The frequency of small oscillations of thin uniform vertical rod of mass 'm' and length 'l' hinged at point O with the help of two springs with spring constants k₁ and k₂ , is

Step-by-step solution

Let the rod be displaced by a small angle from the vertical. The horizontal displacement of the lower end of the rod is x = l . The restoring force exerted by the two springs is F = (k₁ + k₂)x = (k₁ + k₂)l . The restoring torque due to the springs about the hinge O is: _s = -F l = -(k₁ + k₂)l^2 The center of mass of the rod is at a distance l 2 from the hinge O. The restoring torque due to gravity about O is: _g = -mg ( l 2 ) - mgl 2 The total restoring torque on the rod is: = _s + _g = - [ (k₁ + k₂)l^2 + mgl 2 ] T