Physics - Classical Mechanics - Exercises on Fluid Statics (part 2)

about 2 years ago

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 ρ_wood V_wood 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