Physicists like to say that pretty much everything can be modeled as just a mass connected to a spring. I mean, it’s not completely untrue. The mass on a spring appears in lots of different problems. But what about an oscillating mass WITH some type of frictional force? What about a mass-spring with a driving force? This is what we are going to build — in python. It’s going to be awesome.

Mass with a Spring

If you want to build a complicated model, the best option is to start with something simple and make sure that works correctly and then add the complicated parts. That’s what I’m going to do here. So, let’s consider a mass connected to a spring without friction and without a driving force. …

I don’t know about you, but I love data. But not just plain data without a purpose, instead I like to use data to look at trends and make predictions.

So, not too long ago I decided to start a second YouTube channel — Physics Explained. This channel is basically all of my physics explanations and problems. I started off with stuff for introductory-level physics, but I’ve found that more advanced physics (like classical mechanics) is both fun and popular. Here, check out a video.

But here’s the deal. In order to be able to monetize your channel (put ads on there so that you can get paid) YouTube has 2 requirements: 1000 subscribers and 4000 watched hours of video. Right now, I’m just waiting to get to 4000 hours. …

Let me be clear. My main goal is to get to the calculus of variations and use the Euler-Lagrange equation for some cool stuff. However, I think it’s important to start off with some problems that sort of do the same thing but are less complicated.

So, here is the situation. I want to find the path of light as it travels from one medium (say air) into another medium (like water). For this, I will use Fermat’s Principle. This says that light will take a path with that will take the least amount of time. Obviously in a single material, this would also be the shortest distance (which is usually a straight line) since the velocity is constant. …

It’s true. I don’t normally write about politics and stuff like that. But you know what, these aren’t normal times. I have to write something — for me, it’s just basic therapy. So here we go.

Maybe you haven’t heard — but there’s a presidential election coming up. Yup. It’s true. It’s essentially a choice between Biden and Trump. I will let you make up your own mind on these two candidates — but I have to talk about Trump flags.

Yes, you see these flags. People put them on their trucks (but never on their hybrid-electric cars), boats, and houses. I’ve seen several different flags and I think I should give my translation for these things. So, that’s just what I’m going to do. …

It seems like the most common introduction to the Calculus of Variations is to talk about the Brachistochrone problem. It goes like this.

Imagine you have a bead that can slide along a frictionless wire under the influence of gravity only. You want the bead to slide (starting from rest) from point 1 to point 2 along some path. What path would produce the shortest time?

So, let’s do this.

Bead on a Straight Wire

I’m going to start with the simplest path possible — a straight line. It looks like this.

Here is a physics problem. You have an object above the moon and you drop it. What is the final speed before the object hits the surface? Note: this object is NOT the lunar lander since it also has a horizontal motion — but you get the idea. The best way to solve this problem is to use the gravitational potential energy (with respect to infinity) of the object-moon system. This potential looks like this.

There are two kinds of line integrals — those you see in your math course and then there are real line integrals that we use in physics. OK, I’m just joking. But really, let’s do some line integrals.

Definition of Work

Let me start off with the most common example of a line integral in physics — the work. Work is defined as:

Let me be clear. I didn’t invent the idea of having students create videos. Honestly, I don’t even know where I got the idea from — it was probably Andy Rundquist (SuperFly Physics), that guy has tons of great ideas.

The basic idea is that instead of a test (or with a test), students create a short video in which they explain the solution to some problem (physics problem in my case). But how do you implement this in your course? I’m going to go over all the details that I use in my classes. …