Problem 1031 - JEE Physics

Question

The fraction of floating object of volume \(\mathrm{V}_{0}\) and density \(\mathrm{d}_{0}\) above the surface of a Liquid as density \(\mathrm{d}\) will be (A) \(\left(\mathrm{d}_{0} / \mathrm{d}\right)\) (B) \(\left[\left\\{\mathrm{dd}_{0}\right\\} /\left\\{\mathrm{d}+\mathrm{d}_{0}\right\\}\right]\) (C) \(\left[\left\\{d-d_{0}\right\\} / d\right]\) (D) \(\left[\left\\{\mathrm{dd}_{0}\right\\} /\left\\{\mathrm{d}-\mathrm{d}_{0}\right\\}\right]\)

Step-by-Step Solution

Answer

The short answer is: \(\left[\left\\{d-d_{0}\right\\} / d\right]\).

Step 1: Find the weight of the object

We can find the weight of the object (W_object) by multiplying its volume (V0) by its density (d0) and gravitational acceleration (g). So, \(W_\text{object} = V_0 d_0 g \).

Step 2: Find the weight of the liquid displaced

The weight of the liquid displaced (W_liquid) can be found by the volume of the object submerged in the liquid (V_submerged) times the density of the liquid (d) and gravitational acceleration (g). Since we want the fraction of the object above the surface, we find the fraction submerged f_submerged, which is equal to the object's density (d0) divided by the liquid's density (d) i.e., \(f_\text{submerged} = \dfrac{d_0}{d}\). Now, the volume submerged (V_submerged) is the fraction submerged (f_submerged) times the total volume (V0) i.e., \(V_\text{submerged} = V_0 f_\text{submerged}\). So, \(W_\text{liquid} = V_\text{submerged} d g = V_0 f_\text{submerged} d g\).

Step 3: Find the fraction of the object above the surface

According to Archimedes' principle, the weight of the liquid displaced (W_liquid) is equal to the weight of the object (W_object). So, \(V_0 d_0 g = V_0 f_\text{submerged} d g\). We can cancel out V0 and g from this equation, leaving us with \(d_0 = f_\text{submerged} d\). Now, we can solve for the fraction above the surface (f_above) by noting that it's equal to 1 minus the fraction submerged (f_submerged), so \(f_\text{above} = 1 - \dfrac{d_0}{d}\). Comparing the expression for the fraction above the surface with the given options, we find that the correct answer is: (C) \(\left[\left\\{d-d_{0}\right\\} / d\right]\)

Key Concepts

Density

Density is a fundamental property of matter that tells us how much mass exists in a given volume. It is expressed as mass per unit volume, often measured in kilograms per cubic meter (kg/m³).
In equations, density is commonly represented by the symbol \( d \). Knowing the density of an object or substance helps to predict how it will interact with other substances.
An object with higher density than a liquid will sink, while one with lower density will float.

• Formula: \( d = \frac{\text{mass}}{\text{volume}} \)
• Density determines if an object will float or submerge when placed in a liquid.
• It plays a crucial role in applications such as buoyancy and fluid dynamics.

Fraction Submerged

The fraction of an object submerged in a liquid refers to the portion of its volume that is beneath the liquid's surface. This is a key parameter when determining buoyancy.
The submerged fraction depends on the densities of both the object and the liquid.

• Formula: The fraction submerged \( f_\text{submerged} \) is given by \( f_\text{submerged} = \frac{d_0}{d} \), where \( d_0 \) is the object's density and \( d \) is the liquid's density.
• A higher density of the object results in a larger submerged fraction.
• This concept is critical for understanding how much of an object remains above the liquid's surface.

Volume Displaced

When an object is submerged, it displaces a volume of liquid equal to the volume of the object submerged. This is central to the concept of buoyancy.
The displaced volume determines the buoyant force exerted on the object.

• Formula: Volume displaced is given by \( V_\text{submerged} = V_0 \times f_\text{submerged} \).
• The more volume an object displaces, the greater the buoyant force.
• Understanding displaced volume helps us calculate how much of an object will float above the liquid.

Buoyancy

Buoyancy is a force that pushes an object upward when it is placed in a fluid. It's due to the difference in pressure exerted on different parts of the object.
This force is usually equal to the weight of the fluid displaced by the object, which is an idea rooted in Archimedes' Principle.

• Formula: Buoyant force \( F_\text{buoyant} = \text{weight of displaced liquid} = V_\text{submerged} \times d \times g \).
• Archimedes' Principle states that the buoyant force on the object is equal to the weight of the fluid displaced.
• Buoyancy explains why certain objects float on the water surface while others sink.