(II) A \({\rm{0}}{\rm{.48 kg}}\)piece of wood floats in water but is found to sink in alcohol \(\left( {SG = 0.79} \right)\), in which it has an apparent mass of\({\rm{0}}{\rm{.047 kg}}\). What is the SG of the wood

Specific gravity of a fluid is measured as the ratio ofis density of fluid and density of water.

The mass of wood is \({m_{\rm{w}}} = 0.{\rm{48 kg}}\).

The mass of alcohol is \({m_{\rm{a}}} = 0.{\rm{047 kg}}\).

The specific gravity of alcohol is \({\left( {SG} \right)_{\rm{a}}} = 0.79\).

Since the piece of wood floats in water but sinks in alcohol, then there will be a loss in the mass of the piece of wood.The loss in the mass of wood, when it sinks in alcohol is:

\(\begin{aligned}\Delta m &= {m_{\rm{w}}} - {m_{\rm{a}}}\\ &= 0.48\;{\rm{kg}} - 0.047\;{\rm{kg}}\\ &= 0.433\;{\rm{kg}}\end{aligned}\)

Using specific gravity formula, the density of alcohol is calculated as:

\(\begin{aligned}{\rho _{\rm{a}}} &= {\left( {SG} \right)_{\rm{a}}}{\rho _{{\rm{wa}}}}\\ &= 0.79 \times 1000\;{\rm{kg/}}{{\rm{m}}^3}\\ &= 790\;{\rm{kg/}}{{\rm{m}}^3}\end{aligned}\)

Here, \({\rho _{{\rm{wa}}}}\) and \({\rho _{\rm{a}}}\) are the density of water and alcohol.

The relation to calculate the volume of alcohol displaced is:

\(\begin{aligned}V &= \frac{{{m_{\rm{a}}}}}{{{\rho _{\rm{a}}}}}\\ &= \frac{{0.433\;{\rm{kg}}}}{{790\;{\rm{kg/}}{{\rm{m}}^3}}}\\ &= 5.48 \times {10^{ - 4}}\;{{\rm{m}}^3}\end{aligned}\)

Here, volume of alcohol displaced is equivalent to the volume of the wood

The density of wood is calculated as:

\(\begin{aligned}{\rho _{{\rm{wo}}}} &= \frac{{{m_{\rm{w}}}}}{V}\\ &= \frac{{0.48\;{\rm{kg}}}}{{5.48 \times {{10}^{ - 4}}\;{\rm{kg/}}{{\rm{m}}^3}}}\\ &= 875.9\;{\rm{kg}}/{{\rm{m}}^3}\end{aligned}\)

The specific gravity of wood is calculated by using the relation:

\(\begin{aligned}S{G_{{\rm{wo}}}} &= \frac{{{\rho _{{\rm{wo}}}}}}{{{\rho _{{\rm{wa}}}}}}\\ &= \frac{{875.9\;{\rm{kg/}}{{\rm{m}}^3}}}{{1000\;{\rm{kg/}}{{\rm{m}}^3}}}\\ &= 0.876\end{aligned}\)

Hence, the specific gravity of wood is \(0.876\).