Physics - Classical Mechanics - Spherical Mass Distributions
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Introduction
Hey it's a me again @drifter1!
In this article we will continue with Physics, and more specifically the branch of "Classical Mechanics".
Today's topic are Spherical Mass Distributions.
So, without further ado, let's dive straight into it!
Spherical Mass Distribution
Newton proved that, for all spherical mass distributions:
- First Theorem:
A body that is inside a spherical shell of matter experiences no net gravitational force from that shell.
- Second Theorem:
The gravitational force on a body that lies outside a spherical shell of matter is the same as it would be if all of the shell’s matter were concentrated into a point at its center.
Gravity outside a Spherical Shell
A spherical mass is built up of many infinitely thin spherical shells, which are nested inside each other. Let's consider a mass m is at distance r from the center of a spherical shell of mass M and radius R (R < r).
Based on the second theorem, the gravitational potential of the shell is given by:
because the shell can be though of as a point mass.
Proof
Let's proof this equation...
Based on the principle of superposition, the total gravitational attraction is equal to the vector sum of the forces of gravity that all particles of the spherical shells exert on mass m. But, it's simpler to calculate the gravitational potentials, as those are scalar and not vector quantities.
To begin, the shell is cut into rings, with each rings being at a distance l from m.
Each ring has a width Rdθ and radius of Rsinθ.
The surface area of the ring is:
Let's consider that the total mass M of the shell is evenly distributed over its surface.
The surface area of the shell is 4πR2, and thus the mass of each ring is given by:
For infinitely thin rings, the total potential is given by the following integral:
From the Law of Cosines l2 can be replaced by:
Differentiating both sides results in:
Using it, the integral can be rewritten as:
The ring closest to m has a distance l of r - R, whilst the ring that is further has a distance of R + r.
Now the indefinite integral can be turned into a definite integral, which gives:
Gravity within a Spherical Shell
Using the same math as before, inside the shell, the distance of each ring, l, extends from R - r to R + r, giving:
Simple Potential Examples
Hoop
A hoop can be thought of as a spherical shell.
Let's consider a hoop of radius r and mass M
On the x-y plane the hoop can be simplified into a circle:
For an arbitrary point P (0, 0, l), the gravitational potential is given by:
where the denominator is the distance of the point P to the center of the hoop.
Disk
A disk of radius r in the x-y plane, can be described by the following equation:
For a mass density σ the total mass of the disk is πr2σ.
Integrating the formula for the hoop, over a radius a from 0 to r (mass is now 2πaσda) results in:
RESOURCES:
References
- http://www2.yukawa.kyoto-u.ac.jp/~shinsuke.kawai/docs/TeachingMaterial/Gravitation.pdf
- http://astro.utoronto.ca/~bovy/AST1420/notes-2017/notebooks/02.-Spherical-Mass-Distributions.html
- https://www.sparknotes.com/physics/gravitation/potential/section3/
Images
Mathematical equations used in this article, where made using quicklatex.
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
Final words | Next up
And this is actually it for today's post!
Next time we will get into how Earth's Rotation affects Gravity..
See ya!
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