Question

Two solids a and b float on water. It is observed that a float with half its volume immersed and b floats with two thirds of its volume immersed. Find the ration between the densities of a and b.
$\begin{align}
  & a)4:3 \\
 & b)2:3 \\
 & c)3:4 \\
 & d)1:1 \\
\end{align}$

Answer 1

Hint: First take block a. We know when there’s no external force acting, the weight of the block is equal to the force of buoyancy acting on the block. With this condition, find the density of the block in terms of volume immersed now. Similarly, find the density of block b in terms of its volume immersed in water.

Formula used:
${{\rho }_{b}}{{V}_{b}}g={{\rho }_{w}}{{V}_{immersed}}g$

Complete step-by-step answer:
The buoyancy force is an upward force acting on the body when it is in any liquid. As we know, when there’s no net force acting on the body regarding the system, the net forces are equal. The net forces acting on the block a will be the weight due to the gravitation and the buoyancy force.
We can therefore write it as,
$\begin{align}
  & {{\rho }_{b}}{{V}_{b}}g={{\rho }_{w}}{{V}_{immersed}}g \\
 & \Rightarrow {{\rho }_{1}}Vg={{\rho }_{w}}\dfrac{V}{2}g \\
 & \Rightarrow {{\rho }_{1}}=\dfrac{{{\rho }_{w}}}{2}.......(1) \\
\end{align}$
Similarly, if we add the net forces to zero for the block b, we can write,
$\begin{align}
  & \Rightarrow {{\rho }_{b}}{{V}_{b}}g={{\rho }_{w}}{{V}_{immersed}}g \\
 & \Rightarrow {{\rho }_{2}}Vg={{\rho }_{w}}\dfrac{2V}{3}g \\
 & \Rightarrow {{\rho }_{2}}=\dfrac{2{{\rho }_{w}}}{3}........(2) \\
\end{align}$
Now, as we got the relations between the densities and the volume immersed by the blocks in water,
Divide both the densities to get the relation between them,
Dividing equation (1) and (2).,
$\begin{align}
  & \Rightarrow \dfrac{{{\rho }_{1}}}{{{\rho }_{2}}}=\dfrac{{{\rho }_{w}}}{2}\times \dfrac{3}{2{{\rho }_{w}}} \\
 & \Rightarrow \dfrac{{{\rho }_{1}}}{{{\rho }_{2}}}=\dfrac{3}{4} \\
\end{align}$

So, the correct answer is “Option c”.

Additional Information: Buoyancy is the force exerted on the object when the object is partially or fully immersed in a liquid. Buoyancy force is a vector and it states the magnitude and direction of the buoyancy force in the liquid. This buoyancy force is caused by the pressure exerted by fluid in which an object is immersed or partially immersed. It is always pointed upwards because the pressure of fluid increases with depth.

Note: While calculating the buoyancy force, the buoyancy force does not depend on the density of the block. It only depends on the density of the liquid, volume of the block that is immersed in the liquid. Therefore, in the same liquid, no matter what the object density is, the buoyancy force is the same if the volume immersed is the same.