done dW in compressing a fluid by an infinitesimally small amount dV is given by PdV, where P is the pressure acting on the fluid (this comes from the fact that dW = Fdx = PAdx = PdV). The compressibility is defined as . Thus, the work done dW in increasing the pressure by dP over a volume of fluid V, is given by. The energy of elastic deformation stored in water of volume V is the work done in increasing pressure to P from 0 and is given by,



Since, at a depth h, the pressure of water is pgh, where density of water is p.

Irodov Problem 1.313

This problem, in my opinion, should actually be a sub-part of problem 1.312, I am really not sure why Irodov decided to make it a separate problem. As seen from eqn (1) in solution to problem 1.312, at a distance of r from the axis the strain is given by,




Thus, the strain energy per unit volume at a distance r is given by,

Irodov Problem 1.312

Since one end is fixed all points on that end will be at the same place after the distortion. Consider a line OA on the fixed side and a corresponding parallel line O'B on the free end. After the distortion, B moves to a location B' - this induces a shear strain of magnitude BB'/l. Another point X on AB moves to location X' and so this area experiences a shear XX'/l. Thus, the cylinder experiences different shear strains at different distances from the axis.

Suppose that the point X is at a distance x from the axis of the cylinder. The length of XX' is approximately and so the shear strain experienced at point X is given by,



The energy per unit volume due to this shear strain is given by . Consider an infinitesimally thin cylindrical section of thickness dx at point X. The volume of this infinitesimally thin cylindrical section is . The total energy can be computed by integrating the energy contained in all such infinitesimally thin sections and hence this given by,