Mechanical Properties of Fluids: Comprehensive NEET Physics Notes

1. Pressure in Fluids

1.1 Pressure and Its Measurement

Formula: $P=\frac{F}{A}$

• Explanation: Pressure ($P$) is defined as the force ($F$) applied per unit area ($A$). It is a scalar quantity.
• Units: Pascal ($Pa$) where $1Pa=1N/m^2$
• Applications: This concept is fundamental in understanding how fluids exert forces on surfaces, which is crucial for calculating pressures in various contexts, such as in hydraulic systems and atmospheric pressure measurements.

Example Application:

• Problem: A force of 500 N is applied uniformly over an area of $0.5m^2$. Calculate the pressure.
• Solution: $P=\frac{500\text{N}}{0.5{\text{m}}^2}=1000\text{Pa}$

Common Mistakes:

• Confusing force with pressure. Remember that pressure is force per unit area, not just force.

1.2 Pascal’s Law

Formula: $P_1=P_2$

• Explanation: Pascal's Law states that in a fluid at rest, the pressure is the same at all points at the same height. This law is the basis for hydraulic systems, where an applied pressure is transmitted undiminished throughout the fluid.
• Applications: Hydraulic lifts and brakes use Pascal’s Law to amplify force.

Example Application:

• Problem: In a hydraulic lift, a force of 200 N is applied on a small piston of area $0.01m^2$. If the larger piston has an area of $0.1m^2$, calculate the force exerted by the larger piston.
• Solution: By Pascal's Law: $F_2=\frac{A_2}{A_1}\times F_1=\frac{0.1{\text{m}}^2}{0.01{\text{m}}^2}\times 200\text{N}=2000\text{N}$

Common Mistakes:

• Not considering the area ratio when applying Pascal’s law in hydraulic systems.

2. Bernoulli's Principle

2.1 Bernoulli’s Equation

Formula: $P+\frac{1}{2}\rho v^2+\rho gh=\text{constant}$

• Explanation: Bernoulli’s principle relates the pressure, velocity, and height in a moving fluid. It states that an increase in the fluid's speed results in a decrease in pressure or potential energy.
• Applications: Explains the working of airplane wings, Venturi meters, and the lift force on spinning balls.

Example Application:

• Problem: Water flows through a pipe with varying diameters. If the speed of water is 2 m/s in the wider section where the pressure is 1500 Pa, calculate the pressure in the narrower section where the speed is 4 m/s.
• Solution: Applying Bernoulli's equation between the two sections, rearrange to solve for the unknown pressure.

Common Mistakes:

• Neglecting the height term ($\rho gh$) when it's significant, especially in cases where elevation changes are involved.

3. Viscosity and Stokes' Law

3.1 Viscosity

Formula: $\eta =\frac{F}{A}\frac{l}{v}$

• Explanation: Viscosity ($\eta$) is a measure of a fluid's resistance to flow. It describes the internal friction between layers of fluid as they move past each other.
• Units: Pascal-second ($\text{Pa}\cdot \text{s}$)

Example Application:

• Problem: Calculate the viscosity of a fluid if a force of 5 N is needed to move a plate of area 0.2 m² at a velocity of 0.1 m/s over a layer of fluid 0.01 m thick.
• Solution: $\eta =\frac{\frac{5\text{N}}{0.2{\text{m}}^2}\times 0.01\text{m}}{0.1\text{m/s}}=2.5\text{Pa}\cdot \text{s}$

Common Mistakes:

• Forgetting to convert all measurements to SI units before calculating viscosity.

Quick Recap:

• Pressure: $P=\frac{F}{A}$
• Pascal's Law: Pressure applied to an enclosed fluid is transmitted undiminished.
• Bernoulli's Principle: Increase in fluid speed results in a decrease in pressure.
• Viscosity: Describes the internal friction in fluid flow.

NEET Exam Strategy:

• Focus on understanding the applications of Bernoulli’s principle and Pascal’s law, as they are frequently tested in NEET.
• Practice problems involving unit conversions and ensuring correct formula applications.

Practice Questions:

1. A fluid flows through a pipe with varying diameters. If the pressure at a wide section is 1200 Pa and the speed is 2 m/s, what is the pressure at a section where the speed is 4 m/s?
2. Calculate the force required to maintain a steady flow in a viscous fluid where $\eta =3\text{Pa}\cdot \text{s}$, area = 0.1 m², and the velocity gradient is 0.5 m/s per meter.
3. Using Pascal's law, determine the force exerted by a larger piston in a hydraulic lift if the smaller piston is 5 cm² in area and the applied force is 100 N.
4. A metal sphere of radius 2 cm falls through a viscous liquid. Using Stokes' law, calculate the terminal velocity if the density of the sphere is 8000 kg/m³, the fluid density is 1000 kg/m³, and $\eta =0.1\text{Pa}\cdot \text{s}$.

This summary should provide a comprehensive review of key formulas and concepts from the "Mechanical Properties of Fluids" chapter, tailored to NEET exam preparation.