Saturday, April 4, 2009

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.

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