Bug-Rivet Paradox Calculator
Explore the famous relativity paradox demonstrating length contraction and the relativity of simultaneity
The length of the rivet in its rest frame
The length of the hole in its rest frame
Velocity of the rivet relative to the hole (must be less than 299,792,458 m/s)
How to Use This Calculator
Enter the Rest Length of the Rivet
Input the length of the rivet as measured when it's at rest (in meters). This is typically longer than the hole to create the paradox.
Enter the Length of the Hole
Input the length of the hole in its rest frame (in meters). For the paradox, this should be shorter than the rivet's rest length.
Enter the Velocity
Input the velocity of the rivet relative to the hole in meters per second. Must be less than the speed of light (299,792,458 m/s).
Calculate and Interpret Results
Click "Calculate Paradox" to see how length contraction affects the measurements from both reference frames. The results will show whether the rivet fits from each perspective.
Formula
Length Contraction: L = L₀ / γ
Lorentz Factor: γ = 1 / √(1 - v²/c²)
where L₀ is the rest length, v is velocity, and c is the speed of light
Example 1: Moderate Speed
Given: Rivet rest length = 0.1 m, Hole length = 0.08 m, Velocity = 200,000,000 m/s (66.7% c)
γ = 1 / √(1 - 0.667²) = 1.342
Contracted rivet length (from hole's frame) = 0.1 / 1.342 = 0.0745 m
From hole's frame: 0.0745 m < 0.08 m, so rivet fits!
Contracted hole length (from rivet's frame) = 0.08 / 1.342 = 0.0596 m
From rivet's frame: 0.1 m > 0.0596 m, so rivet does not fit!
Example 2: High Speed
Given: Rivet rest length = 0.2 m, Hole length = 0.1 m, Velocity = 260,000,000 m/s (86.7% c)
γ = 1 / √(1 - 0.867²) = 2.028
Contracted rivet length = 0.2 / 2.028 = 0.0986 m
From hole's frame: 0.0986 m < 0.1 m, so rivet fits!
Contracted hole length = 0.1 / 2.028 = 0.0493 m
From rivet's frame: 0.2 m > 0.0493 m, so rivet does not fit!
About the Bug-Rivet Paradox
The Bug-Rivet Paradox (also known as the Rivet Paradox) is a variation of the Barn-Pole Paradox that demonstrates the same fundamental principles of special relativity: length contraction and the relativity of simultaneity. It involves a rivet that appears to fit through a hole from one reference frame but not from another.
The Paradox Scenario
Imagine a rivet with rest length L₀ that is longer than a hole with length L. When the rivet moves at relativistic speeds toward the hole, length contraction causes it to appear shorter from the hole's reference frame. From the hole's perspective, the rivet might fit entirely through. However, from the rivet's reference frame, it's the hole that appears contracted, making it seem impossible for the rivet to fit.
When to Use This Calculator
- Physics Education: Teaching students about special relativity and length contraction
- Understanding Relativity: Visualizing how measurements depend on the observer's frame
- Paradox Resolution: Exploring how simultaneity resolves apparent contradictions
- Relativistic Calculations: Calculating Lorentz factors and contracted lengths
- Research: Analyzing relativistic scenarios in physics research
Why Use Our Calculator?
- ✅ Instant Results: Calculate length contraction and Lorentz factors immediately
- ✅ Dual Perspectives: See measurements from both reference frames simultaneously
- ✅ Educational: Clear explanations of the physics behind the paradox
- ✅ 100% Free: No registration or payment required
- ✅ Accurate: Uses precise relativistic formulas
- ✅ Interactive: Experiment with different velocities and lengths
The Resolution
Like the Barn-Pole Paradox, this paradox is resolved by understanding that simultaneity is relative. Events that occur simultaneously in one reference frame do not occur simultaneously in another frame. From the hole's frame, the rivet can fit through. From the rivet's frame, the timing of events at different positions is different, allowing the paradox to be resolved without contradiction.
Common Applications
Teaching Special Relativity: The Bug-Rivet Paradox is a compact version of the Barn-Pole Paradox, useful for illustrating length contraction in physics courses.
Understanding Relativistic Effects: This calculator helps visualize how objects appear shorter when moving at relativistic speeds, which is crucial for understanding particle physics and high-energy experiments.
Research Applications: Scientists use similar calculations when studying relativistic particles and cosmic rays.
Tips for Best Results
- Use velocities close to but less than the speed of light to see significant effects
- Make the rivet's rest length longer than the hole to create the paradox
- Try different velocities to see how the Lorentz factor changes
- Compare results from both frames to understand the relativity of measurements
- Remember that velocities must always be less than 299,792,458 m/s
Frequently Asked Questions
What is the Bug-Rivet Paradox?
The Bug-Rivet Paradox is a thought experiment in special relativity that demonstrates how a rivet can appear to fit through a hole from one reference frame while appearing too long from another frame. It's a variation of the Barn-Pole Paradox that illustrates the same principles.
How does it differ from the Barn-Pole Paradox?
The Bug-Rivet Paradox is essentially the same as the Barn-Pole Paradox but uses a rivet and hole instead of a pole and barn. The physics and resolution are identical—both demonstrate length contraction and the relativity of simultaneity.
Why is it called the "Bug-Rivet" Paradox?
The name comes from the original formulation where a bug sits on the rivet. Some versions of the paradox include a bug that gets crushed or doesn't, depending on the frame, adding another layer to illustrate the relativity of simultaneity.
How is the paradox resolved?
The paradox is resolved by recognizing that simultaneity is relative. Events that are simultaneous in one frame are not simultaneous in another frame. This allows both perspectives to be correct without contradiction.
What practical applications does this have?
While this is primarily a teaching tool, the underlying principles of length contraction are crucial for understanding particle physics, cosmic rays, GPS systems, and any system involving relativistic speeds.
Can I use this for velocities at or above the speed of light?
No. According to special relativity, nothing with mass can travel at or above the speed of light. The calculator will reject velocities ≥ c because they would result in undefined or imaginary values for the Lorentz factor.