What volume of lead must be fastened underneath it to sink the wood in calm water so that its top is just even with the water level?

The bouyancy force is:  

Since the wood-lead system is completely submerged, the bouyancy force 

 is FB = ĎwgVl + ĎwgVb, where Ďw is the density of water,Vl 

 is the volume of 

 the piece of lead and Vb is the volume of the wooden block. The weight of the

  combined lead and wooden block is: W = ĎlgVl + ĎbgVb. Since the system is 

 in equilibrium, the bouyancy force must be equal to the total weight: 

 ĎwgVl + ĎwgVb = ĎlgVl + ĎbgVb 

 now we can solve for the volume of lead: 

 ĎwgVl â’ ĎlgVl = ĎbgVb â’ ĎwgVb 

 Vl(Ďw â’ Ďl) = Vb(Ďb â’ Ďw) 

 Vl =   

 Ďbâ’Ďw 

 Ďwâ’Ďl   

 Vb 

 Now we substitute the values for the density of lead Ďl = 11.3 Ă— 103kg/m3  , 

 the density of the wood and the density of water Ďw = 1000kg/m3 

 . We get: 

 Vl =   

 600â’1000 

 1000â’11300 

 (0.6m Ă— 0.25m Ă— 0.08m) = 4.66 Ă— 10â’4m3