Question

A fire wood board floats in fresh water with $60\%$ of its volume under water. The density of the wood in is $g/c{{m}^{3}}$
(a) $0.4$:
(b) $0.5$
(c) $0.6$
(d) $< 0.4$

Answer 1

In order to calculate the density of the wood in $g/c{m^3}$. It is known that density is inversely proportional to the volume. To calculate the density of the wood directly, there is a formula i.e.
$\dfrac{{{d_1}}}{{{d_2}}} = \dfrac{{{v_1}}}{{{v_2}}}$ where ${{d}_{1}}$ is the density of wood outside the water, ${d_2}$ is the density of wood inside the water, ${{v}_{1}}$ is the volume of wood outside the water and ${v_2}$ is the volume of wood inside the water.

Complete step by step solution
Now, from the question,
$\dfrac{{{d}_{1}}}{{{d}_{2}}}=\dfrac{{{v}_{1}}}{{{v}_{2}}}$
$=\dfrac{{{d}_{1}}}{{{d}_{2}}}=\dfrac{40}{60}$
$\dfrac{{{d_1}}}{{{d_2}}} = 0.66g/c{m^3}$
As it is the same material i.e. firewood, the density of the fire wood is $0.66g/c{{m}^{3}}$.

Additional Information: Density and Volume are inversely proportional to each other. Mass and volume are not the same. Thus, we can say that, For a fixed mass, when density increases, volume decreases.Density expresses the amount of mass an object holds within its volume.Volume is the space confined within a closed three-dimensional space. It can be a space that a substance holds or shape occupies.Density and volume are scientific concepts pertaining to physical properties and characteristics of matter. Volume refers to the measurement of the amount of three-dimensional space occupied by an object. Unlike mass, volume changes according to the external conditions. Density refers to the mass contained in a substance for a given volume. It determines how densely molecules of a given object are packed into a given volume.

Note: This question can also be solved using different method i.e. We know, Buoyant force = mass of water displaced and $p=~\text{mass}/\text{volume}$. Therefore, $p=v\prime /V\times$density of water which was assumed to be $1g/c{{m}^{3}}$.