Physics - Classical Mechanics - Gravitational Potential Energy

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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 Gravitational Potential Energy. So, without further ado, let's get straight into it!

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Gravitational Potential Energy

In the Work and Energy chapter we saw the usefulness of Work and Potential.

Work W is defined as:

The measure of how much energy is being transferred by one object when this object is moved over a distance by an external force which is applied in the direction of displacement.

Potential Energy (in general) can be thought of as:

The energy that is stored on a system.

Potential Energy U only depends on the relative position of the various parts of a system and the initial and final configuration of those parts.

We also saw that Gravitational Potential Energy is defined as:



where:

• U : Gravitational Potential Energy (in Joules)
• w : Force of Gravity (Weight) (in Newtons)
• h = h₂ - h₁ : Height Difference (in meters)
• m : Mass of the attracted object (in Kg)
• g : Gravitational Acceleration (in m/s²)

But, how do we end up with this equation?

Proof

The Gravitational Potential Energy is equal to the work done by the Force of Gravity, while moving a mass m from height h₁ to h₂:



The force F_h is considered to be pointing from the Earth's center towards the surface (inverse) making the value negative:



We want to end up with W = U₁ - U₂, which means that we define U as:



Using this definition ΔU = U₂ - U₁ will be negative when h₁ > h₂, which is what we expected.

In gravity problems we will put U = 0 at the best possible point, so that calculations become easier.

By combining the equation of U with the equation of Gravitational Acceleration g we end up with:

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Gravitational Potential

The Gravitational Potential V is defined as the Gravitational Potential Energy U that a unit mass m possesses:



The value of V is always negative and maximum at infinity. The S.I. unit of V is the J/Kg.

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From the Earth to the Moon (Escape Velocity)

Using the Law of Conservation of Energy, we can easily calculate the velocity at which we have to be launched at to escape the Earth's Gravity and reach the moon.

Let's suppose that the final velocity is 0 and that U₂ = 0, meaning that the height is thought as infinite.



This velocity is called escape velocity.

In the case of the Earth the value is 11.186 Km/s.

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RESOURCES:

References

1. https://opentextbc.ca/universityphysicsv1openstax/chapter/13-3-gravitational-potential-energy-and-total-energy/
2. https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_9A__Classical_Mechanics/7%3A_Gravitation/7.3%3A_Energy_in_Gravitational_Systems
3. https://byjus.com/jee/gravitational-potential-energy/
4. http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html

Images

1. 

Mathematical equations used in this article, where made using quicklatex.

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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

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Final words | Next up

And this is actually it for today's post!

Next time we will start getting into Exercises around Newtonian Gravity (I'm currently thinking of 2 parts)

See ya!

Keep on drifting!