Find the fundamental natural frequency of vibration of a uniform beam of length l that is fixed...

Find the fundamental natural frequency of vibration of a uniform beam of length l that is fixed at one end and simply supported at the other end. Assume the deflection shape of the beam to be same as the static deflection curve under its self weight. Hint: The static deflection of a uniform beam under self weight is governed by

Where is the density, g is the acceleration due to gravity, and A is the area of cross section of the beam. This equation can be integrated for any known boundary conditions of the beam.