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:
