Pendulum#

Welcome to the Pendulum Simulation!#

Have you ever watched a swing moving back and forth and wondered why it behaves that way? This pendulum simulation is like a digital swing you can play with on your computer. It’s a simple tool that shows how a weight (called a “bob”) hanging from a string moves when you let it go from an angle. Even if you’ve never used an app like this or studied physics before, this guide will help you understand and enjoy it.

The simulation lets you change things like the length of the string, the weight of the bob, gravity (like on Earth or another planet), and how far you pull the bob to start. You can watch it swing and see graphs that show what’s happening. We’ll explain everything step by step, including some basic physics ideas with simple math equations. Don’t worry—we’ll use easy algebra, no fancy calculus here!

Getting Started: How to Use the Simulation#

This simulation is easy to use, like playing a game. Here’s how to get going.

Basic Buttons

  • Start: Click this to make the pendulum begin swinging.

  • Pause: Click to stop the motion temporarily, like freezing time.

  • Reset: Click to set everything back to the beginning and clear any graphs.

Things You Can Change

You can adjust these settings to see how they affect the pendulum’s swing:

  • Length (\(L\)): This is how long the string is. Make it longer or shorter to see changes.

  • Mass (\(m\)): The weight of the bob at the end of the string. Try different weights.

  • Gravity (\(g\)): This is the pull of gravity, like on Earth (about 9.8 m/s²). Change it to pretend you’re on the Moon or Mars!

  • Initial Angle (\(θ\)): The starting tilt of the pendulum from straight down. A small angle is like a gentle push; a big one is more dramatic.

Seeing the Data with Graphs

The app draws graphs in real time as the pendulum swings. You pick what goes on the bottom (x-axis, like time) and the side (y-axis, like the angle). Choices include:

  • Time (how long it’s been swinging)

  • Angle (how tilted it is)

  • Angular velocity (how fast the angle is changing)

  • Kinetic energy (energy from moving)

  • Potential energy (energy from height)

For example, choose “time” on the bottom and “angle” on the side to see a wavy line showing the back-and-forth motion.

Hint

Try graphing the angle over time. Count how long it takes for one full swing (that’s the “period,” or \(T\)). Change the length or gravity and see if the period gets longer or shorter!

The Science Behind the Swing: Physics Basics#

Let’s dive into why the pendulum moves the way it does. We’ll start with simple ideas and build up using basic equations. Imagine a ball tied to a string hanging from a hook—that’s our pendulum. When you pull it to the side and let go, gravity pulls it back down, and it swings past the middle and up the other side, then repeats.

Fundamental Forces: Gravity and Tension

Everything starts with forces, which are pushes or pulls. The two main forces here are:

  • Gravity: Pulls the bob downward with force \(F_g = m \times g\), where \(m\) is mass and \(g\) is gravity’s strength. This is a basic equation: force equals mass times acceleration.

  • Tension (\(T\)): The string pulls the bob upward toward the hook. It keeps the bob from falling straight down.

When the pendulum is tilted at an angle \(θ\), gravity can be split into two parts:

  1. One part along the string: \(mg \cos(θ)\) (this balances some tension).

  2. One part sideways (tangential): \(mg \sin(θ)\) (this makes it swing back).

The sideways part is the “restoring force” because it always tries to straighten the pendulum out.

Building the Motion Equation

To understand the swing, we think about rotation around the hook. We use torque, which is like a twisting force. The torque from the sideways gravity is:

\[τ = -L \times (mg \sin(θ))\]

The negative sign means it twists back toward the center. (L is the string length.)

For rotation, Newton’s second law says torque equals moment of inertia times angular acceleration: \(τ = I \times α\). Here:

  • \(I = m L^2\) (moment of inertia for a point mass at distance L).

  • \(α\) is how fast the angle speeds up or slows down.

Plug in the torque:

\[-L m g \sin(θ) = m L^2 \times α\]

Now, simplify by dividing both sides by \(m L^2\) (assuming m and L aren’t zero):

\[α = -\frac{g}{L} \sin(θ)\]

This equation shows the angular acceleration depends on gravity, length, and the sine of the angle.

Simplifying for Small Angles

For small tilts (like less than 15 degrees), \(sin(θ)\) is almost equal to \(θ\) (if \(θ\) is in radians). So we approximate:

\[α ≈ -\frac{g}{L} θ\]

This looks like simple harmonic motion (SHM), where acceleration is opposite and proportional to position. In SHM, things oscillate regularly, like a perfect swing.

Finding the Period: How Long for One Swing?

In SHM, the time for one full cycle (period \(T\)) comes from the angular frequency \(ω\), where \(ω^2\) matches the constant in the equation. From above, \(ω^2 = \frac{g}{L}\), so:

\[ω = \sqrt{\frac{g}{L}}\]

Then, \(T = \frac{2π}{ω}\). Plugging in:

\[T = 2π \sqrt{\frac{L}{g}}\]

We derived this by starting from forces, getting to acceleration, approximating for small angles, and using the SHM formula. Notice: \(T\) doesn’t depend on mass or starting angle (for small angles)!

Energy: It Swings Without Stopping (Almost)

Energy is another way to see the motion. The bob has:

  • Potential Energy (\(U\)): Higher when up high. If zero at the bottom, \(U = m g h\), where \(h = L (1 - \cos(θ))\).

  • Kinetic Energy (\(K\)): From speed, \(K = \frac{1}{2} m v^2\) (v is speed).

Total energy \(E = K + U\) stays the same if no friction. At the top (max \(θ\)), speed is zero, so \(E = m g L (1 - \cos(θ_{max}))\). At the bottom, all potential turns to kinetic.

This conservation explains why it keeps swinging—energy just swaps forms!

What to Watch For

For small starting angles, swings are regular and match our equations. For big angles, the motion is less perfect, and the period gets longer than predicted. Play with the simulation to spot these differences!

Fun Ways to Learn with This Simulation#

This app is great for discovering physics:

  • See what makes motion “harmonic” (regular back-and-forth).

  • Test the period formula by measuring \(T\) and comparing to calculations.

  • Watch energy swap between potential and kinetic—add them up and see they stay constant!

  • Push the limits: Try huge angles and see when our simple equations don’t fit anymore.

If Something Goes Wrong: Troubleshooting#

Note

Swings taking too long or acting weird? Hit Reset to start fresh.

Make sure your settings make sense (like positive lengths and gravity). If the app freezes, reset or reload the page. Have fun exploring!