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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 article will be about **Exercises on Newtonian Gravity**, and splitted into two parts.

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

## Recap of Useful Formulas

The following formulas will be very useful for solving Problems around Gravity (in part 1).### 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 Attraction of the Earth in Various Heights (based on Ref1 Sample Problems)

Let's consider a person of mass *m = 60 Kg* is inside the Gravitational Field of the Earth (which has a mass of *M = 5.98 x 10 ^{24} Kg* and radius of

*r = 6.38 x 10*).

^{6}mCalculate the Force of Gravity that the Earth exerts on the person when:

- The person is standing at sea-level (distance to the center of the earth equal to the radius
*r*) - The person is standing at the peak of Mount Everest (highest altitude above mean sea level is
*h = 8,848 m* - The person is on an airplane that flies at about
*h = 12 km*above sea-level

### 1. Sea-Level

### 2. Mount Everest

### 3. Airplane

The difference is minimal on so even using the value

*g = 9.8 m/s*for Earth would be sufficient enough as it gives:

^{2}## Calculating the Mass of the Earth

Knowing that the equatorial radius of the Earth is *R _{E} = 6.380 Km* and the Gravitational acceleration is

*g = 9.8 m/s*, can we somehow calculate the mass of the Earth?

^{2}### Solution

Let's consider a test-mass of mass *m = 1 Kg*.

Solving the equation of gravitational acceleration *g* for the mass of the Earth *M _{E}* we have:

## Multiple Attractors and Force of Gravity

### One-Dimensional Problem

Let's consider the following problem:[Custom Figure using draw.io]

Calculate the total force of gravity excelled on sphere *m* by the two spheres with equal mass *M* if *d _{1}* is greater than

*d*.

_{2}The result will be an expression of

*m*,

*M*,

*d*and

_{1}*d*.

_{2}#### Solution

Because *d _{1} > d_{2}* we already know that

*F*.

_{1}< F_{2}Thus, the total force will be

*ΣF = F*.

_{2}- F_{1}Using the Universal Law of Gravitation we have:

The difference of squares

*a*is useful when more than 2 masses attract the object, as it simplifies the result by a lot.

^{2}- b^{2}= (a + b)(a - b)### Two-Dimensional Problem

Let's consider the following problem:[Custom Figure using GeoGebra]

In this figure, *A (0, 0)*, *B (-10, -10)*, *C (5, 8.66)* and *D (8, -4.618)* represent the centers of mass of the point masses: *m _{A} = 4 Kg*,

*m*,

_{B}= 11 Kg*m*and

_{C}= 9 Kg*m*.

_{D}= 7 KgThe distance of mass A to each of the other masses is

*d*,

_{AB}= 14.14 m*d*and

_{AC}= 10 m*d*.

_{AD}= 9.24 mThe Force of Gravitational Attraction from the masses B, C and D towards A, has an angle of

*β = 45°*,

*γ = 60°*and

*δ = 30°*correspondingly, always in respect to the horizontal or x-axis.

#### Solution

Using the Universal Law of Gravitation we can calculate the force of attraction of each mass B, C and D towards A:

In order to calculate the sum of those forces we have to split the forces into two components: one for the horizontal axis and one for the vertical axis.

The components are defined easily using Trigonometric Functions:

In the case of the force *F _{B}* which attracts mass A towards mass B we have the following components:

(The results are negative because sin and cos have negative values in the 3rd Quadrant)

Similarly, for force

*F*from mass C we have:

_{C}Lastly, for force

*F*from mass D we have:

_{D}(The trigonometric function sin is negative in the 4th Quadrant, whilst cos is positive)

The signs shows us where the vectors are pointing (left-right and up-down).

Now that we have the components we can easily calculate the components of *ΣF*: *ΣF _{x}* and

*ΣF*as:

_{y}Thus, the total force is:

and with an angle of:

## 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
- https://www.grc.nasa.gov/www/k-12/airplane/wteq.html
- 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

## Final words | Next up

And this is actually it for today's post!

In part 2 we will get into exercises around Gravitational Fields and Gravitational Potential Energy...

See ya!