Question

(III)A \(3.65\;{\rm{kg}}\) block of wood\(\left( {SG = 0.50} \right)\)floats on water. What minimum mass of lead, hung from the wood by a string, will cause the block to sink?

Short answer

The minimum mass of lead, hung from the wood by a string, causing it to sink is \(4\;{\rm{kg}}\).

Step-by-step solution

Concept of Buoyant force

In order to determine the minimum mass of lead, use the relation of buoyant forces and density of fluid. Also, use the law of equilibrium of forces.

Given data

The mass of the block of wood is \({m_{{\rm{wood}}}} = 3.65\;{\rm{kg}}\).

The specific gravity of the block of wood is \(S{G_{{\rm{wood}}}} = 0.50\).

The specific gravity of the lead is \(S{G_{{\rm{Pb}}}} = 11.3\).

Relation of Buoyant force and density of fluid

The Buoyant force is calculated as:

\({F_{{Bouyant}}}{ = }\rho Vg\)

Here,\(\rho \)is density of fluid, \(V\) is volume occupied and \({\rm{g}}\) is acceleration due to gravity.

The density of a fluid is calculated as:

\(\rho = \frac{m}{V}\)

Here, \(m\) is the mass of the fluid.

Calculation of minimum mass of lead

Using law of equilibrium of forces,

\(\begin{aligned}{F_{\rm{W}}} &= {F_{\rm{B}}}\\\left( {{m_{{\rm{wood}}}} + {m_{{\rm{Pb}}}}} \right)g &= {\rho _{{\rm{water}}}}\left( {{V_{{\rm{wood}}}} + {V_{{\rm{Pb}}}}} \right)g\\\left( {{m_{{\rm{wood}}}} + {m_{{\rm{Pb}}}}} \right) &= {\rho _{{\rm{water}}}}\left( {\frac{{{m_{{\rm{wood}}}}}}{{{\rho _{{\rm{water}}}}}} + \frac{{{m_{{\rm{Pb}}}}}}{{{\rho _{{\rm{Pb}}}}}}} \right)\\{m_{{\rm{Pb}}}}\left( {1 - \frac{{{\rho _{{\rm{water}}}}}}{{{\rho _{{\rm{Pb}}}}}}} \right) &= {m_{{\rm{wood}}}}\left( {\frac{{{\rho _{{\rm{water}}}}}}{{{\rho _{{\rm{wood}}}}}} - 1} \right)\end{aligned}\)

Here,\({m_{{\rm{wood}}}}\)is the mass of wood,\({m_{{\rm{Pb}}}}\)is the mass of lead,\({\rho _{{\rm{wood}}}}\)is the density of wood,\({V_{{\rm{wood}}}}\)is the volume of wood,\({V_{{\rm{Pb}}}}\)is the volume of lead and\({\rho _{{\rm{Pb}}}}\)is the density of lead.

On rearranging the above relation.

\(\begin{aligned}{m_{{\rm{Pb}}}} &= {m_{{\rm{wood}}}}\frac{{\left( {\frac{{{\rho _{{\rm{water}}}}}}{{{\rho _{{\rm{wood}}}}}} - 1} \right)}}{{\left( {1 - \frac{{{\rho _{{\rm{water}}}}}}{{{\rho _{{\rm{Pb}}}}}}} \right)}}\\ &= {m_{{\rm{wood}}}}\frac{{\left( {\frac{1}{{S{G_{{\rm{wood}}}}}} - 1} \right)}}{{\left( {1 - \frac{1}{{S{G_{{\rm{Pb}}}}}}} \right)}}\end{aligned}\)

Here,\(S{G_{{\rm{wood}}}}\)is the specific gravity of wood and\(S{G_{{\rm{Pb}}}}\)is the specific gravity of lead.

Substitute the values in the above equation to find the mass of Lead.

\(\begin{aligned}{m_{{\rm{Pb}}}} &= 3.65\;{\rm{kg}}\frac{{\left( {\frac{1}{{0.50}} - 1} \right)}}{{\left( {1 - \frac{1}{{11.3}}} \right)}}\\ &= 4.00\;{\rm{kg}}\end{aligned}\)

Hence, the mass of lead is \(4\;{\rm{kg}}\).