Irodov Problem 1.311

As the steel band shown beside is bent into a hoop, the deformation causes strain in the hoop. Suppose that the steel band was made into a hoop of radius ro. Then we have,



Consider an infinitesimally thin section of radius r and thickness dr in the hoop. The length of this section of the hoop is . Hence the change in length experienced by this section is given by . The strain seen by this section of the hoop is given by,



















The volume of this infinitesimally thin ring is given by . The elastic energy per unit volume is given by . Hence, the total elastic energy contained in this infinitesimally thin hoop section is given by,



The total energy contained in the hoop is thus given by,








The work done on the steel band to deform it into the hoop will be exactly equal to this elastic energy due to deformation.

Irodov Problem 1.310

The part of the cylinder AX will be pulled downward at point X (which is at a height x from the bottom end B), by the weight of the cylinder in part XB given by where p is the density of the cylinder. Hence, the stress experienced at the cross-section of point X will be



Thus, at each height the cylinder experiences a different amount of stress. The strain at this point X is then given by,



The net elongation of the cylinder can be found as,










The elastic energy per unit volume at the location X is given by . Consider an infinitesimally thin disc at location X of thickness dx. The volume of this infinitesimally thin disc will be The elastic energy in an infinitesimally thin disc of thickness dx is given by given by,




The total elastic energy contained in the cylinder is given by,