Irodov Problem 1.294

The stress F in the wire opposes the weight of the hanging mass mg and keeps it suspended as shown in the figure. The stress in the wire comes from the fact that as the length of the wire has increased from its original state due to the mass being suspended at a distance x below the horizontal as depicted in the figure (AO <>).

The length of AO' is given by using Pythagoras theorem, thus the change in length is . The stress induced in the string is given by,





The force in the string due to the stress is the stress times the area of the thickness of the string and is given by,




The total vertical component of the forces due to the both the halves of the string are responsible for holding the weight in place and so we have,












(4) is not easy to solve and so we can get an approximate solution by using Maclaurin's series as follows,

Irodov Problem 1.293

Each point in the ring experiences a centrifugal force acting radially outwards . If we cut the ring into two halves as shown in the figure, this centrifugal force threatens to sever the ring into two halves. The stress in the ring (red arrows) must support the net force centrifugal force acting on each of the halves of the ring in order for the ring to remain intact.

Consider an infinitesimally small piece of the ring that subtends an angle . The length of this infinitesimally small piece will be . If the area of cross-section of the ring is a, and the density of the material is p, then the mass of this infinitesimally small piece will be . As the ring rotates, the infinitesimally small piece of the ring will experience a centrifugal force that acts radially outwards given by,



Only the component contributes to the net centrifugal force on the half (since the other component will cancel itself out). The net centrifugal force is the integral of over the entire semi-ring and this is equal to,






This force is to be balanced by the force due to the stress in the string acting over the two cross-sections each of area a and is given by . Hence we have,











Here, is the tensile strength (the maximum stress that the material can generate before falling apart) of the material.