Physics - Classical Mechanics - Spherical Mass Distributions

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.

Based on these two theorems, a spherical shell can be safely thought of as a point mass (all mass concentrated in 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

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!

Keep on drifting!