End B of a thin rod of length ‘l’ is hinged so that it can swing in a vertical plane. ‘S’ is a stopper at a distance ‘x’ below the hinge B. Initially the rod is held in horizontal position by an external agent as shown in figure. After the rod is released it swings in a vertical plane and when it becomes vertical it strikes the stopper S and stops. If during the course of impact the horizontal component of reaction at the hinge B is zero, find x.

 

detailed solution

Correct option is B

During the course of impact moment of all external forces about point S is zero and hence angular momentum of the rod about S is conserved.Just after the impact angular momentum of the rod about S, Lf = 0.Linear momentum of the element of length dr is given by, dP=dm × V= ml  dr   ωr = mωl   rdrTherefore angular momentum of the rod about S before impact is given byLi=  ∫0x  x−r  dP           − ∫xl  r−x  dP=  x  ∫0x  mωl  rdr     −∫0x  mωl  r2dr   −∫xl  mωl  r2dr     +   x   ∫xl  mωl  rdr  =  mωl    x32  −  x33  −  l33  +   x33  +  xl22  −   x32=  mωl    xl22  −  l33Now   Li= Lf  ⇒   mωl   xl22  −  l33 =  0  ⇒   x  =  2l3