Problem 63 - Master Resource Book in JEE Main Physics

Question

A body floats with one-third of its volume consider water and \(3 / 4\) of its volume outside another liquid. The density of other liquid is (a) \(\frac{9}{4} \mathrm{~g} / \mathrm{cc}\) (b) \(\frac{4}{9} \mathrm{~g} / \mathrm{cc}\) (c) \(\frac{8}{3} \mathrm{~g} / \mathrm{cc}\) (d) \(\frac{3}{8} \mathrm{~g} / \mathrm{cc}\)

Step-by-Step Solution

Answer

The density of the other liquid is (b) \(\frac{4}{9} \text{ g/cc} \).

Step 1: Understanding the Scenario

The problem states that a body floats with one-third of its volume submerged in water and three-quarters of its volume outside another liquid. We need to find the density of this other liquid. We will use the concepts of buoyancy and the principle of flotation to solve this.

Step 2: Principle of Flotation in Water

According to the principle of flotation, the weight of the liquid displaced by the submerged volume of the body is equal to the weight of the body. Since one-third of the volume (V) is submerged in water, we have: \[ \rho_{body} \cdot V = \rho_{water} \cdot \frac{1}{3}V \]Assuming \(\rho_{water} = 1 \text{ g/cc} \), this simplifies to:\[ \rho_{body} = \frac{1}{3} \text{ g/cc} \]

Step 3: Principle of Flotation in Other Liquid

Given that three-quarters of the body is outside the other liquid, this implies one-quarter is submerged. The principle of flotation dictates: \[ \rho_{body} \cdot V = \rho_{liquid} \cdot \frac{1}{4}V \]Substitute \(\rho_{body} = \frac{1}{3} \text{ g/cc}\) (from Step 2):\[ \frac{1}{3} \cdot V = \rho_{liquid} \cdot \frac{1}{4}V \] Solving for \(\rho_{liquid}\), we get: \[ \rho_{liquid} = \frac{4}{3} \times \frac{1}{3} = \frac{4}{9} \text{ g/cc} \]

Step 4: Selection of the Correct Option

From the calculations, we found that the density of the other liquid is \(\frac{4}{9} \text{ g/cc} \). Comparing this with the given options, the correct answer is (b) \(\frac{4}{9} \text{ g/cc} \).

Key Concepts

Buoyancy

Buoyancy is the upward force that a fluid exerts on an object immersed in it. This force occurs because the fluid pressure is more at the bottom than at the top due to gravity pulling the fluid downwards.
This upward force is what allows objects to float or sink in a fluid. The greater the buoyant force, the more likely an object is to float. If the weight of the object is less than the buoyant force, the object will float to the surface.

• Buoyancy is why ships float and why balloons rise.
• It is often calculated using the object's submerged volume and the density of the fluid.

Understanding buoyancy is essential in determining whether an object will stay afloat or sink in a liquid.

Density calculation

Density is a measure of how much mass is contained in a given volume of a substance. It's calculated as mass divided by volume, expressed often in grams per cubic centimeter (g/cc) or kilograms per cubic meter (kg/m³).
To solve problems like the one presented, proper density calculation is crucial. Knowing the density of an object and the fluid it is in helps us predict flotation:

• If an object's density is less than that of the liquid, it will float.
• If it's more, it will sink.

In the exercise, calculating the densities allows us to compare and conclude whether the body will float or not. Mastering density calculations also helps in identifying the characteristics of an unknown liquid, as in our exercise where the unknown liquid's density was calculated to be \(\frac{4}{9} \text{ g/cc}\).

Archimedes' Principle

Archimedes' Principle is a fundamental concept in fluid mechanics, stating that the upward buoyant force on a body immersed in a fluid is equal to the weight of the fluid displaced by the body.
This principle explains why boats and logs can float on water. If the weight of the water displaced is equal to or greater than the weight of the object, it will float.

• This principle is critical in solving flotation problems.
• It allows us to relate the volume of submerged object with the densiy of the fluid to determine buoyancy.

In our exercise, Archimedes' Principle helped establish the relationship between the volume of the body submerged in the water and other liquid, enabling the calculation of the density of the unknown liquid. Using this principle, we derived the density of the new liquid to be \(\frac{4}{9} \text{ g/cc}\). This is key to understanding how objects behave in different fluids.