a metallic rod of length l is tied to a sting of length 2l and made to rotate with angular velocity w on a horizontal table with one end of the sting fixed.if there is a vertical magnetic field Bin the region, the induced emf across the ends of the rods is

The metallic rod is tied to one end of a rod and the other end is left free.  The other end of the string is fixed at the center of a horizontal table.  The rod rotates on the horizontal table about an axis perpendicular to it and passing through its center.  The angular speed = ω.    Assume that there is a metallic frame on which the ends of the rod are travelling.  The ends of the rod are always on the metallic points.  So a circular loop is established.  

   Magnetic field  B is perpendicular to the metallic rod.  Area covered by the rod and the loop:  A.   To find A, we take the radius of the revolution of the center of the rod, which is 2L + L/2 = 2.5 L.    Then the angle of revolution theta = ω * time t.    The r * theta gives the length through which the rod moves.  Its length is L.  

         A = [ 2.5 L * (ω t) ] L = 2.5 L² ω t
 EMF induced = -dΦ / dt  = - d/dt [Φ] = - d/dt [B A] 
           = - d/dt [ B 2.5 L² ω t ] 
           = - 2.5 B L² ω