Alight string can support a station- ary hanging load of 25.0 kg before breaking. an object ofmassm53.00kg attached to the string rotates on a friction- less, horizontal table in a circle of radius r50.800m, and the other end of the string is held fixed as in figure p6.1. what range of speeds can the object have before the string breaks? what is the free body diagram

25.0kg would give a tension force in the string of about 250N (taking g=10, from 9.81)For a mass of 53kg rotating horizontally in a circle of radius 50.8, the string has to be able to supply a centripetal force to keep the mass moving in circular motion.Centripetal force F = m v^2/r. If the string breaks at about 250N, then it will break  if250=53 v^2/50.8 m/ssqrt(250x50.8/53) = vSince there don't seem to be any "easy numbers" in the q, I guess it wants the 9.81 version of g.

Where are the options ?

Look it up you feel me