The simple pendulum is described by the differential equation: \[ \frac{d^2 \theta}{dt^2} = -\frac{g}{L}\,\sin(\theta), \] where \(\theta\) is the oscillation angle, \(g\) is the gravitational acceleration, and \(L\) is the length of the string. For small angles (\(\sin \theta \approx \theta\)), this equation simplifies, and the period is given by: \[ T = 2\pi \sqrt{\frac{L}{g}}. \] This highlights the exchange between potential energy \(\bigl(PE = m\,g\,h\bigr)\) and kinetic energy \(\bigl(KE = \tfrac{1}{2}m\,v^2\bigr)\) during the oscillation.
Real-Time Results
| Formula | Result |
|---|---|
| Equation of Motion: \( \theta'' = -\frac{g}{L} \sin(\theta) \) | |
| Period (T): \( T = 2\pi \sqrt{\frac{L}{g}} \) | -- s |
| Maximum Height (h): \( h = L \left(1 - \cos(\theta_0)\right) \) | -- m |
| Potential Energy (PE): \( PE = m \cdot g \cdot h \) | -- J |
| Kinetic Energy (KE): \( KE = \frac{1}{2} m (L \cdot \omega)^2 \) | -- J |
| Total Energy (TE): \( TE = PE + KE \) | -- J |
| Linear Velocity X (Vx): \( V_x = L \cdot \omega \cdot \cos(\theta) \) | -- m/s |
| Linear Velocity Y (Vy): \( V_y = -L \cdot \omega \cdot \sin(\theta) \) | -- m/s |
| Current Amplitude: | --° |
| Angular Velocity: | -- rad/s |
| Energy Difference: | -- J |
| Kinetic Energy Before Equilibrium: | -- J |