Buoyancy problem: find the mass of lead needed to sink a wood block (Archimedes' Principle)

We apply Archimedes' Principle to calculate the mass of lead needed to sink a wood block, specifically the mass required for the wood and lead to stay submerged in perfect equilibrium.

To start things out, we make complete force diagrams for the wood block and the unknown lead mass. The block has buoyant force pointing up and weight and tension pulling down. The lead mass has buoyant force and tension pointing up, and weight pointing down. Note that it is a very common mistake to ignore the buoyant force on the lead -- it is displacing some water, so by Archimedes' Principle it must feel a buoyant force pointing upward.

Applying Newton's Second Law to each object, we arrive at a system of two equations and two unknowns, where the unknowns are the mass of lead required to sink the block and the tension in the string that ties the lead to the block. We eliminate T from this system and set up an equation in which the mass of the lead is the only unknown.

We give a quick note on the manipulation of the definition of density here: density is mass per unit volume, so we can compute mass as density times volume or volume as mass divided by density. We apply Archimedes' Principle to write the buoyant force on each object in terms of the volume of each object (since that is the volume of displaced water as well). Then we write each volume as mass divided by density and arrive at a single equation in which the mass of the lead is the only unknown. We solve for the mass of the lead weight symbolically, then plug numbers in to obtain the answer.

Finally, we go back to our original equation describing the balanced forces on the wood block and we solve for the tension.