Question

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

Answer 1

Hint: We know that the specific gravity of an object is equal to the ratio of density of an object to the density of water. Taking density of water as unity, this formula reduces to mass per unit volume of the object. The volume of the water displaced is equal to the volume of the block plus the volume of the lead piece.

Complete answer:
We are given a cubical block of wood whose mass is given as 200 g. A lead piece fastened underneath it and let its mass be m.
Now when the block is put into water then the mass of water displaced by it is equal to the sum of the mass of wood and that of lead.
\[\therefore \]mass of displaced water $ = m + 200$
The specific gravity of an object is defined as the mass of the object divided by its volume. We are given that the specific gravity of wood is 0.8 and that of lead is 11.3.
For the wooden block and the lead piece, we can write
Volume of wooden block $ = \dfrac{{{\text{mass of the wooden block}}}}{{{\text{specific gravity of wood}}}} = \dfrac{{200}}{{0.8}} = 250{m^3}$
Volume of lead $ = \dfrac{{{\text{mass of lead piece}}}}{{{\text{specific gravity of lead}}}} = \dfrac{m}{{11.3}}$
We can write the volume of water in the following way as well where the specific gravity of water is 1.
Volume of water displaced by block$ = \dfrac{{{\text{mass of displaced water}}}}{{{\text{specific gravity of water}}}} = \dfrac{{m + 200}}{1}$
So, the total volume of water that will be displaced is equal to the volume of the wooden block plus the volume of the lead piece. Therefore, we can write

$\begin{align}
 & m + 200 = \dfrac{m}{{11.3}} + 250 \\
 & \Rightarrow m - \dfrac{m}{{11.3}} = 250 - 200 = 50 \\
 & \Rightarrow m \times \dfrac{{10.3}}{{11.3}} = 50 \\
 & \Rightarrow m = 50 \times \dfrac{{11.3}}{{10.3}} \\
 & \Rightarrow m = 54.85g \\
\end{align}$

This is the required mass of the lead peace which will just allow the block to float in water..

Note:
 1. The specific gravity of an object has no dimensions as it is the ratio of two same quantities.
2. The specific gravity can also be called the relative density because it is defined with reference to a certain material, in this case, water.