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A test tube of mass ‘ \[m\] ’ sinks up to mark ‘ \[X\] ’ in water and the same test tube sinks below the mark ‘ \[X\] ’ in a given liquid. Then the density of water is:
A. Less than the density of the given liquid.
B. More than the density of the given liquid.
C. Equal to the density of the given liquid.
D. Product of depths of immersion in water and given liquid.

Last updated date: 23rd Jul 2024
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Hint: First of all, we will take into account the fluid displaced by the test tube when it is dipped in it. The weight lost by the test tube is equal to the fluid displaced.

Complete step by step answer:In the given question, we are supplied the following data:
The mass of the test tube is provided as ‘ \[m\] ’.
When it is dipped in water, it dips up to a certain mark indicated as ‘ \[X\] ’.
However, when the same test tube is dipped in a different liquid, it is below the mark indicated as ‘ \[X\] ’.
We are asked to compare the densities of the two different liquids.

To begin with,
We bring the Archimedes principle, which notes that the upward buoyant force applied on a body immersed in a fluid is proportional to the weight of the fluid that the body displaces, whether completely or partly submerged. The theory of Archimedes is a law of physics which is essential to the dynamics of fluids.
Here in the case of the test tube, as you all know, the weight of the test tube acts vertically downwards while at the same time the water provides an upward thrust to the test tube to balance in its position. This upward thrust is called buoyant force.
But, in the second liquid, the test tube dips below the mark, which clearly indicates that the test tube dips more in the second liquid. It means, to balance the test tube it has to provide more upward thrust than the previous case. In order to do so, more volume of liquid has to be displaced.
We can now conclude that, higher is the density of the fluid, less is the fluid displaced or vice-versa.
The density of the water is more than that of the liquid, so it displaces less water.
The correct option is B.

Note:To solve this problem, we must have the theoretical knowledge of the Archimedes principle. Many students seem to make mistakes by taking into account that, more the fluid displaced, higher is the density, which is wrong. A ship dips less in marine water than in freshwater, as the density of marine water is more than that of fresh water.