Question

A cubical block of wood weighing 200 g has a lead piece fastened underneath. Find the mass of the lead piece which will just allow it to float in water. The specific gravity of wood is 0.8 and that of wood is 11.3.

Answer 1

Hint In this solution, we will use the property of the buoyancy of different objects. When an object is immersed in water, a buoyant force will act on it upwards which will balance the gravitational acceleration downwards.

Complete step by step answer
We’ve been given that the block of wood has a lead piece fastened to it underneath such that the combination just floats in water. Since it is just floating in the water, we can say that the complete volume of the block is immersed inside the water completely such that the combination still floats. The volume of the lead block can be calculated using the relation,
$\Rightarrow {V_{lead}} = \dfrac{m}{{{\text{Specific gravity of lead}}}}$
$\Rightarrow {V_{lead}} = \dfrac{m}{{11.3}}$
Similarly, the volume of the wooden block is,
$\Rightarrow {V_{water}} = \dfrac{{200}}{{0.8}}$
$\Rightarrow {V_{water}} = 250$
The specific volume of water displaced can be calculated as the,
$\Rightarrow {V_{total}} = \dfrac{{{\text{mass of water displaced}}}}{{{\text{specific gravity of water}}}}$
$\Rightarrow {V_{total}} = \dfrac{{200 + m}}{1}$
Now, Archimedes's principle tells us that the amount of water displaced when an object is immersed in the water is equal to the volume of the object which is immersed in it. So we can equalize the specific volume of the water displaced with the sum of the specific volume of the wood and the lead block.
$\Rightarrow 200 + m = \dfrac{m}{{11.3}} + 250$
Taking the terms containing $m$ on one side, we get
$\Rightarrow \dfrac{{10.3}}{{11.3}}m = 50$
Solving for the mass of the lead block, we get
$\Rightarrow m = 54.89\,g$.

Note
Since the lead block is attached below the wooden block, when the wooden block is immersed in the water, so will the lead block so we must also take the volume of water displaced by the lead block into consideration. Since lead is much heavier than wood, we can see that we only need about 54 grams of it to submerge a wooden block of mass 200 grams.