Simple Harmonic Motion Calculator
Calculate displacement, velocity, acceleration, period, frequency, and energy in simple harmonic motion using mass-spring or pendulum formulas.
simple harmonic motion calculator:
Use this Simple Harmonic Motion Calculator to find displacement, velocity, acceleration, angular frequency, or energy for oscillatory systems. Enter values for amplitude, mass, spring constant, and time — the tool applies SHM formulas like \( x = \textrm{Asin}(\omega t + \Phi), \nu = \omega \sqrt{A^{2} - x^{2}}, and \alpha = - \omega^{2}x \) It instantly computes period, frequency, and total energy for both spring and pendulum systems.
simple harmonic motion tool Formula:
The Simple Harmonic Motion Calculator helps you analyze oscillatory systems such as springs, pendulums, or vibrating bodies. Using standard SHM equations, it computes displacement (x), velocity (v), acceleration (a), and angular frequency (ω), period (T), and energy (E). You can enter parameters such as mass (m), spring constant (k), amplitude (A), and phase angle (φ), and time (t). The calculator applies the fundamental relations:
- \( x = \textrm{Asin}(\omega t + \Phi) \)
- \( \nu = \omega \sqrt{A^{2} - x^{2}} \)
- \( \alpha = - \omega^{2}x \)
- \( T = 2\pi\sqrt{\frac{m}{k}} \)
- \( E = \frac{1}{2}KA^{2} \)
It supports both mass-spring and pendulum models and converts between SI and common lab units (cm, g, N/cm, Hz). Ideal for students, teachers, and engineers studying vibration or resonance, the calculator also shows step-by-step solutions and optional plots of motion vs. time.
⚡ Work & Installation Input to Output:
Input: Input: amplitude (A), mass (m), spring constant (k), phase (φ), time (t), or gravity (g for pendulum). User selects SHM type — spring or pendulum — and desired output (x, v, a, ω, T, E).
Processing:
- Determine motion type.
- For a mass-spring system: \( \omega = \sqrt{\frac{k}{m}} \)
- For a pendulum: \( \omega = \sqrt{\frac{g}{L}} \)
- Apply selected formulas for x, v, a, and energy. \( \)
- Auto-convert inputs to SI units (m, kg, s, N/m).
- Perform sanity checks for negative or zero values.
Output:
- Displacement (x)
- Velocity (v)
- Acceleration (a)
- Angular frequency (ω)
- Period (T), Frequency (f)
- Energy (KE, PE, Total)
- Step-by-step calculations with optional plot (x–t graph).
Testing and Final Adjustments
To ensure accuracy, test standard SHM cases:
- Spring system: m = 0.2 kg, k = 80 N/m → T ≈ 0.314 s.
- Pendulum: L = 1 m, g = 9.81 m/s² → T ≈ 2.01 s.
Validate trigonometric evaluations of x, v, a at various times (t = 0, T/4, T/2). Confirm that energy E = ½kA 2 remains constant, and that KE + PE = E. Test unit conversions (cm→m, g→kg) and boundary cases (A = 0, k = 0 → error). UX testing: ensure inputs are numeric, clear field labels (A, m, k, φ, t), and responsive layout. Add preset examples and tooltip hints with equations. Finally, verify graph accuracy using time vs displacement plotting and confirm correct handling of radians vs degrees. Ensure SEO metadata includes “simple harmonic motion calculator,” “oscillation,” and “spring system.”
Frequently Asked Questions - Simple Harmonic Motion Calculator:
What is simple harmonic motion?
Simple harmonic motion (SHM) is periodic oscillatory motion where acceleration is directly proportional to displacement and opposite in direction.
How do I calculate SHM displacement?
Use x = A sin(ωt + φ), where A is amplitude, ω is angular frequency, and φ is phase angle.
How is angular frequency found?
For a spring system, ω = √(k/m); for a pendulum, ω = √(g/L).
What is the SHM period formula?
The time period is T = 2π√(m/k) for a spring or T = 2π√(L/g) for a pendulum.
How do I find velocity in SHM?
Velocity is v = ω√(A² - x²), the rate of change of displacement.
What is acceleration in SHM?
Acceleration is a = -ω²x, directed toward the equilibrium position.
What is the total energy in SHM?
Total energy E = ½kA² remains constant; it converts between potential and kinetic forms.
Can SHM calculator handle pendulum motion?
Yes, select the pendulum mode to calculate period and frequency using length and gravity.
Does amplitude affect frequency?
For ideal SHM, frequency depends only on system parameters (k, m, or L, g), not amplitude.
Who uses SHM calculators?
Students, physicists, and engineers studying vibration, acoustics, or mechanical resonance use SHM calculators.
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