Question

A metallic sphere weighs 210g in air, 180g in water, and 120g in an unknown liquid. Find the density of metal and liquid.

Answer 1

Hint: To solve this type of question we use the following formula of relative density. Relative density is directly proportional to the weight of an object in air and inversely proportional to the change in weight of an object in the water.

Formula used:
${\text{Relative density = }}\dfrac{{{\text{Weight of the object in air}}}}{{{\text{Change in weight of object in water}}}}$

Complete answer:
Let us first write the information given in the question.
We are provided with, Weight of metallic sphere in air = $210g$, weight in water = $180g$, and an unknown liquid = $120g$.
Let us use the relative density formula to find the relative density of the metallic sphere,
We know that the density of the metallic sphere is given as,
${\text{Density of metallic sphere = }}\dfrac{{{\text{Weight in air}}}}{{{\text{Weight in water}}}}$
Let us substitute the value in the equation of density of metallic sphere, we get,
Thus, Density of metallic sphere = $\dfrac{{210}}{{210 - 180}} = \dfrac{{21}}{3} = 7 g/cm^3$
 Hence, the density of the metallic sphere is equal to 7g/cm3
Now know that the change in weight in water is equal to the upthrust in liquid.
Upthrust is the upward force exerted by the liquid on the body floating in it. We know that,
$upward thrust \propto density$
Therefore we can write the following relation, which is as follows,
$\Rightarrow \dfrac{{\Delta {W_{liquid}}}}{{\Delta {W_{air}}}} = \dfrac{{{\rho _{liquid}}}}{{{\rho _{air}}}}$
Let us substitute the values and find the density of the liquid.
Rearrange the formula for the density of the liquid, as follows
$\Rightarrow {\rho _{liquid}} = {\rho _{air}} \times \dfrac{{\Delta {W_{liquid}}}}{{\Delta {W_{air}}}}$
$\Rightarrow {\rho _{liquid}} = 1 \times \dfrac{{210 - 120}}{{210 - 180}} = \dfrac{9}{3} = 3g/{m^3}$

$\therefore$ The required values of density of the metallic sphere and the density of liquid are $7g/m^3$ and $3g/m^3$.

Note:
i) Archimedes’ Principle states that the upthrust acting on an object floating in the fluid is equal to the weight of the liquid displaced by its weight. So the volume of the object multiplied by the density of the fluid.
ii) Density is the mass per unit volume. Its unit is $g/m^3$.
Relative density is the ratio of the density (mass of a unit volume) of a substance to the density of given reference material.
If relative densities ;
Case 1) when relative density <1, then the substance is less dense than the reference
Case 2) when relative density >1, then the substance is denser than the reference
Case 3) when relative density =1,then substance densities are equal.