Problem 27 - Master Resource Book in JEE Main Physics

Question

A solid of density \(D\) is floating in a liquid of density \(d\). If \(v\) is the volume of solid submerged in the liquid and \(V\) is the total volume of the solid, then \(v / V\) is equal to (a) \(\frac{d}{P}\) (b) \(\frac{D}{d}\) (c) \(\frac{D}{(D+d)}\) (d) \(\frac{D+d}{D}\)

Step-by-Step Solution

Answer

The fraction \( \frac{v}{V} \) is equal to option (b) \( \frac{D}{d} \).

Step 1: Understanding Buoyancy

For a floating object, the weight of the object is balanced by the buoyancy force. This can be represented by the equation: \[ \text{Weight of solid} = \text{Buoyant force} \] The weight of the solid is given by \( D \times V \times g \) and the buoyant force is \( d \times v \times g \) where \( g \) is the acceleration due to gravity.

Step 2: Setting the Equation for Equilibrium

At equilibrium, the weight of the solid (\( D \times V \times g \)) is equal to the buoyant force (\( d \times v \times g \)). Therefore, we can write:\[ D \times V \times g = d \times v \times g \] Since \( g \) is common, it cancels out, and we simplify the equation to:\[ D \times V = d \times v \]

Step 3: Solving for Submerged Volume Fraction

To find the fraction of the volume of the solid that is submerged, we need to solve for \( \frac{v}{V} \). From the equation \( D \times V = d \times v \), dividing both sides by \( V \) and rearranging gives:\[ \frac{v}{V} = \frac{D}{d} \]

Step 4: Conclusion and Answer

Comparing this result to the options provided, we see that the correct answer is option (b) \( \frac{D}{d} \). Therefore, the fraction of the solid submerged is equal to the ratio of the solid's density to the liquid's density.

Key Concepts

Density

Density is a crucial concept in understanding how objects interact with fluids. It is essentially how much mass an object has in a given volume. Mathematically, it can be expressed as:\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]Density determines whether an object will sink or float in a fluid:

• If an object's density is greater than the fluid's density, it will sink.
• If it's less, the object will float.
• If densities are equal, it will remain suspended in the fluid.

Knowing the densities of both the object and the fluid allows us to predict their interaction. Thus, density is not just a property of solids but of liquids and gases as well. Understanding density provides the basis to solve many other fluid dynamics problems, such as buoyancy, which comes into play in the next sections.

Floating objects

When an object is placed in a fluid and it floats, there are several forces in action. This scenario is a classic example of equilibrium in fluids. For any object to float:

• The buoyant force must equal the object's weight.
• This balance is what allows an object to float or remain partially submerged in a fluid.

For example, consider a wooden block floating on water. If the block's density is less than that of water, it'll float. It doesn't mean the entire object is above the surface; rather, part of it is submerged. This ensures that the buoyant force equalizes with the force of gravity acting on the object. Understanding how objects float leads us to the concept of volume submerged, as the part of an object below the fluid's surface contributes to the buoyancy.

Volume submerged

The volume of an object submerged in a fluid is directly linked to the object’s ability to float. This concept focuses on how much of the object's volume is below the surface of the fluid. For floating objects:

• The ratio \( \frac{v}{V} \) represents the fraction of the object submerged.
• This ratio is determined by comparing the density of the object to the fluid density.
• A higher fluid density leads to less volume being submerged.

Mathematically, if an object is floating in a fluid, you can express the submerged volume as:\[\frac{v}{V} = \frac{D}{d}\]where \( D \) is the density of the object, and \( d \) is the density of the fluid. By maintaining equilibrium in the forces involved, the submerged volume helps ensure the object neither sinks nor floats away entirely. This balance is a satisfying example of physics at work.

Equilibrium in fluids

Equilibrium in fluids is achieved when the forces acting on an object within the fluid are balanced. This concept is essential when examining how objects float or sink. For an object floating in a fluid:

• The gravitational force acting downward must be balanced by the buoyant force pushing upwards.
• The key equation at play here is \( D \times V = d \times v \), where both terms represent the equilibrium of forces.
• This results in the object floating at a stable position, dictated by its density relative to the fluid.

Achieving equilibrium allows for predictable behaviors in fluid dynamics, enabling engineers and scientists to design ships, submarines, and other floating structures appropriately. Understanding these principles is fundamental, as equilibrium ensures the proper functioning and safety of many fluid-based systems.