If the rod were translating as well as rotating, what will be its emf? Assume that the centre of mass has a velocity v and the rod is rotating with an angular velocity omega about its centre of mass.

When the rod is translation motion  than the value of emf produced  is e₁ = BvL  .........1>
(where B is the magnetic field , v = velocity of centre of mass and L is the length of conductor)
now  when the body is rotated about a fixed point (no translation ) than the value of emf induced is 
e₂ = BwL²/2     (here w = angular speed of rotation, B = magnetic field, L = radius of rotation)   .......2>
now the body has both motion means it has both translation as well as rotation motion than the value of emf is sum of both above emf
e =  e₁+e₂
  =  ​BvL +​ BwL²/2
  =B (vL +​wL²/2) volt