Physics - Classical Mechanics - Force and Potential Energy

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    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:


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:


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:






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

Previous articles of the series

Rectlinear motion

Plane motion

Newton's laws and Applications

Work and Energy

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!

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