Physics - Classical Mechanics - Force and Potential Energy

Introduction

Hey it's a me again @drifter1! After a big pause, today we finally continue with Physics and more specifically the branch "Classical Mechanics" to talk about Force and Potential Energy.

More specifcally, today's topics are:

Force as the Potential Energy Derivative (one-dimension)
Force as the Potential Energy Gradient (three-dimensions)

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

Force as the Potential Energy Derivative (one-dimension)

Thinking about the two types of conservative forces and energies that we talked about previously (elastic and gravitational) you might have noticed something!
The gravity force is given by:

whilst the gravitational potential energy is given by:

In the same way, based on Hooke's law the elastic force is given by:

whilst the corresponding potential energy is:

Knowing some Mathematical Analysis, you can clearly see that the force can be found using a Differentiation on the potential energy equation. Having both the force and the potential energy in the same dimension and remembering that the work of a force is equal to the negative of the change in potential energy (W = -ΔU), we can easily talk about infinitesimal changes in position (dx).

So, in the end the force in one dimension is given by:

Example:

The electric potential energy of a specific system is given by:

Calculating the electric force is pretty simple and looks like this:

Force as the Potential Energy Gradient (three-dimensions)

Expanding our analysis into 2, 3 or even more dimension is quite easy! The force along the other dimensions (y and z) can be found using a differentiation on the equation using the variable of that dimension(dy and dz). This clearly gets us into partial differentiation, which is based on exactly that. So, the 3 equations for F look like this:

Using the unit vectors for each dimension (i, j and k) we can write:

In the end, the force is equal to the negative gradient of the potential energy:

Example:

Consider the following Potential energy equation:

Let's calculate the applied force equation in vector form!
Using partial differentiation we calculate the equation for each of the two axes:

So, the total force is:

RESOURCES:

References

https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_7A/2%3A_Applying_Models_to_Mechanical_Phenomena/2.5%3A_Force_and_Potential_Energy
http://hyperphysics.phy-astr.gsu.edu/hbase/pegrav.html

Images

https://phys.libretexts.org/Courses/University_of_California_Davis/UCD%3A_Physics_7A/2%3A_Applying_Models_to_Mechanical_Phenomena/2.5%3A_Force_and_Potential_Energy

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, accelaration and averages of those
Rectlinear motion with constant accelaration and free falling -> const accelaration 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 accelaration 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

Final words | Next up

This is actually it for today's post! Next time we will continue with even more stuff around Potential Energy.

See ya!

Keep on drifting!