Moveable Pendulum - Frames of Reference Physics Simulation

Executive Summary

The Moveable Pendulum simulator tackles one of physics’ most conceptually challenging topics—reference frames and relative motion—through an elegantly simple yet profoundly instructive demonstration. By allowing you to drag the pivot point of a swinging pendulum, this tool reveals how the same physical motion appears completely different depending on your frame of reference.

The Power of Perspective

In Einstein’s physics, no absolute frame of reference exists. Motion is inherently relative—we can only describe how objects move relative to other objects or chosen reference frames. A passenger sitting still in a moving train is stationary in the train’s reference frame but moving at high speed in the ground’s reference frame. Both descriptions are valid; neither is “more correct” than the other.

This relativity extends beyond simple velocity. When reference frames accelerate (non-inertial frames), observers in those frames experience fictitious forces—forces with no identifiable source that arise purely from the frame’s acceleration. The centrifugal force you feel in a turning car or the Coriolis effect deflecting projectiles on Earth’s rotating surface are fictitious forces, real in effect but absent from inertial frames.

The Moveable Pendulum makes these abstract concepts concrete. Watch a pendulum swing in its natural path. Now grab and drag the pivot point—suddenly the pendulum’s motion transforms as you’ve shifted to a moving reference frame. Release the pivot and drag it periodically at the pendulum’s natural frequency—you can pump energy into the system, making the pendulum swing higher, just as you pump a playground swing by shifting your weight rhythmically.

Core Capabilities

• Interactive Pivot Control: Click and drag the pendulum’s pivot point anywhere on the screen in real-time
• Multiple Reference Frame Views: Toggle between ground frame (stationary observer) and pivot frame (moving with the pivot) perspectives
• Fictitious Force Visualization: Display arrows showing fictitious forces (centrifugal, Coriolis) experienced in accelerating reference frames
• Energy Analysis: Track how moving the pivot transfers energy into or out of the pendulum system
• Motion Tracing: Leave trails showing pendulum paths in different reference frames for direct comparison
• Customizable Parameters: Adjust pendulum length, mass, gravity, and damping to explore various scenarios
• Split-Screen Comparison: View ground and pivot frames simultaneously to observe the same motion from different perspectives
• Zero Installation: Browser-based physics simulation running entirely client-side with complete privacy

Educational Value

Physics Students studying classical mechanics encounter reference frames in textbooks but often struggle to visualize how frame choice affects motion descriptions. This simulator provides the missing experiential component, transforming abstract mathematics into tangible understanding.

Educators use the Moveable Pendulum to demonstrate principles difficult or impossible to show with physical apparatus. Smoothly accelerating a real pendulum’s pivot while minimizing external disturbances requires sophisticated equipment; here, it’s as simple as dragging with your mouse.

Engineering Students learning dynamics and vibration theory see practical applications of reference frame transformations—essential skills for analyzing mechanisms, vehicles, robotics, and rotating machinery.

Curious Minds fascinated by relativity (special or general), gyroscopic effects, or how things behave in accelerating environments (elevators, vehicles, rotating space stations) gain intuitive foundations that demystify these phenomena.

The Moveable Pendulum exemplifies how interactive simulations can illuminate concepts that static diagrams and equations struggle to convey. Motion is inherently dynamic; understanding it requires dynamic tools that let you manipulate, observe, and explore from multiple perspectives.

Feature Tour

Basic Pendulum Controls

Initial Setup: The simulation begins with a pendulum hanging vertically at rest from a stationary pivot point. Click and drag the pendulum mass to pull it away from equilibrium. Release to start swinging. Standard pendulum behavior emerges—back and forth oscillation with period determined by length and gravity (T = 2π√(L/g)).

Damping and Friction: Toggle air resistance to observe motion with or without energy dissipation. Without damping, the pendulum swings indefinitely at constant amplitude. With damping, amplitude gradually decreases until the pendulum hangs at rest. Adjust damping coefficient from 0 (frictionless) to 0.5 (heavily damped) to explore how dissipation affects motion.

Parameter Customization: Set pendulum length from 0.5 to 3 meters, mass from 0.1 to 5 kg, and gravitational acceleration from 1 to 15 m/s². These controls enable exploration of different scenarios—pendulums on the Moon (g = 1.62 m/s²), Earth (9.81 m/s²), or Jupiter (24.79 m/s²).

Pivot Movement Controls

Free Dragging: The key innovation—click the pivot point and drag it freely across the screen. While dragging, the pendulum responds as its support point moves, creating complex motion patterns. Drag smoothly along horizontal lines, circular paths, or erratic trajectories to observe different effects.

Controlled Motion Profiles: Select from predefined pivot motion patterns:

• Horizontal Oscillation: Pivot moves sinusoidally left-right, simulating a pendulum on a moving cart
• Circular Motion: Pivot follows circular path at constant angular velocity
• Vertical Motion: Pivot oscillates up-down, modulating effective gravity
• Custom Paths: Record your own mouse motion for repeatable experiments

Speed and Amplitude Control: For predefined motions, adjust the frequency and amplitude of pivot movement. Match the frequency to the pendulum’s natural frequency to observe resonance—dramatic amplitude growth as energy pumps into the system efficiently.

Reference Frame Visualization

Ground Frame View: The default perspective shows both pivot and pendulum motion relative to a stationary background. This represents an inertial reference frame—no fictitious forces appear, and Newton’s laws apply in their standard form.

Pivot Frame View: Switch to a frame moving with the pivot. In this view, the pivot appears stationary at the center while the background scrolls. The pendulum’s motion in this non-inertial frame looks different from the ground frame perspective. When the pivot accelerates, fictitious forces appear to explain the pendulum’s motion from this frame’s perspective.

Split-Screen Mode: Display both frames simultaneously side-by-side. Watch identical physics produce completely different visual motion descriptions. This dual view powerfully demonstrates that “what motion looks like” depends fundamentally on frame choice.

Force Vector Display: Enable visualization of forces acting on the pendulum:

• Gravity: Constant downward force (green arrow)
• Tension: Force from the rod/string supporting the pendulum (blue arrow)
• Fictitious Forces (pivot frame only): Centrifugal force (red) and Coriolis force (purple) arising from frame acceleration

Energy Tracking and Analysis

Energy Graphs: Real-time plots display:

• Kinetic Energy: Motion energy (½mv²)
• Potential Energy: Gravitational energy (mgh)
• Total Mechanical Energy: Sum of kinetic and potential

In the ground frame with stationary pivot, total energy remains constant (no damping) or decreases (with damping). When you move the pivot, you do work on the system, adding or removing energy. The energy graphs reveal this transfer quantitatively.

Power Measurement: Display instantaneous power (rate of energy transfer) as you drag the pivot. Positive power means you’re adding energy (pendulum amplitude increases); negative means you’re extracting energy (amplitude decreases). Zero average power corresponds to non-resonant driving where energy alternately flows in and out.

Work Calculation: The simulation integrates power over time to calculate total work done by pivot motion. This feature demonstrates the work-energy theorem: the change in pendulum energy equals the work you performed by moving the pivot.

Motion Tracing and Comparison

Trajectory Trails: Enable path tracing to draw the pendulum mass’s trajectory. In the ground frame, this shows the actual path through space—often complex curves when the pivot moves. In the pivot frame, the path appears different, reflecting the frame transformation.

Multi-Trial Overlay: Run the same pivot motion multiple times with different initial pendulum positions or parameters. Overlay the resulting paths with different colors to compare outcomes directly.

Lissajous Figures: When pivot motion combines perpendicular oscillations (e.g., horizontal and vertical sinusoidal motions at different frequencies), the pendulum traces Lissajous figures—beautiful geometric patterns revealing the frequency relationships.

Advanced Features

Phase Space Plots: For advanced users, display pendulum state in phase space (angle vs. angular velocity). Stationary pivot produces closed ellipses (periodic motion). Moving pivot creates more complex phase portraits revealing resonances, stability boundaries, and chaotic transitions.

Resonance Detection: The simulation automatically identifies when pivot motion frequency matches pendulum natural frequency (or integer multiples/fractions thereof). Visual indicators highlight resonance conditions where small driving amplitudes produce large responses.

Data Export: Save time series data (position, velocity, energy, forces) as CSV files for external analysis in Python, MATLAB, or spreadsheet software. Export rendered animations as MP4 videos for presentations or educational materials.

Preset Scenarios: Load predefined demonstrations:

• Playground Swing: Pivot oscillates horizontally at pendulum frequency, simulating pumping action
• Foucault Pendulum: Slow rotation demonstrating Coriolis effect
• Elevator Pendulum: Vertical pivot acceleration modifying apparent gravity
• Coupled Oscillators: Compare with the Interactive Pendulum Lab for contrast

Usage Scenarios

Scenario 1: Understanding Playground Swings

Emma wonders how she makes playground swings go higher by shifting her weight rhythmically. Her physics teacher suggests exploring the Moveable Pendulum:

1. Emma launches the pendulum swinging with small amplitude
2. She drags the pivot point back and forth horizontally, timing her motion to match the pendulum’s swing
3. When she moves the pivot forward as the pendulum swings forward (and backward as it swings backward), the amplitude grows dramatically
4. When she reverses this timing (pivot forward as pendulum swings backward), amplitude decreases
5. Switching to the energy graph, she sees energy steadily increasing with matched timing (resonance) and decreasing with opposite timing

Emma realizes playground swinging works by doing work on the system at its natural frequency—transferring energy efficiently through resonant driving. The Moveable Pendulum made this abstract principle tangible.

Scenario 2: Demonstrating Non-Inertial Reference Frames

Professor Lee teaches classical mechanics and wants to illustrate fictitious forces. He uses the Moveable Pendulum in lecture:

1. Projects the simulator in split-screen mode (ground frame left, pivot frame right)
2. Starts the pendulum swinging in both views
3. Drags the pivot steadily to the right while students watch both frames
4. In the ground frame, the pendulum continues swinging with slight distortion
5. In the pivot frame, the pendulum swings differently and appears to experience an additional leftward force
6. Professor Lee enables force visualization in the pivot frame, revealing a fictitious force (centrifugal/inertial force) pointing left
7. He explains this force has no source—it arises because the pivot frame is accelerating

Students visually grasp that fictitious forces aren’t physical pushes but consequences of observing motion from accelerating frames. The side-by-side comparison makes the concept clear in a way equations alone cannot.

Scenario 3: Engineering Student Analyzing Vibration Isolation

Marcus studies mechanical vibration and needs to understand how oscillating bases affect pendulum-like systems (relevant to seismometers, vehicle suspensions, and building dynamics during earthquakes):

1. He models an earthquake as horizontal sinusoidal pivot motion with 2 Hz frequency and 5 cm amplitude
2. Tests pendulum responses with different natural frequencies (by varying length)
3. Discovers that when pendulum natural frequency matches driving frequency (2 Hz), amplitude becomes very large (resonance)
4. When pendulum natural frequency differs significantly from driving, response amplitude stays small
5. Exports data and creates amplitude-vs-frequency plots (transfer functions) showing resonance peaks

Marcus applies this understanding to design pendulum systems with natural frequencies far from expected disturbance frequencies, minimizing unwanted vibrations—exactly how engineers design vibration isolation systems.

Scenario 4: Physics Enthusiast Exploring the Coriolis Effect

Aiden read about the Coriolis effect causing hurricanes to rotate and wondered how it works. He uses the Moveable Pendulum with slow circular pivot motion to simulate a rotating reference frame:

1. Sets the pivot to rotate slowly in a circle (simulating Earth’s rotation, greatly exaggerated)
2. Launches the pendulum swinging in a straight line (from ground frame perspective)
3. Switches to pivot frame view, where the background rotates
4. Observes that in this rotating frame, the pendulum’s path curves—it doesn’t swing back and forth in a straight line
5. Enables Coriolis force visualization, seeing purple arrows perpendicular to the pendulum’s velocity
6. Recognizes this deflection in rotating frames explains why hurricanes spiral and why projectiles drift sideways on Earth

The simulation connects abstract Coriolis mathematics to visual, interactive experience.

Scenario 5: Comparing Inertial and Non-Inertial Frames

A homeschool educator teaches her student about relativity’s principle that no privileged reference frame exists:

1. They launch the pendulum and drag the pivot in complex patterns
2. Switch between ground and pivot frames, observing the drastically different motion descriptions
3. Discuss: Which frame is “correct”? The educator emphasizes both are equally valid
4. In the ground frame, Newton’s laws work straightforwardly (F = ma with just gravity and tension)
5. In the pivot frame, Newton’s laws still work but require adding fictitious forces to account for the frame’s acceleration
6. The key lesson: Physics laws work in any frame, but you must account for that frame’s motion

This exploration builds foundational understanding for later studying Einstein’s general relativity, where gravity itself becomes a fictitious force in appropriately chosen frames.

Code Examples and Technical Integration

URL Parameters for Specific Scenarios

Direct link to predefined configurations using URL parameters:

https://gray-wolf.tools/tools/education/moveable-pendulum?length=1.5&mass=1&theta0=45&pivotMotion=horizontal&frequency=0.5&amplitude=0.3

Parameters:

• length: Pendulum length in meters
• mass: Pendulum mass in kg
• theta0: Initial angle in degrees
• pivotMotion: Type (horizontal, vertical, circular, custom)
• frequency: Driving frequency in Hz
• amplitude: Driving amplitude in meters
• damping: Air resistance coefficient
• frame: View mode (ground, pivot, split)

Embedding in Educational Platforms

Integrate into learning management systems:

<iframe 
  src="https://gray-wolf.tools/tools/education/moveable-pendulum?embed=true&frame=split&pivotMotion=horizontal"
  width="900" 
  height="600"
  frameborder="0"
  title="Moveable Pendulum Reference Frame Demonstration"
  allow="accelerometer">
</iframe>

Exported Data Format

CSV export provides comprehensive state information:

time,pivot_x,pivot_y,theta,omega,pendulum_x,pendulum_y,KE,PE,total_energy,fictitious_force_x,fictitious_force_y
0.00,0.00,0.00,0.785,0.000,1.061,-1.061,0.000,5.895,5.895,0.000,0.000
0.02,0.012,0.000,0.779,0.288,1.073,-1.055,0.041,5.849,5.890,0.250,0.000
0.04,0.048,0.000,0.761,0.565,1.108,-1.038,0.160,5.716,5.876,0.485,0.143
...

Columns include pivot position, pendulum angle, angular velocity, Cartesian coordinates, energies, and fictitious forces (in pivot frame).

Accessibility Features

WCAG 2.1 compliance ensures broad usability:

// Example ARIA implementation
<button 
  aria-label="Drag pivot point to move reference frame"
  aria-pressed="false"
  role="button">
  Enable Pivot Dragging
</button>

<div 
  role="img" 
  aria-label="Pendulum at 45 degrees with pivot moving horizontally at 0.5 meters per second">
  <canvas id="simulation-canvas"></canvas>
</div>

Keyboard Controls:

• Arrow keys: Move pivot in small increments
• Space: Start/stop predefined pivot motion
• Tab: Navigate between controls
• Enter: Activate buttons and toggles

Screen Reader Announcements: State changes, parameter values, and force magnitudes announced for visually impaired users.

Troubleshooting

Pivot Won’t Drag or Responds Erratically

Symptoms: Clicking pivot doesn’t enable dragging, or cursor and pivot position don’t align.

Solutions:

• Ensure “Enable Pivot Dragging” mode is activated (check button state)
• Verify mouse/trackpad is functioning properly (test in other applications)
• Try different browser—some older browsers have canvas interaction issues
• Disable browser extensions that might interfere with mouse events
• On touch devices, use long-press then drag gesture
• Check that simulation isn’t paused (pivot won’t move while paused unless in step mode)

Forces Don’t Appear in Pivot Frame

Symptoms: Switching to pivot frame doesn’t show fictitious forces despite pivot motion.

Solutions:

• Verify “Show Forces” visualization is enabled (check toggle button)
• Ensure pivot is actually accelerating—constant velocity motion produces no fictitious forces
• Fictitious forces may be small for slow pivot motions—increase amplitude or acceleration
• Check that you’re viewing the pivot frame, not ground frame (fictitious forces only appear in non-inertial frames)
• Refresh page if visualization rendering has glitched

Energy Appears to Violate Conservation

Symptoms: Total energy increases or decreases when it shouldn’t.

Solutions:

• Energy should change when you move the pivot—you’re doing work on the system, adding or removing energy
• In ground frame with stationary pivot and no damping, energy should remain constant—if it doesn’t, report as potential numerical error
• With damping enabled, energy should decrease due to friction—this is correct physics
• Very large or very fast pivot motions can cause numerical instability—try smaller amplitudes or slower speeds
• Ensure you’re viewing the correct energy graph (ground frame vs. pivot frame energy accounting differs)

Split-Screen Mode Looks Confusing or Misaligned

Symptoms: Two frames show motion that seems contradictory or doesn’t align properly.

Solutions:

• This is expected! The two frames show the same physics from different perspectives—they should look different
• Verify that you understand which frame is which (usually labeled “Ground Frame” and “Pivot Frame”)
• Enable force vectors to see how fictitious forces in pivot frame explain the different-looking motion
• Start with simple cases: stationary pivot (frames identical), then gentle horizontal motion (frames differ slightly)
• Use motion tracing to see path differences clearly
• Consult the knowledge article for conceptual explanation of reference frames

Mobile Device Performance Issues

Symptoms: Laggy dragging, slow frame rates, or touch controls don’t work smoothly.

Solutions:

• Disable split-screen mode to reduce rendering load
• Turn off motion trails and force vectors
• Reduce pendulum length or mass (simpler physics calculations)
• Use single frame view instead of simultaneous multi-frame rendering
• Ensure device isn’t running many background apps
• Consider using desktop/laptop for smoothest experience, especially for precision experiments

Cannot Achieve Resonance/Amplitude Won’t Grow

Symptoms: Attempting to pump energy by moving pivot, but amplitude doesn’t increase.

Solutions:

• Verify timing: pivot motion frequency must match pendulum natural frequency (f = 1/(2π)√(g/L))
• Check phase: move pivot in sync with pendulum motion (forward when pendulum swings forward)
• Ensure damping isn’t too high—excessive friction dissipates energy faster than you can add it
• Try automated horizontal oscillation mode with frequency set to calculated natural frequency
• Increase pivot motion amplitude if it’s too small to transfer significant energy
• Be patient—resonance builds gradually; amplitude grows over multiple cycles

Frequently Asked Questions

What are reference frames and why do they matter?

A reference frame is a perspective from which you observe and measure motion. You might describe a car’s speed relative to the ground (ground frame) or relative to another car (car’s frame). Physics recognizes that motion is inherently relative—there’s no absolute “correct” frame. However, different frames have different properties. Inertial frames (not accelerating) allow Newton’s laws to work in their simplest form. Non-inertial frames (accelerating) require adding fictitious forces. Understanding reference frames is essential for analyzing motion in vehicles, rotating systems, and ultimately for grasping relativity theory.

What are fictitious forces and are they “real”?

Fictitious forces (also called inertial forces or pseudo-forces) arise when observing motion from accelerating reference frames. Common examples include centrifugal force (felt when turning in a car) and Coriolis force (causes hurricanes to rotate). They’re called “fictitious” because they have no physical source—no object pushing or pulling. They arise from the reference frame’s acceleration. However, their effects are completely real if you’re in that frame. A passenger in a turning car genuinely experiences outward force. So fictitious forces are “real” in effect but “fictitious” in that they vanish when you switch to an inertial frame.

How is this different from the regular Interactive Pendulum Lab?

The Interactive Pendulum Lab focuses on pendulum motion with a fixed pivot, exploring energy conservation, damping, and simple harmonic motion. The Moveable Pendulum adds the ability to drag the pivot, introducing reference frame physics and energy transfer. Use the Interactive Pendulum Lab to understand basic pendulum physics, then advance to the Moveable Pendulum to explore how motion depends on reference frame choice and how driving oscillators transfers energy. They complement each other, covering different aspects of pendulum physics.

Can this simulate the Foucault pendulum that shows Earth’s rotation?

Yes, with limitations. A true Foucault pendulum swings freely while Earth rotates beneath it, causing the swing plane to appear to rotate slowly. You can approximate this by setting circular pivot motion with very long period (many hours) to simulate Earth’s rotation. However, the simulation uses a 2D model, whereas Foucault pendulums require 3D geometry and Coriolis forces from rotation about a vertical axis. The Moveable Pendulum can demonstrate the concept and show how rotating frames produce deflection, but it’s a simplified model rather than an exact Foucault simulation.

Why does moving the pivot at the right frequency make the pendulum swing higher?

This is resonance—a fundamental physics phenomenon where driving a system at its natural frequency efficiently transfers energy. The pendulum has a natural frequency determined by its length and gravity. When you move the pivot at this frequency with correct phase (timing), each push adds energy when the pendulum is already moving in that direction, maximizing energy transfer. It’s like pushing a playground swing at the right moments—tiny pushes timed correctly produce large amplitudes. Resonance appears throughout physics: musical instruments, bridges (famously the Tacoma Narrows collapse), electrical circuits, and quantum systems.

What’s the relationship between this and Einstein’s relativity?

Classical mechanics (this simulation) deals with reference frames moving at ordinary speeds without considering light speed. Special relativity extends these concepts to high-speed frames, showing that time and space measurements depend on frame choice. General relativity goes further, treating gravity itself as a fictitious force arising from curved spacetime—similar to how centrifugal force arises from rotating frames. The Moveable Pendulum builds foundational understanding of how physics depends on frame choice, essential preparation for understanding relativity’s deeper insights. It’s not relativistic physics, but it’s excellent preparation for studying relativity.

Can I use this to understand vehicle dynamics or earthquake effects on buildings?

Absolutely! The moving pivot models situations where a pendulum-like system is attached to an accelerating base:

• Vehicles: Hanging objects in cars (dice, air fresheners) swing when the car accelerates—modeled by horizontal pivot motion
• Earthquakes: Buildings sway when their foundations shake—modeled by horizontal sinusoidal pivot motion at earthquake frequencies
• Elevators: Pendulums in elevators experience modified effective gravity—modeled by vertical pivot acceleration
• Ships: Objects hanging in ships oscillate due to wave-induced motion

Engineers use similar models (often more complex) to design suspension systems, analyze seismic building response, and understand vibration isolation.

How do I determine the pendulum’s natural frequency for resonance experiments?

The pendulum’s natural frequency (for small angles) is f = (1/(2π))√(g/L), where g is gravitational acceleration and L is length. For Earth (g ≈ 9.81 m/s²) and L = 1 meter, f ≈ 0.498 Hz (period ≈ 2 seconds). The simulator displays the natural frequency in the info panel. To achieve resonance, set horizontal pivot oscillation frequency to match this value. You’ll see amplitude grow dramatically over several cycles as energy accumulates. Deviation by even 10-20% from natural frequency significantly reduces resonance effectiveness.

Does this work for understanding gyroscopes and rotating coordinate systems?

Partially. The Moveable Pendulum demonstrates some principles of rotating frames (Coriolis forces, fictitious forces from acceleration). However, gyroscopes involve angular momentum conservation and torque in 3D, which this 2D pendulum simulation doesn’t fully capture. For gyroscopic effects, a 3D simulation is needed. Still, the conceptual foundation—how rotation creates fictitious forces and how motion looks different in rotating frames—transfers directly to understanding gyroscopes. Use this as a stepping stone; explore the Physics Simulation Lab for additional mechanics simulations.

What should I explore next after mastering this tool?

Continue your physics journey with:

• Interactive Pendulum Lab: Deep dive into pendulum physics with energy visualization and damping
• Chaotic Double Pendulum: See how adding one joint transforms predictable motion into chaos
• Physics Simulation Lab: Explore 40+ simulations covering mechanics, waves, oscillations, and more
• Interactive Projectile Motion Lab: Study 2D kinematics and trajectory physics

Read the companion reference frames knowledge article for deeper theoretical exploration.

References and Additional Resources

Related Gray-wolf Tools

• Interactive Pendulum Lab: Comprehensive pendulum simulation with energy visualization
• Chaotic Double Pendulum: Explore chaos theory through double pendulum dynamics
• Physics Simulation Lab: Broad collection of physics simulations across mechanics, waves, and oscillations
• Interactive Projectile Motion Lab: 2D kinematics and trajectory analysis

Learning Resources

• The Physics Classroom: Excellent tutorials on reference frames and relative motion (www.physicsclassroom.com)
• Khan Academy: Video lessons covering inertial vs. non-inertial frames and fictitious forces
• MIT OpenCourseWare: University-level classical mechanics courses with reference frame treatments
• Feynman Lectures on Physics: Volume I, Chapter 12 covers characteristics of force and reference frames

Academic Standards Alignment

The Moveable Pendulum supports learning objectives from:

• AP Physics C: Mechanics: Oscillations, rotating coordinate systems, non-inertial frames
• International Baccalaureate (IB) Physics: Core topics on mechanics and reference frames
• Next Generation Science Standards (NGSS): HS-PS2 (Motion and Stability: Forces and Interactions)

Citation

For academic use, cite as:

Gray-wolf Tools Team. (2025). Moveable Pendulum - Frames of Reference Physics Simulation. Gray-wolf Tools. https://gray-wolf.tools/tools/education/moveable-pendulum