Two non-mixing liquids of densities $\rho$ and $ n\rho (n > 1)$ are put in a container. The height of each liquid is h. A solid cylinder of length L and density d is put in this container. The cylinder floats with its axis vertical and length $ pL (p < 1)$ in the denser liquid. The density d is equal to :
- $\{2 + (n + 1) p\} \rho$
- $\{2 + (n - 1) p\} \rho$
- $\{1 + (n - 1) p\} \rho$
- $\{1 + (n + 1) p\} \rho$
The Correct Option is C
Solution and Explanation
Density and Volume Relations
The equations provided represent the relationship between different variables, including density (\( \rho \)), length (\( L \)), and the volume fractions of two materials (represented by \( p \) and \( n \)). Let's go through each equation step by step:
First Equation:
\(L_{Adg} = (1 - p) L_A \rho g + p L_A n \rho g\)
This equation expresses the total force \( L_{Adg} \) as a combination of two parts: one where the material is characterized by \( p \), and another characterized by \( n \). The force is proportional to density \( \rho \), gravitational acceleration \( g \), and the material's length \( L_A \).
Second Equation:
\(d = (1 - p) \rho + np \rho\)
This equation expresses the overall density \( d \) as a weighted sum of two densities, where the fraction \( p \) corresponds to the second material, and \( (1 - p) \) corresponds to the first material.
Third Equation:
\(d = \rho [L - p + np]\)
Final Equation:
\(d = \rho [1 + (n - 1) p]\)
In this final equation, \( d \) represents the overall density, and it is a function of the density \( \rho \) and the fraction \( p \), along with the material property \( n \). This shows how the overall density depends on the proportion of the two materials and their respective characteristics.
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Concepts Used:
Viscosity
Viscosity is a measure of a fluid’s resistance to flow. The SI unit of viscosity is poiseiulle (PI). Its other units are newton-second per square metre (N s m-2) or pascal-second (Pa s.) The dimensional formula of viscosity is [ML-1T-1].
Viscosity: Formula
Viscosity is measured in terms of a ratio of shearing stress to the velocity gradient in a fluid. If a sphere is dropped into a fluid, the viscosity can be determined using the following formula:
η = [2ga2(Δρ)] / 9v
Where ∆ρ is the density difference between fluid and sphere tested, a is the radius of the sphere, g is the acceleration due to gravity and v is the velocity of the sphere.
Viscosity: Types
- Dynamic viscosity: When the viscosity is measured directly by measuring force. It is defined as the ratio of shear stress to the shear strain of the motion. Dynamic viscosity is used to calculate the rate of flow in liquid.
- Kinematic viscosity: There is no force involved. It can be referred to as the ratio between the dynamic viscosity and density of the fluid. It can be computed by dividing the dynamic viscosity of the fluid with fluid mass density.
- Laminar flow: Laminar flow is the type of flow in which the fluid moves smoothly or in a regular path from one layer to the next. Laminar flow occurs in lower velocities.