Undergraduate
  1. Classical mechanics  1.1. dynamics  1.1.1. Motion in one dimension  1.1.2. Motion in two dimensions  1.1.3. projectile motion  1.1.4. Relative speed  1.1.5. Uniform circular motion  1.1.6. Non-uniform circular motion  1.1.7. Reference frames and transformations  1.2. Newton's Laws of Motion  1.2.1. First law of motion  1.2.2. Second law of motion  1.2.3. Third Law of Motion  1.2.4. Applications of Newton's Laws  1.2.5. Friction and its types  1.2.6. Free Body Diagram  1.2.7. Constraints and pseudo-forces  1.3. Work and Energy  1.3.1. work done by the force  1.3.2. Work–energy theorem  1.3.3. Kinetic energy  1.3.4. Potential energy in classical mechanics  1.3.5. Conservative and non-conservative forces  1.3.6. Energy Conservation  1.3.7. Power and efficiency  1.4. Speed and collisions  1.4.1. Linear momentum  1.4.2. Conservation of momentum  1.4.3. Impulse and impact force  1.4.4. Elastic and inelastic collision  1.4.5. Center of mass and speed  1.4.6. Rocket Propulsion  1.5. Rotational motion  1.5.1. Torque and Angular Momentum  1.5.2. Moment of inertia  1.5.3. Rotational Kinematics  1.5.4. Rotational Energy  1.5.5. Rolling Motion  1.5.6. Precession and gyroscopic motion  1.6. Gravitational force  1.6.1. Newton's law of universal gravitation  1.6.2. Gravitational potential energy  1.6.3. Orbital mechanics  1.6.4. Kepler's laws of planetary motion  1.6.5. Gravitational field and potential  1.7. Fluid mechanics  1.7.1. Pressure and Pascal's Principle  1.7.2. Buoyancy and Archimedes' principle  1.7.3. Fluid Dynamics and Bernoulli's Principle  1.7.4. Viscosity and Poiseuille's law  1.7.5. Surface tension and capillarity  1.8. Oscillations and waves  1.8.1. Simple Harmonic Motion  1.8.2. Damped and driven oscillations  1.8.3. Coupled oscillations and common modes  1.8.4. Wave properties and types  1.8.5. Sound Waves and the Doppler Effect  1.8.6. Wave interference and superposition
  2. Electromagnetism  2.1. Electrostatics  2.1.1. Electric charge and properties  2.1.2. Coulomb's law  2.1.3. Electric field and electric potential  2.1.4. Gauss's Law  2.1.5. Capacitance and Dielectric  2.1.6. Electric dipole and potential energy  2.2. Electric circuits  2.2.1. Current and Resistance  2.2.2. Ohm's law  2.2.3. Kirchhoff's Laws  2.2.4. RC Circuit  2.2.5. AC Circuits and Reactance  2.2.6. Electric power and energy  2.3. Magnetism  2.3.1. Magnetic Fields and Forces  2.3.2. Ampere's Law  2.3.3. Magnetic dipole moment  2.3.4. Biot-Savart law  2.3.5. Magnetic Materials and Hysteresis  2.4. Electromagnetic induction  2.4.1. Faraday's Law  2.4.2. Lenz's law  2.4.3. Self and mutual induction  2.4.4. Transformers and Inductive Circuits  2.5. Maxwell's equations  2.5.1. Gauss's law for electricity  2.5.2. Gauss's law for magnetism  2.5.3. Faraday's law of induction  2.5.4. Ampere-Maxwell law  2.5.5. Electromagnetic waves
  3. Thermodynamics  3.1. Laws of Thermodynamics  3.1.1. Zeroth law of thermodynamics  3.1.2. First law of thermodynamics  3.1.3. Second Law  3.1.4. Third Law  3.2. Heat and work  3.2.1. Heat transfer (conduction, convection, radiation)  3.2.2. Thermal expansion  3.2.3. Heat engines and refrigerators  3.2.4. Carnot cycle and efficiency  3.3. Statistical mechanics  3.3.1. Maxwell–Boltzmann distribution  3.3.2. Entropy and Probability  3.3.3. Partition function  3.3.4. Fermi–Dirac and Bose–Einstein statistics
  4. Optics  4.1. Geometrical Optics  4.1.1. Reflection and Refraction  4.1.2. Mirrors and lenses  4.1.3. Optical Instruments  4.1.4. Fermat's principle  4.2. Wave optics  4.2.1. Interference in wave optics  4.2.2. Diffraction  4.2.3. Polarization  4.2.4. Coherence and holography
  5. Quantum mechanics  5.1. Wave–particle duality  5.1.1. Blackbody radiation in wave–particle duality in quantum mechanics  5.1.2. Photoelectric effect  5.1.3. Compton scattering  5.1.4. De Broglie wavelength  5.2. Schrödinger Equation  5.2.1. Time-independent Schrödinger equation  5.2.2. Particle in a box  5.2.3. Quantum Tunneling  5.2.4. Potential wells and obstacles  5.3. Quantum States  5.3.1. Wave function  5.3.2. Quantum operators  5.3.3. Heisenberg uncertainty principle  5.3.4. Angular momentum and spin
  6. Relativity  6.1. Special relativity  6.1.1. Lorentz transformations  6.1.2. Time expansion and length contraction  6.1.3. Relativistic energy and momentum  6.2. General relativity  6.2.1. Principle of Equivalence  6.2.2. Schwarzschild metric  6.2.3. Gravitational waves  6.2.4. Black holes and event horizons
  7. Solid state physics  7.1. Crystal structure  7.1.1. Bravais lattices  7.1.2. X-ray diffraction  7.1.3. Band theory  7.1.4. Phonons and lattice vibrations  7.2. Electrical and Magnetic Properties  7.2.1. Conductors, Semiconductors and Insulators  7.2.2. Superconductivity  7.2.3. Hall effect
  8. Nuclear and particle physics  8.1. Atomic Structure  8.1.1. Atomic Model  8.1.2. Nuclear binding energy  8.2. Radioactivity  8.2.1. Alpha, beta, gamma decay  8.2.2. Half life  8.2.3. Nuclear Fission and Fusion  8.3. Particle physics  8.3.1. Standard Model  8.3.2. Quarks and Leptons  8.3.3. antimatter  8.3.4. Fundamental interactions
  9. Astrophysics and cosmology  9.1. Stellar evolution  9.1.1. Star formation  9.1.2. Black holes and neutron stars  9.1.3. White dwarfs and supernovae  9.2. Cosmology  9.2.1. The Big Bang Theory  9.2.2. Dark matter and dark energy  9.2.3. Cosmic microwave background

Undergraduate → Classical mechanics → Fluid mechanics ↓


Pressure and Pascal's Principle


Fluid mechanics is an essential topic in classical mechanics, exploring the behavior of fluids at rest and in motion. An important part of this field is understanding pressure and how it affects fluid dynamics. One of the fundamental principles associated with pressure is Pascal's principle, named after French mathematician, physicist, and inventor Blaise Pascal. This principle is a cornerstone of fluid mechanics and has a profound impact on the behavior of fluids in various systems.

Withstanding the pressure

Pressure is a concept we encounter in everyday life, from the pressure in car tires to the atmospheric pressure that affects weather patterns. In the context of fluid mechanics, pressure is defined as the force applied perpendicular to the surface of an object divided by the area over which the force is distributed. Mathematically, pressure (P) can be expressed as:

P = F / A

Where:

  • P is the pressure.
  • F is the applied force.
  • A is the area over which the force is distributed.

In the International System of Units (SI), pressure is measured in units of pascals (Pa), where 1 pascal is equal to 1 newton per square meter.

Visualizing pressure

Imagine that you are squeezing a balloon with your hand. The pressure applied by your hand is evenly distributed over the surface of the balloon. The smaller the contact area of your hand, the greater the pressure, because the same force is applied over a smaller area. This is why squeezing with a finger can cause a balloon to burst more easily than using the palm of your hand.

The diagram above shows a balloon with a force applied at a single point. The force is best represented by using lines to show the direction and magnitude, where a smaller area leads to higher pressure.

Pascal's principle

Pascal's principle, or Pascal's law, states that a change in pressure applied to a closed fluid is transmitted without loss to every point in the fluid and to the walls of its container. In simple terms, if you apply pressure to a fluid in a closed system, the increase in pressure is felt equally throughout the fluid. This principle enables hydraulic systems to work.

Mathematically, Pascal's law can be represented as follows:

dP = Delta P

Where dP is the change in pressure applied to an enclosed fluid and Delta P represents the pressure change experienced at any other point in the fluid.

Real-world application of Pascal's principle

Pascal's principle has many real-world applications, most notably in hydraulic systems, which are used in brakes, hydraulic jacks, and other machinery. Imagine a simple hydraulic system with two pistons, a small one and a large one, connected by a tube filled with oil. When you apply force to the smaller piston, the pressure is transmitted through the fluid, causing a greater force to be applied to the larger piston because the larger piston area amplifies the applied force.

Consider this hydraulic press:

Here, the smaller piston on the left is pressed down, increasing the pressure in the fluid and the larger piston on the right is pushed upward with more force.

Textual examples

Consider a car's hydraulic brake system. When you press the brake pedal, it increases the pressure in the hydraulic fluid, which transfers that increased pressure to the brake cylinders at each wheel. This results in the brake pads pressing powerfully against the wheel rotors, effectively stopping the vehicle.

Pressure in liquids

Fluids exert pressure on an object due to the weight of the fluid above it. This type of pressure is called hydrostatic pressure. The pressure at a certain depth in a fluid column is given by the formula:

P = P_0 + rho gh

Where:

  • P_0 is the pressure at the surface of the fluid.
  • rho is the density of the liquid.
  • g is the acceleration due to gravity.
  • h is the height of the fluid above the point.

This equation helps explain why pressure increases as you go deeper underwater. When there is more fluid above, the pressure exerted on objects below increases proportionately, impacting the behavior and implementation of engineering solutions for submarines, water towers, and other fluid-based systems.

Conclusion

Understanding pressure and Pascal's principle is vital in fluid mechanics. From everyday applications such as car brakes to complex hydraulic machinery, the principles governing fluid pressure and its transmission answer many questions about the behavior of fluids in closed systems. Mastering these concepts is crucial to taking advantage of the properties of fluids in practical and innovative engineering solutions.


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Pressure and Pascal's Principle

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