Problem 68 - Master Resource Book in JEE Main Physics

Question

The relative velocity of two parallellayers of water is \(8 \mathrm{cms}^{-1}\). If the perpendicular distance between the layers is \(0.1 \mathrm{em}\), then velocity gradient will be (a) \(40 \mathrm{~s}^{-1}\) (b) \(50 \mathrm{~s}^{-1}\) (c) \(60 \mathrm{~s}^{-1}\) (d) \(80 \mathrm{~s}^{-1}\)

Step-by-Step Solution

Answer

The velocity gradient is 80 s⁻¹, option (d).

Step 1: Understanding Velocity Gradient

The velocity gradient is defined as the change in velocity per unit distance between the layers of a fluid.

Step 2: Identify Given Values

The relative velocity between the two layers is given as \(8 \text{ cm/s}\), and the distance between these layers is \(0.1 \text{ cm}\).

Step 3: Apply Velocity Gradient Formula

The formula to calculate the velocity gradient is \(\text{Velocity Gradient} = \frac{\Delta v}{\Delta y}\), where \(\Delta v\) is the change in velocity and \(\Delta y\) is the distance between layers.

Step 4: Insert Known Values

Substitute the known values into the formula: \(\text{Velocity Gradient} = \frac{8 \text{ cm/s}}{0.1 \text{ cm}}\).

Step 5: Calculate the Velocity Gradient

Perform the division: \(\frac{8}{0.1} = 80 \text{ s}^{-1}\).

Step 6: Conclusion and Answer

The calculated velocity gradient is \(80 \text{ s}^{-1}\), which corresponds to option (d).

Key Concepts

Relative Velocity

Relative velocity is a crucial concept in understanding how two objects move in relation to each other. When discussing fluids, we often deal with materials like layers of water moving at different speeds. In this context, relative velocity refers to the speed difference between these layers.

For example, imagine two parallel layers of water flowing in the same direction. If one layer is moving faster than the other, then the difference in their speeds is the relative velocity. This concept helps us understand how fluid particles interact and how stress is distributed across different layers.

Important points to remember about relative velocity:

• It measures the velocity of one object (or layer) with respect to another.
• In fluid mechanics, it's vital for calculating other properties like velocity gradient and shear stress.
• Understanding relative velocity aids in predicting how fluids will behave under different conditions.

Fluid Mechanics

Fluid mechanics is the branch of physics that studies fluids (liquids and gases) and the forces acting on them. This field is essential for engineers and scientists as it applies to various practical applications, from designing pipelines to predicting weather patterns.

In fluid mechanics, several laws and equations describe how fluids move and interact. Key areas include:

• Hydrostatics: The study of fluids at rest.
• Hydrodynamics: The study of fluids in motion.
• Continuity Equation: Describes the conservation of mass in fluid flow.
• Bernoulli's Principle: Relates the pressure, velocity, and height in a moving fluid.

Understanding fluid mechanics requires knowing how to apply these principles to solve problems, such as calculating the velocity gradient. This particular term refers to the rate of change of velocity in a fluid based on the perpendicular distance between layers, which is critical for understanding flow behavior.

Viscosity

Viscosity is a fundamental property of fluids that measures their resistance to deformation or flow. Think of honey and water: honey has a much higher viscosity than water because it flows more slowly and offers more resistance to movement.

Viscosity is influenced by temperature and is a key factor in how fluids interact with surfaces and themselves.
Some key aspects of viscosity include:

• Dynamic Viscosity (Absolute Viscosity): Measures the internal resistance to flow and is usually denoted by the Greek letter \( \eta \).
• Kinematic Viscosity: The ratio of dynamic viscosity to fluid density, symbolized by \( u \).
• Higher viscosity means greater resistance to flow and slower movement rates.
• Understanding viscosity helps in designing systems where fluid flow is critical, such as lubrication in engines or drug delivery in medical devices.

In practical terms, viscosity affects how easily a fluid can be pumped through pipes, how it spreads on a surface, and how it interacts with objects it comes into contact with. For engineers and scientists, mastering the concept of viscosity helps in predicting and managing fluid behavior in various environments.