🕰️ Simple Harmonic Motion & Gravitational Oscillations
Pendulum Period & Frequency Calculator
Compute harmonic pendulum period ($T = 2\pi\sqrt{L/g}$), oscillation frequency ($f$), large-amplitude Borda angle corrections, and planetary gravitational acceleration presets.
🌍 Earth ($g = 9.81\text{ m/s}^2$)
🌕 Moon ($g = 1.62\text{ m/s}^2$)
🔴 Mars ($g = 3.72\text{ m/s}^2$)
🪐 Jupiter ($g = 24.79\text{ m/s}^2$)
Pendulum Specifications
m/s²
degrees (°)
Angles > 15° trigger Borda non-linear series correction.
Full Period of Oscillation (T)
2.007 s
Small Angle Period ($T_0$)
2.006 s
Large Angle Error
+0.05%
Seconds Pendulum
0.994 m
For T = 2.0 s
💡 Physical Insight:
A 1.0 meter pendulum on Earth completes a full two-way swing in almost exactly 2 seconds, forming the historical basis for grandfather clock escapements.
The Mathematics of Pendulum Motion
For a point mass suspended by a massless string:
- Small-Angle Harmonic Approximation ($\theta \le 15^\circ$): $$T_0 = 2\pi\sqrt{\frac{L}{g}}, \quad f = \frac{1}{T_0} = \frac{1}{2\pi}\sqrt{\frac{g}{L}}$$
- First-Order Borda Amplitude Correction: $$T \approx T_0 \left(1 + \frac{1}{16}\theta_0^2 + \frac{11}{3072}\theta_0^4\right) \quad (\theta_0 \text{ in radians})$$