In this paper we look for nonuniform rotating beams that are isospectral to a given uniform nonrotating beam. A rotating nonuniform beam is isospectral to the given uniform nonrotating beam if both the beams have the same spectral properties, i.e., both the beams have the same set of natural frequencies under a given boundary condition. The Barcilon-Gottlieb type transformation is proposed that converts the governing equation of a rotating beam to that of a uniform nonrotating beam. We show that there exist rotating beams isospectral to a given uniform nonrotating beam under some special conditions. The boundary conditions we consider are clamped-free and hinged-free with an elastic hinge spring. An upper bound on the rotation speed for which isospectral beams exist is proposed. The mass and stiffness distributions for these nonuniform rotating beams which are isospectral to the given uniform nonrotating beam are obtained. We use these mass and stiffness distributions in a finite element analysis to show that the obtained beams are isospectral to the given uniform nonrotating beam. A numerical example of a beam having a rectangular cross section is presented to show the application of our analysis.