Question

A piece of steel has a weight $W$in air, ${W_1}$when completely immersed in water and ${W_2}$ when completely immersed in an unknown liquid. The relative density (specific density) of liquid is
A. $\dfrac{{W - {W_1}}}{{W - {W_2}}}$
B. $\dfrac{{W - {W_2}}}{{W - {W_1}}}$
C. $\dfrac{{{W_1} - {W_2}}}{{W - {W_1}}}$
D. $\dfrac{{{W_1} - {W_2}}}{{W - {W_2}}}$

Answer 1

Hint:In the question, we had weight of body in water and weight of body in other liquid and we have to find the relative density. For that we can use Archimedes principle which states that the buoyant force is equal to the displaced fluid where buoyant force is upward force exerted by the fluid when any weight is immersed in any liquid.

Complete step by step answer:
Buoyancy is an upward force exerted by the fluid that opposes the weight of partially or fully immersed fluid. According to the Archimedes principle,
Buoyant force = weight of displaced fluid
Now, going back to the question,
Consider the weight of the body in any water be given by ${W_1}$ , weight in air be given by $W$ and buoyant force by ${F_1}$. Using the Archimedes principle ${W_1} = W - {F_1}$
Now, writing the formula for buoyant force ${F_1} = V{\rho _W}g$ where $V$ is the volume of displaced liquid, ${\rho _W}$ is the density of the water and $g$ be the gravitational acceleration.
Putting value in above equation,
${W_1} = W - V{\rho _W}g$
$\Rightarrow Vg{\rho _W} = W - {W_1}$ $ \ldots \left( 1 \right)$
And weight of body in any liquid be given by ${W_2}$, again ${W_2} = W - {F_2}$
Again, buoyant force of any liquid be given by ${F_2} = V{\rho _l}g$ where ${\rho _l}$ is the density of the liquid.
Putting the value
${W_2} = W - V{\rho _l}g$
$\Rightarrow V{\rho _l}g = W - {W_2}$ $ \ldots \left( 2 \right)$
Dividing the above two marked equations
$\therefore\dfrac{{{\rho _W}}}{{{\rho _l}}} = \dfrac{{W - {W_1}}}{{W - {W_2}}}$
Which is our required relative density.
So, the correct option is A.

Note:The center of buoyancy of an object is the center of gravity of the displaced volume of the fluid. There are three types of buoyancy that are positive buoyancy, negative buoyancy and neutral buoyancy.