Calculator Inputs
Example Data Table
| Mode | Rest Mass (kg) | Amplitude (m) | Input Parameter | vmax/c | γmax | γavg | Approx Period Shift (%) |
|---|---|---|---|---|---|---|---|
| Spring constant | 0.001 | 0.002 | k = 1.0e18 N/m | 0.210964 | 1.023024 | 1.011411 | 0.568937 |
| Spring constant | 0.005 | 0.003 | k = 5.0e17 N/m | 0.100069 | 1.005045 | 1.002517 | 0.125773 |
| Angular frequency | 0.002 | 0.0015 | ω = 5.0e10 rad/s | 0.250173 | 1.032843 | 1.016219 | 0.807667 |
These examples are intentionally high-speed teaching cases. Everyday laboratory oscillators usually produce much smaller relativistic shifts.
Formula Used
Classical harmonic motion
Displacement: x(t) = A cos(ωt)
Velocity: v(t) = -Aω sin(ωt)
Angular frequency from stiffness: ω = √(k / m₀)
Frequency relation: ω = 2πf
Classical period: T₀ = 2π / ω
Relativistic terms
Speed ratio: β(t) = v(t) / c
Lorentz factor: γ(t) = 1 / √(1 - β(t)²)
Teaching convention for relativistic mass: m_rel(t) = γ(t)m₀
Peak relativistic kinetic energy: K_rel,max = (γ_max - 1)m₀c²
Approx corrected period: T_rel ≈ T₀√(γ_avg)
This approximation is useful for comparison work. Exact relativistic oscillator dynamics need a full relativistic equation of motion, not only an effective mass substitution.
How to Use This Calculator
- Select the input mode that matches your known parameter.
- Enter rest mass and amplitude using the desired units.
- Provide spring constant, frequency, or angular frequency.
- Choose graph cycles, plot points, and decimal precision.
- Press Calculate to show the result block above the form.
- Review peak gamma, peak relativistic mass, and period shift.
- Use the chart to compare displacement, speed ratio, and mass increase.
- Export the computed report with the CSV or PDF buttons.
Frequently Asked Questions
1. What does this calculator estimate?
It estimates how high-speed harmonic motion changes Lorentz factor, relativistic mass, energy, and oscillator timing. It also compares classical and approximate relativistic behavior on one page.
2. Why does the page use relativistic mass?
The calculator follows a common teaching convention because your requested topic is mass increase. It labels the method clearly and also reports gamma directly, which is the more fundamental quantity.
3. Why can the period increase?
As speed rises, gamma increases. The effective inertia rises with gamma, so oscillation can slow slightly. This page models that effect through an average-gamma correction to the classical period.
4. Which input mode should I choose?
Use spring mode when stiffness is known. Use frequency mode when ordinary frequency is measured. Use angular frequency mode when your source already gives values in radians per second.
5. What happens if maximum speed reaches light speed?
The page stops the calculation and shows an error. Harmonic motion with classical sinusoidal speed cannot exceed light speed, so the entered combination must be reduced first.
6. Are the results exact for relativistic oscillators?
No. They are practical approximations for comparison and teaching. Exact relativistic oscillators require solving the full relativistic equation of motion with a consistent force model.
7. Why show both classical and relativistic kinetic energy?
The side-by-side comparison reveals how strongly classical energy underestimates motion when speed becomes a meaningful fraction of light speed. That makes the shift easier to interpret.
8. What units work best here?
SI units are easiest to interpret. The page still supports common mass, length, frequency, and angular-frequency choices, then converts them internally before performing every calculation.