[Image1]
Introduction
Hey it's a me again @drifter1!
Today we continue with Physics, and more specifically the branch of "Classical Mechanics", in order to get into Exercises on Fluid Statics. This is part 2, where we will cover the remaining topics.
So, without further ado, let's get straight into it!
Hydraulic Car Jack Example
Consider a hydraulic car jack with a piston of 20 cm diameter. What's the required gauge pressure for lifting a car of 1000 Kg?
A piston is a cylinder and its area of contact with the fluid is a circle. Additionally, the radius of a circle is half its diameter and so r = d / 2 = 10 cm = 0.1 m. Thus, the area A where the pressure will be applied is:
The only forces acting upon the system are the atmospheric pressure, weight of the car and the pressure that needs to be applied for lifting the car. Those are related in the following way:
And so, in the end, the required gauge pressure is:
Buoyancy Examples
Next up, let's apply Archimedes' principle for various cases.
Wooden Plank on Water
Consider a wooden plank that floats above a lake. A 20 Kg box is added on top of it. What's the minimum required volume for the plank, so that it doesn't sink. The density of the lake water is 1 x 103 Kg/m3, whilst the wood has a density of 0.7 x 103 Kg/m3.
First of all, because the plank is already floating and will be floating, we don't need to know or use its mass. We will just substitute the mass with the product of density and volume.
In order to be in equilibrium, the buoyant force exerted by the lake water must equal the weight of the plank and box:
The volume of water that is displaced equals the volume of the plank, and so the buoyant force is:
Substituting the mass of the plank with the corresponding ρwood Vwood yields:
All quantities are known, and so the minimum required volume of the wood is:
Bonus: Knowing the volume, we can now also calculate the mass of the wooden plank which is:
Fraction of Iceberg Below Water
[Image 2]
There's a difference in density between the iceberg and the seawater. The ice in icebergs is formed from freshwater and has a density of about 0.9 x 103 Kg/m3. On the other hand, seawater has a density of 1.03 x 103 Kg/m3.
From their densities we can already conclude that icebergs float, because their density is smaller than that of the seawater. So, what fraction is submerged?
In the post about Buoyancy we related Archimedes' principle with the density and showed that the fraction submerged is given by:
So, approximately 87% or 7/8 of the iceberg is submerged below the water.
Surface Tension Example
Lastly, let's also get into a surface tension application, as we only discussed the topic from a theoretical perspective in the corresponding post.
Let's determine the pressure within a soap bubble. The soap bubble has a diameter of 2 cm and the surface tension coefficient is γ = 0.025 N/m.
A soap bubble is filled with air on the inside and has a thin layer of liquid on the outside, which separates that air from the atmosphere. That liquid is what causes the phenomenon of surface tension.
Atmospheric pressure is applied to the overall spherical area of the bubble, which has a surface area of:
Because the bubble is in equilibrium, the net force is zero, and so the force due to surface tension and force exerted by the atmosphere are equal:
The force due to surface tension on each of the two surfaces of the thin liquid layer is γ(2πr) and so the total force is 2γ(2πr), which gives us:
This is the difference in pressure between the inside and outside of the soap bubble. The pressure within the soap bubble is thus:
RESOURCES:
Images
Mathematical equations used in this article, where made using quicklatex.
Visualizations were made using draw.io.
Previous articles of the series
Rectlinear motion
- Velocity and acceleration in a rectlinear motion -> velocity, acceleration and averages of those
- Rectlinear motion with constant acceleration and free falling -> const acceleration motion and free fall
- Rectlinear motion with variable acceleration and velocity relativity -> integrations to calculate pos and velocity, relative velocity
- Rectlinear motion exercises -> examples and tasks in rectlinear motion
Plane motion
- Position, velocity and acceleration vectors in a plane motion -> position, velocity and acceleration in plane motion
- Projectile motion as a plane motion -> missile/bullet motion as a plane motion
- Smooth Circular motion -> smooth circular motion theory
- Plane motion exercises -> examples and tasks in plane motions
Newton's laws and Applications
- Force and Newton's first law -> force, 1st law
- Mass and Newton's second law -> mass, 2nd law
- Newton's 3rd law and mass vs weight -> mass vs weight, 3rd law, friction
- Applying Newton's Laws -> free-body diagram, point equilibrium and 2nd law applications
- Contact forces and friction -> contact force, friction
- Dynamics of Circular motion -> circular motion dynamics, applications
- Object equilibrium and 2nd law application examples -> examples of object equilibrium and 2nd law applications
- Contact force and friction examples -> exercises in force and friction
- Circular dynamic and vertical circle motion examples -> exercises in circular dynamics
- Advanced Newton law examples -> advanced (more difficult) exercises
Work and Energy
- Work and Kinetic Energy -> Definition of Work, Work by a constant and variable Force, Work and Kinetic Energy, Power, Exercises
- Conservative and Non-Conservative Forces -> Conservation of Energy, Conservative and Non-Conservative Forces and Fields, Calculations and Exercises
- Potential and Mechanical Energy -> Gravitational and Elastic Potential Energy, Conservation of Mechanical Energy, Problem Solving Strategy & Tips
- Force and Potential Energy -> Force as Energy Derivative (1-dim) and Gradient (3-dim)
- Potential Energy Diagrams -> Energy Diagram Interpretation, Steps and Example
- Internal Energy and Work -> Internal Energy, Internal Work
Momentum and Impulse
- Conservation of Momentum -> Momentum, Conservation of Momentum
- Elastic and Inelastic Collisions -> Collision, Elastic Collision, Inelastic Collision
- Collision Examples -> Various Elastic and Inelastic Collision Examples
- Impulse -> Impulse with Example
- Motion of the Center of Mass -> Center of Mass, Motion analysis with examples
- Explaining the Physics behind Rocket Propulsion -> Required Background, Rocket Propulsion Analysis
Angular Motion
- Angular motion basics -> Angular position, velocity and acceleration
- Rotation with constant angular acceleration -> Constant angular acceleration, Example
- Rotational Kinetic Energy & Moment of Inertia -> Rotational kinetic energy, Moment of Inertia
- Parallel Axis Theorem -> Parallel axis theorem with example
- Torque and Angular Acceleration -> Torque, Relation to Angular Acceleration, Example
- Rotation about a moving axis (Rolling motion) -> Fixed and moving axis rotation
- Work and Power in Angular Motion -> Work, Work-Energy Theorem, Power
- Angular Momentum -> Angular Momentum and its conservation
- Explaining the Physics behind Mechanical Gyroscopes -> What they are, History, How they work (Precession, Mathematical Analysis) Difference to Accelerometers
- Exercises around Angular motion -> Angular motion examples
Equilibrium and Elasticity
- Rigid Body Equilibrium -> Equilibrium Conditions of Rigid Bodies, Center of Gravity, Solving Equilibrium Problems
- Force Couple System -> Force Couple System, Example
- Tensile Stress and Strain -> Tensile Stress, Tensile Strain, Young's Modulus, Poisson's Ratio
- Volumetric Stress and Strain -> Volumetric Stress, Volumetric Strain, Bulk's Modulus of Elasticity, Compressibility
- Cross-Sectional Stress and Strain -> Shear Stress, Shear Strain, Shear Modulus
- Elasticity and Plasticity of Common Materials -> Elasticity, Plasticity, Stress-Strain Diagram, Fracture, Common Materials
- Rigid Body Equilibrium Exercises -> Center of Gravity Calculation, Equilibrium Problems
- Exercises on Elasticity and Plasticity -> Young Modulus, Bulk Modulus and Shear Modulus Examples
Gravity
- Newton's Law of Gravitation -> Newton's Law of Gravity, Gravitational Constant G
- Weight: The Force of Gravity -> Weight, Gravitational Acceleration, Gravity on Earth and Planets of the Solar System
- Gravitational Fields -> Gravitational Field Mathematics and Visualization
- Gravitational Potential Energy -> Gravitational Potential Energy, Potential and Escape Velocity
- Exercises around Newtonian Gravity (part 1) -> Examples on the Universal Law of Gravitation
- Exercises around Newtonian Gravity (part2) -> Examples on Gravitational Fields and Potential Energy
- Explaining the Physics behind Satellite Motion -> The Circular Motion of Satellites
- Kepler's Laws of Planetary Motion -> Kepler's Story, Elliptical Orbits, Kepler's Laws
- Spherical Mass Distributions -> Spherical Mass Distribution, Gravity Outside and Within a Spherical Shell, Simple Examples
- Earth's Rotation and its Effect on Gravity -> Gravity on Earth, Apparent Weight
- Black Holes and Schwarzschild Radius -> Black Holes (Creation, Types, How To "See" Them), Schwarzschild Radius
Periodic Motion
- Periodic Motion Fundamentals -> Fundamentals (Period, Frequency, Angular Frequency, Return Force, Acceleration, Velocity, Amplitude), Simple Harmonic Motion, Example
- Energy in Simple Harmonic Motion -> Forms of Energy in SHM (Potential, Kinetic, Total and Maximum Energy, Maximum Velocity), Simple Example
- Simple Harmonic Motion Equations -> SHM Equations (Displacement, Velocity, Acceleration, Phase Angle, Amplitude)
- Simple Harmonic Motion and Reference Circle -> SHM and Smooth Circular Motion, Reference Circle
- Simple Harmonic Motion Exercises -> 2 Complete Examples on Simple Harmonic Motion
- Simple Pendulum -> Simple Pendulum (Return Force, Small Angle Approximations, More Accurate Period, Gravity Approximation)
- Physical Pendulum -> Physical Pendulum (Return Torque, Small Angle Approximations, Estimating Moment of Inertia)
- Exercises around Pendulums -> Complete Examples on the 2 types of Pendulums (Simple, Physical)
- Damped Oscillation -> Damping Force, Total Force and Differential Equation, Motion Equations, Special Cases
- Forced Oscillation and Resonance -> Forced Oscillation (Differential Equation, Amplitude, Resonance)
- Exercises around Damped and Forced Oscillation -> Complete Examples on Damped Oscillation and Forced Oscillation
- Chaos and Chaotic Oscillation -> Chaos, Unpredictability and Randomness, Chaotic Oscillation
Fluid Mechanics
- Density and Pressure -> Fluids and Fluid Mechanics, Density, Specific Gravity, Pressure
- Measuring Pressure in Fluids -> Pressure in Fluids (Variation with Depth), Absolute and Gauge Pressure, Measuring Pressure
- Pascal's Principle and Hydraulics -> Static Equilibrium, Pascal's Principle, Hydraulic Systems
- Archimedes' Principle and Buoyancy -> Buoyant Force, Archimedes' Principle, Relation with Density
- Surface Tension -> Surface Tension (Cohesive and Adhesive Forces, Unit, Definition), Capillarity
- Exercises on Fluid Statics (part 1) -> Various Density and Pressure Examples
Final words | Next up
And this is actually it for today's post!
Next time we will start with Fluid Dynamics...
See ya!
Keep on drifting!