Question

A boy has 60kg weight, he wants to swim in a river with the help of a wooden log. If the relative density of wood is 0.6, what is the minimum volume of wooden log? (density of the river water is $1000\dfrac{{kg}}{{{m^3}}}$)
A. $0.66{m^3}$
B. $150{m^3}$
C. $\dfrac{3}{1}{m^3}$
D. $\dfrac{3}{{20}}{m^3}$

Answer 1

When a body at rest in a fluid is acted upon by a force pushing upward which is equal to the weight of the fluid that body displaces. If the body is partially submerged then the volume of the fluid displaced is equal to the volume of the part of the body submerged.
In this question since both the boy and the wooden log is floating on the river so the weight of the water displaced will be equal to the sum of the weight of the boy and the wooden log.

Complete step by step answer:
Mass of the boy, ${w_{boy}} = 60kg$
Relative density of wood is ${\rho _w} = 0.6$
Now as we know the relative mass density of a body is given by the formula$w = \rho \times V$, hence the weight of the wooden log will be

$${w_{\log }} = \rho \times V \\\ = 0.6 \times {10^3} \times V \\\$$

Now by applying the Archimedes principle which says that the total weight of the body submerged is equal to the weight of the water displaced we can write
${w_{boy}} + {w_{\log }} = {w_{water}} - - (i)$
Hence by substituting the values in equation (i) we get

$$\left( {60 + 0.6 \times {{10}^3} \times V} \right) = \left( {1000 \times V} \right) \\\ \Rightarrow 60 + 600 \times V = 1000 \times V \\\ \Rightarrow 1000 \times V - 600 \times V = 60 \\\ \Rightarrow 400 \times V = 60 \\\ \Rightarrow V = \dfrac{3}{{20}}{m^3} \\\$$

Therefore the minimum volume of wooden log will be $V = \dfrac{3}{{20}}{m^3}$

So, the correct answer is “Option D”.

Note:
It is interesting to note here that if the weight of the water displaced is less than the weight of the object then the object will sink and if the weight of the water displaced is equal to the weight of the object then the object will float. The Archimedes principle totally depends on the volume of the water displaced by the object and not on the total quantity of the water.