[Image1]

## 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 article is part two of **Exercises on Newtonian Gravity**.
Part 1 can be found here.

So, without further ado, let's dive straight into it!

## Recap of Useful Formulas

The following formulas will be very useful for solving Problems around Gravity.### Universal Law of Gravitation

where:

*d*is the distance of the COMs of the two masses*m*and_{1}*m*_{2}*G*is the Gravitation Constant which equals*6.674 × 10*^{-11}N

### Weight or Force of Gravity

### Gravitational Acceleration

where:

*r*can be thought of as the median radius of a planet*h*as the height of an object in respect to the surface of the planet

### Gravitational Potential Energy

or

*ΔU = U*is thus negative when the height decreases (

_{2}- U_{1}*h*).

_{1}> h_{2}### Gravitational Potential

## Two-Dimensional Problem's Field Acceleration

Let's start by calculating the Gravitational Acceleration *g* of the two-dimensional gravity problem that we solved during Part 1.

The total force aplied on mass *m _{A} = 4 Kg* was calculated to be

*ΣF = 2.059 x 10*.

^{-11}
Thus, the Gravitational acceleration *g* at the point where *m _{A}* is in the field of the three other masses is:

## Gravitational Field on the Axis of a Ring (based on Ref1)

Let's consider the following gravitational field along the axis of a uniform ring:

[Custom Figure using draw.io]

The ring has a mass of *M* and a radius of *a*, and the points *P* are taken at a distance *b* along the axis of the ring.
At the center of the ring the Gravitational field of course has a strengh of zero.

Let's find the maximum strength of the Gravitational Field along the axis of the ring in respect to the distance from the center *O*.

### Solution

In order to calculate the field of a ring, the ring has to be split into small masses *dM*, whose fields can then be summed up together.

The distance towards each of those small masses is equal to *c*, which can be easily calculated using Pythagoras's Theorem:

The total gravitational field strengh *dg* of each of those masses *dM* is:

Because each field *dg* points towards another direction a split of each vector into two components is necessary.

Doing that its easy to notice that there is another mass *dM'* that cancels out one of the components completely, which is something that happens for all masses *dM*:

[Custom Figure using draw.io]

Thus, only the component parallel to the axis of the ring will contribute towards the field strength.

Using the trigonometric function of *cosinus*, this component is defined as:

Summing up the contributions of all those small masses *dM*, the total gravitational field *g* along the axis of the ring is calculated to be:

The maximum value is reached at the point(s) where the derivative of *g*, *g'*, is zero.

The derivative is equal to:

The nominator is zero when:

Therefore, the maximum value is at a distance of *b = a / √2*, which gives us a field strength of:

The plot of the gravity field strength *g* in respect to the distance *b*, supposing that *G = M = a = 1*, looks as following:

[Custom Figure using GeoGebra]

## Speed of a Rollercoaster (Inversed Example 2 of Reference 2)

Consider a rollercoaster starts going down from an unknown height *h* and reaches a final speed of *v = 25 m / s*.

Find the value of *h* if friction is negligible and *g = 9.8 m/s ^{2}*.

### Solution

The initial velocity of the rollercoaster is zero (*K _{initial} = 0*) and thus the rollercoaster initially has only gravitational potential energy.

The final potential energy is zero (

*U*), and thus, from Energy conservation, the initial potential energy equals the final kinetic energy, or mathematically:

_{final}= 0The mass is cancelled out and the rest is known and thus the initial height

*h*is:

## RESOURCES:

### References

- https://phys.libretexts.org/Bookshelves/Astronomy__Cosmology/Book%3A_Celestial_Mechanics_(Tatum)/05%3A_Gravitational_Field_and_Potential/5.04%3A_The_Gravitational_Fields_of_Various_Bodies/5.4.02%3A_Field_on_the_Axis_of_a_Ring
- https://courses.lumenlearning.com/physics/chapter/7-3-gravitational-potential-energy/

### 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

## Final words | Next up

And this is actually it for today's post!

Next time we will get into a Physics explanation of the circular motion of Satellites...

See ya!