Damped Harmonic Oscillator Calculator

About the Damped Harmonic Oscillator Calculator

The Damped Harmonic Oscillator Calculator computes the angular frequency, period, frequency, or amplitude decay of a damped harmonic oscillator using the formulas \( \omega = \sqrt{\omega_0^2 - \gamma^2} \), \( T = \frac{2\pi}{\omega} \), \( f = \frac{\omega}{2\pi} \), and \( A(t) = A_0 e^{-\gamma t} \), where \( \omega_0 = \sqrt{\frac{k}{m}} \), \( \gamma = \frac{b}{2m} \). This tool is ideal for physics students and educators studying oscillatory motion with damping.

• Features:
  • Calculates angular frequency, period, frequency, or amplitude decay based on mass, spring constant, damping coefficient, initial amplitude, and time.
  • Supports unit conversions for mass (kg, g) and amplitude (m, cm).
  • Validates inputs: positive mass, spring constant; non-negative damping coefficient, amplitude, time.
  • Identifies damping type: underdamped (\( \omega_0 > \gamma \)), critically damped (\( \omega_0 = \gamma \)), or overdamped (\( \omega_0 < \gamma \)).
  • Keypad includes digits, decimal point, pi constant, and typical spring constant (100 N/m).
  • Clear, backspace, and copy functionality.
• Practical Applications: Useful in physics education, engineering, and experimental physics for analyzing damped spring-mass systems, vibrations, and oscillatory motion with energy dissipation.
• How to Use:
  • Enter the mass (e.g., 1) and select unit (kg, g).
  • Enter the spring constant (e.g., 100 N/m).
  • Enter the damping coefficient (e.g., 0.5 kg/s).
  • Enter the initial amplitude (e.g., 0.1) and select unit (m, cm).
  • Enter the time (e.g., 1 s) for amplitude decay calculation.
  • Select calculation type: Angular Frequency, Period, Frequency, or Amplitude at Time t.
  • Use the keypad to insert digits, decimal point, pi, or typical spring constant (k).
  • Click "Calculate" to compute the result and view steps, then use "Copy" to copy the result.
  • Use "Clear" to reset, or "⌫" to delete the last character.
  • Share or embed the calculator using the action buttons.
• Helpful Tips:
  • Ensure mass and spring constant are positive; damping coefficient, amplitude, and time are non-negative.
  • Use realistic values (e.g., spring constant 10–1000 N/m, damping coefficient 0–10 kg/s).
  • Angular frequency and period/frequency calculations assume underdamped motion (\( \omega_0 > \gamma \)).
  • Formulas assume linear damping and ideal conditions.
• Examples:
  • Example 1: Angular Frequency:
    • Input: Mass = 1 kg, Spring Constant = 100 N/m, Damping Coefficient = 2 kg/s
    • Steps: \( \omega_0 = \sqrt{\frac{k}{m}} = \sqrt{\frac{100}{1}} = 10 \, \text{rad/s} \); \( \gamma = \frac{b}{2m} = \frac{2}{2 \cdot 1} = 1 \, \text{s}^{-1} \); \( \omega = \sqrt{\omega_0^2 - \gamma^2} = \sqrt{10^2 - 1^2} \approx 9.95 \, \text{rad/s} \)
    • Result: Angular Frequency ≈ 9.95 rad/s (Underdamped)
  • Example 2: Period:
    • Input: Mass = 500 g, Spring Constant = 200 N/m, Damping Coefficient = 1 kg/s
    • Steps: Convert 500 g to 0.5 kg; \( \omega_0 = \sqrt{\frac{200}{0.5}} \approx 20 \, \text{rad/s} \); \( \gamma = \frac{1}{2 \cdot 0.5} = 1 \, \text{s}^{-1} \); \( \omega = \sqrt{20^2 - 1^2} \approx 19.97 \, \text{rad/s} \); \( T = \frac{2\pi}{\omega} \approx 0.314 \, \text{s} \)
    • Result: Period ≈ 0.31 s
  • Example 3: Amplitude Decay:
    • Input: Mass = 2 kg, Spring Constant = 50 N/m, Damping Coefficient = 4 kg/s, Initial Amplitude = 20 cm, Time = 1 s
    • Steps: Convert 20 cm to 0.2 m; \( \gamma = \frac{4}{2 \cdot 2} = 1 \, \text{s}^{-1} \); \( A(t) = A_0 e^{-\gamma t} = 0.2 \cdot e^{-1 \cdot 1} \approx 0.0736 \, \text{m} \)
    • Result: Amplitude at t = 1 s ≈ 7.36 cm

Calculate the properties of a damped harmonic oscillator with detailed steps using this calculator. Share or embed it on your site!