Question

A wooden plank immerses up to \[50\% \] in water. Then what percentage of its volume is immersed in a liquid of density $0.5g.c{m^{ - 3}}$ .
(A) \[50\% \]
(B) $25\% $
(C) $75\% $
(D) $100\% $

Answer 1

Hint: In order to answer this question, first we will write the density of water and then the density of liquid (given). And then we will apply the formula of net force in the downward force in terms of weight and buoyancy force. And then we will compare the condition of the existing volume and the new one.

Complete step by step solution:
Given that-
Density of liquid, ${\rho _L} = 0.5g.c{m^{ - 2}} = 500kg{m^{ - 3}}$
Now, as we know that,
Density of water, ${\rho _W} = 1000kg.{m^{ - 3}}$
Density of plank, ${\rho _p} = ?$
Now,
We know that-
$Net\,force\,downward\,force = Weight - buoyancy\,force$
And,
$Net\,force\,downward\,force = Total\,Weight - weight\,of\,water\,displaced$
So, for equilibrium conditions, net downward force is zero.
$ \Rightarrow 0 = {\rho _p}{V_g} - {\rho _w}(0.5V)g$
And, ${\rho _p} = 0.5{\rho _W} = 0.5 \times 1000 = 500kg.{m^{ - 3}}$
So, $Net\,force\,downward\,force = Weight - buoyancy\,force$
$
  \because 0 = {\rho _p}{V_g} - {\rho _L}{V_1}g \\
   \Rightarrow 0 = 500 \times {V_g} - 500 \times {V_1}g \\
   \Rightarrow {V_1} = V \\
 $
Therefore, 100% volume immersed in liquid.
Hence, the correct option is (D) $100\% $ .

Note: A measure of a substance's compactness. The amount of mass per unit of volume is known as density. In general, when pressure rises and temperature falls, density rises.
Density is the mass of a material substance per unit volume. Density is calculated using the formula \[d = \dfrac{M}{V}\] , where d represents density, M is mass, and V is volume. The density of a substance is usually measured in grammes per cubic centimetre.