Fluid Mechanics: Pascal & Bernoulli Theorems

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Questions and Answers

What is the formula for pressure in a liquid according to Pascal's law?

P = A/F

P = F + S

P = F/A (correct)

P = F*v

In a hydraulic system, if the area of the first piston is 21 cm² and the force applied is 38 N, what is the pressure exerted on the liquid?

1.81 Pa

1800 Pa (correct)

180 Pa

0.20 Pa

What does the fundamental theorem of hydrostatics state about pressure differences?

Pressure depends solely on the area of the surfaces.

Pressure decreases with height.

Pressure increases with depth. (correct)

Pressure remains constant at all depths.

What is the relationship expressed by the equation P + ρgZ = Constant?

It describes the balance of forces in a static fluid. Signup and view all the answers

How is the equilibrium condition of forces in a hydraulic system expressed?

F1 + F2 - P = 0 Signup and view all the answers

If the second piston in a hydraulic system has an area of 100 cm², what effect would increasing the area have on the force exerted?

The force will increase. Signup and view all the answers

In hydrostatics, what does ρ represent?

Density of the fluid. Signup and view all the answers

What causes pressure in a motionless liquid to act equally in all directions?

Pascal's law. Signup and view all the answers

What happens to the velocity of fluid when the cross-sectional area increases?

Velocity decreases Signup and view all the answers

If the radius of a tube is halved, what effect does this have on the flow velocity when the flow rate remains constant?

Flow velocity quadruples Signup and view all the answers

According to Bernoulli's principle, which of the following statements is true for a static fluid?

Pressure plus gravitational potential energy is constant Signup and view all the answers

In the context of the equation of continuity, if the area of section S1 is twice that of section S2, what can be said about the velocities v1 and v2?

v2 is half of v1 Signup and view all the answers

What does Bernoulli's theorem state about the relationship between pressure, velocity, and height in a fluid?

Constant total energy is maintained as pressure increases when velocity decreases. Signup and view all the answers

Which terms are included in the Bernoulli equation for a fluid particle in motion?

Pressure energy, kinetic energy, and potential energy. Signup and view all the answers

How does Bernoulli's theorem apply when a fluid moves from a higher altitude to a lower altitude?

Kinetic energy decreases as the fluid's pressure increases. Signup and view all the answers

Which of the following statements best describes the hydrostatic principle?

Pressure increases with depth and is consistent across horizontal planes in equilibrium. Signup and view all the answers

Which relationship holds true between two points in a fluid according to Bernoulli's theorem?

If pressure decreases, velocity must increase. Signup and view all the answers

Study Notes

Fluid Mechanics: Pascal's Law

Pascal's Law: A force (F) on a stationary liquid creates equal pressure (P) in all directions. Pressure is perpendicular to the liquid's boundary surface. P = F/S (N/m² = Pa).
Pascal's Law dimensions: [P] = [M¹L⁻¹T⁻²]
All liquid force and pressure processes are based on this law.

Hydrostatics: Fundamental Theorem

Fundamental Theorem of Hydrostatics: In a fluid at equilibrium, the pressure (P) plus the gravity pressure (ρgZ) is constant at any point (ρ = density, g = gravitational acceleration, Z = level). P + ρgZ = Constant
This implies that pressure is uniform across a horizontal plane in a static fluid at equilibrium.

Hemodynamics: Bernoulli's Theorem

Bernoulli's Theorem for Ideal Fluids: In an ideal fluid, the sum of static pressure (P), dynamic pressure (1/2ρV²), and gravity pressure (ρgZ) is constant at every point. P + ½ρV² + ρgZ = C (C is a constant)
Bernoulli's Theorem and Hydrostatics: If velocity (v) = 0 (static fluid), Bernoulli's equation simplifies to P + ρgZ = Constant, demonstrating that it's applicable to hydrostatic fluids.
Understanding pressure changes: A decrease in altitude results in a decrease in potential energy, an increase in passage section leads to reduced kinetic energy increase, causing the pressure to increase to maintain a constant total energy.

Equation of Continuity

Equation of Continuity: In a fluid flow, the volume flow rate (D or Q) is constant. D = Sv (S = cross-sectional area, v = velocity)
Variable cross-sections: If the cross-sectional area (S) increases, velocity (v) decreases; if S decreases, v increases.

Exercises: Applications of Principles

Exercise 1 (Pascal's Law): A hydraulic system with pistons of different areas demonstrates the application of Pascal's principle (force amplification).
Exercise 2 (Continuity Equation): Calculates the output velocity (VB) given the input velocity (VA) and areas (SA and SB) using the continuity equation.
Exercise 3 (Continuity Equation): Calculates VB for different scenarios with varying cross-sectional areas.
Exercise 4 (Continuity Equation): Determines a tube output radius (RB) to achieve a target velocity (VB) given an input velocity (VA) and radius (RA).
Exercise 5 (Bernoulli's Equation): Calculates relevant parameters in a system involving blood flow through vessels of different cross-sectional areas using Bernoulli's equation.

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Description

This quiz covers fundamental principles in fluid mechanics, including Pascal's Law, Hydrostatics, and Bernoulli's Theorem. Explore how these theories apply to pressure and fluid behavior in various contexts. Test your understanding of the relationships between force, pressure, and fluid dynamics.