Master the Toughest Topic in Physics: Helical Motion! Break Free from the Textbook with 3D Graphs (Grapher & GeoGebra)
I am Ken Kuwako, your Science Trainer. Every day is an experiment.
Have you ever wanted to take those static diagrams from your physics textbook and make them dance in 3D right before your eyes? On a flat page, a phenomenon might look like nothing more than a tangled mess of lines. But when you step back and view it from a new perspective, you’ll find the hidden, breathtakingly beautiful, and dynamic laws of the universe.
Today, you don’t need any specialized knowledge or expensive software! I’m going to show you how to master one of the trickiest concepts in physics—helical motion—using a hidden gem of an app that comes standard on every Mac. First, take a look at this site to get a feel for what we are aiming for:
「High School Physics CG Video Materials」
The Best Tool You Never Knew You Had: Mac’s Grapher
Do you think creating 3D graphs requires complex coding or professional analysis software? Think again. If you’re a Mac user, you already have a powerhouse tool sitting in your Applications folder for free. It’s called Grapher.
It’s a bit of an unsung hero, but with just a few simple equations, it can render stunning 3D visualizations in seconds. Today, we’re going to recreate the helical motion of a charged particle, a classic topic in electromagnetism. When a particle enters a uniform magnetic field at an angle, it experiences a mysterious force called the “Lorentz force,” which pushes it into a path that circles and moves forward simultaneously.
This exact motion is what creates the Aurora Borealis (the Northern Lights). Charged particles from the sun get trapped by Earth’s magnetic field, spiraling down toward the poles. As they crash into our atmosphere, they release the brilliant light we see from the ground.
While beautiful, this concept is often a nightmare for students because it’s hard to visualize 3D movement on a 2D sheet of paper. That is exactly where 3D graphing saves the day!
Turn Equations into Living Physics
Here is the final result. Seeing it move and rotate is truly impressive:
The magic of Grapher is that you can type in equations exactly as they are. To create this spiral, you only need these simple “magic words”:
Type these in, click, and—voila!—the spiral appears.
The logic is simple: we move the particle at a constant speed in the x-direction (x=0.2t) while combining sine and cosine for the y and z directions. This creates a circle that moves forward—a helix! Once it’s drawn, you can use your mouse to orbit the graph and see it from any angle.
View the whole picture from an angle!
From the front (x-axis), it’s a perfect circle!
From the side, it’s a steady sine wave!
From the top, it’s a steady cosine wave!
If you aren’t on a Mac, I also highly recommend GeoGebra, a free browser-based app. Much like Grapher, you can just input your formulas and watch the math come to life. It’s a fantastic digital canvas.
Here is an even more detailed version created by another teacher—truly impressive work:
「charged particle in magnetic field」https://www.geogebra.org/m/afphxyqj
Since it’s in 3D, you can tweak the parameters and observe the motion from any perspective you like.
Looking from above…
What do you think? I was blown away when I first discovered this. Huge thanks to the creators! These apps make math and physics so much more intuitive. We really do live in a wonderful era for learning.
A New Perspective Changes Everything
When solving physics problems, the most important skill is looking at things from multiple angles. When you spin a 3D graph and see exactly where to “slice” the motion to analyze it, the math starts to make sense.
Physics is the study of the “rules of the world.” There’s no reason not to use a tool like Grapher to visualize those rules so easily. I encourage you all to play around with functions and physical phenomena in 3D. It will turn your black-and-white textbook world into full color!
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