Question

The base area of a boat is $2{{m}^{2}}$. A man weighing 76kg steps into the boat. Calculate the depth up to which the boat sinks further into water.

Answer 1

When the man steps on the boat, the boat sinks into the water by displacing the water below it. According to Archimedes principle, the weight of the man should be equal to the weight of water displaced. And the weight of water displaced can be calculated using the buoyant force of water over the given surface area of the boat. We shall proceed with this analogy in our problem.

Complete step-by-step answer:
Let us first assign some terms that we are going to use later in our solution.
Let the base area of boat be given by ‘A’, where A is equal to:
$\Rightarrow A=2{{m}^{2}}$
Also, let the depth that the boat further sinks be given by ‘h’.
And, the density of water be ‘d’, where ‘d’ is equal to:
$\Rightarrow d=1000kg{{m}^{-3}}$
Now, the weight of man acting downwards on the boat will be balanced by the buoyant force acting on the boat in upward direction. Mathematically, this could be written as:
$\Rightarrow 76\times g=d\times (Ah)\times g$
Putting the values of all the respective terms and solving for depth, we get:
$\begin{aligned} & \Rightarrow 76\times g=1000\times (2h)\times g \\\ & \Rightarrow h=\dfrac{38}{1000}m \\\ & \Rightarrow h=\dfrac{38}{1000}\times 100cm \\\ & \therefore h=3.8cm \\\ \end{aligned}$
Hence, the depth up to which the boat sinks further into water after the man steps on it is equal to 3.8cm .

Note: In these types of problem, when there is very less data available in our equation, we shall use the standard values of these data-sets, like we did for the density of water. Also, we should always make sure the units of all the terms are appropriate and correct, so that they can be used in the calculations.