Question

A boat having a length 3m and breadth 2m is floating on a lake. The boat sinks by 1 cm when a man gets on it. The mass of man is
(a) 12 kg
(b) 60 kg
(c) 72 kg
(d) 96 kg

Answer 1

Hint: When the man gets into the boat, the boat sinks by 1 cm. So, a certain volume of the boat is inside the water when the man gets into the boat. This means the boat will now experience an upward force from the water and this force is called buoyant force. Since the boat is in equilibrium, balance all the forces in the vertical direction.

Complete step-by-step answer:
Before proceeding with the question, we must know all the formulas that will be required to solve this question.
Whenever an object has some of his volume inside the water, this object will experience an upward force called buoyant force which is given by the formula,
${{F}_{b}}={{\rho }_{w}}vg..........\left( 1 \right)$
Here, ${{\rho }_{w}}$ is the density of the water, g is the acceleration due to gravity and v is the volume of the object that is inside the water.
It is given in the question that the boat having the length of 3 m and the breadth of 2 m sinks by 1cm i.e. 0.01 m when the man gets on it. So, the volume inside the water is,
$\begin{align}
  & v=3\times 2\times 0.01 \\
 & \Rightarrow v=0.06{{m}^{3}} \\
\end{align}$
Also, the density of water ${{\rho }_{w}}=1000kg/{{m}^{3}}$. Substituting the volume and density in formula $\left( 1 \right)$, the buoyant force experience by the boat is equal to,
$\begin{align}
  & {{F}_{b}}=\left( 1000 \right)\left( 0.06 \right)g \\
 & \Rightarrow {{F}_{b}}=60g...........\left( 2 \right) \\
\end{align}$
Let us assume the mass of man is m kg.
Since man is inside the boat, the boat will experience a downward force. This downward force is equal to,
$F=mg...........\left( 3 \right)$
Since the boat is in equilibrium, we can equate the upward force in equation $\left( 2 \right)$ and the downward force in equation $\left( 3 \right)$. So, we get,
$\begin{align}
  & 60g=mg \\
 & \Rightarrow m=60 \\
\end{align}$
Hence, the answer is option (b).

Note: There is a possibility that one may commit a mistake while finding the volume of the boat that is inside the water. One may not convert the depth of the boat from cm to m which may lead to an incorrect answer.