Question

The total weight of a piece of wood is $6\text{ kg}$, in the floating state in water, its $\dfrac{1}{3}$ part remains inside water. On this floating solid, what maximum weight is to be put such that the whole of the piece of wood is drowned in the water?
A.$12\text{ kg}$
B.$10\text{ kg}$
C.$14\text{ kg}$
D.$15\text{ kg}$

Answer 1

Hint: When the solid is in floating state then its weight that is acting in the downwards direction is balanced by the buoyant force acting on it in the upwards direction. The amount of buoyant force that acts on the solid determines how much of the volume of solid will float above water and how much of it will remain submerged.

Complete answer:
To solve this question, let us first balance the weight and buoyant force acting on the solid that makes its $\dfrac{1}{3}$ part remains inside water. The buoyant force acting on the solid is equal to the volume of water displaced by the solid. Hence,
$mg=\dfrac{V}{3}\times {{\rho }_{w}}\times g$
Here, the total weight of a piece of wood is $mg=6\text{ kg}$,
$V$ is the total volume of the solid,
${{\rho }_{w}}$ is the density of water,
$g$ is the acceleration due to gravity.
The equation turns out be as follows:
$\begin{align}
  & m=\dfrac{V}{3}\times {{\rho }_{w}} \\
 & \Rightarrow 3m=V\times {{\rho }_{w}} \\
\end{align}$
Now, let us assume that the maximum weight to be put on top of the piece of wood such that the whole of the piece is drowned in the water be \[m'g\]. Then
$\begin{align}
  & m'g+mg=V\times {{\rho }_{w}}\times g \\
 & \Rightarrow m'+m=V\times {{\rho }_{w}} \\
 & \Rightarrow m'+m=3m \\
 & \Rightarrow m'=3m-m \\
 & \Rightarrow m'=2m \\
 & \Rightarrow m'=2\times 6 \\
 & \therefore m'=12\text{ kg} \\
\end{align}$

Thus, the correct option is $A$.

Note:
Buoyant force is also known as upthrust and it is an upward force that a body experiences on being partly or fully submerged in a liquid. If we considered some other liquid other than water in this case then we will have to consider the density of that liquid. The upthrust makes a body appear lighter than its weight when it is placed in a liquid.