Question

An object of mass m is floating in a liquid of density $\sigma$. If the object is made up of density $\rho$, then apparent weight of the object in the liquid is

$$A)\,mg\\\ B)\,mg\left( {1 - \dfrac{\sigma }{\rho }} \right)\\\ C)\,mg\left( {1 - \dfrac{\rho }{\sigma }} \right)\\\ D)\,Zero$$

Answer 1

When we are dealing with such a problem, we have to apply the concept of Archimedes principle. We have to think of the condition of the apparent weight. This is a conceptual question and we have to use the theorems and conditions to take out the answer.

Complete step by step answer:
If we think of the Archimedes principle, Archimedes principle states that when a body is submerged in a liquid, a buoyant force acts on it in the upward direction which is equal to the weight of the liquid displaced. The Archimedes principle is applied for both the floating as well as submerged bodies.
For the above question we have to apply the condition of floatation.
In the condition of Archimedes Law of floatation,
We have to balance the weight of the body submerged to the buoyant force acting on the body.
The weight of the body submerged is given by $mg$, where m is the mass and g is the acceleration due to gravity.
The buoyant force acting on the body is given by,
${F_b} = V\sigma g$
Now we have to calculate the apparent weight of the body, the apparent weight of a body submerged in a liquid is given by: -
Apparent weight of the body = Actual weight of body – buoyant force
now we will apply the above calculated formulas in the equation of apparent weight,
Apparent weight = $mg\, - \,V\rho g$
Apparent weight = $mg - \dfrac{m}{\sigma }\rho g$
Apparent weight = $mg\left( {1 - \dfrac{\rho }{\sigma }} \right)$
Hence, we come to know that the correct option is option (C).

Note: While considering the solution of the above question, we have to read the question very carefully if in the question the condition of floating and completely submerged was mentioned then the answer would be option (D) because the densities of the body and the fluid becomes equal. Therefore, we have to consider all the conditions very carefully.