🚀 Simple Pendulum Period Calculator & Simulator
- Adjust Pendulum Length & Mass: Set the rod length (0.5m to 4m) of the suspended pendulum and the actual mass (kg) of the bob.
- Specify Initial Angle (θ) & Damping Coefficient (b): Adjust the initial release angle (degrees) and the damping coefficient that causes loss of kinetic energy through air resistance and friction.
- Select Gravitational Body: Choose the celestial gravity environment in which the pendulum swings, such as Earth (9.81), Jupiter (24.79), or the Moon (1.62).
- Monitor Tension & Energy in Real-Time: Observe the real-time dynamic changes in the induced tension and the potential/kinetic energy (PE/KE) bar charts as the pendulum swings.
📚 Detailed Explanation of Simple Pendulum Motion Period and Physical Energy Formulas ▼
1. Simple Harmonic Motion and the Nonlinear Equation of Pendulum Motion
While the period of a pendulum typically taught in physics is linearized using the small-angle approximation (θ ≈ sinθ), this simulator performs real-time numerical integration (Euler-Cromer integration) of the following nonlinear second-order differential equation to ensure accuracy even at large oscillation angles.
d²θ/dt² + (b/m)·dθ/dt + (g/L)·sin(θ) = 0
Where b is the damping coefficient, m is the mass, L is the pendulum length, and g is the gravitational acceleration. When damping is zero, total energy is conserved indefinitely. However, when damping is introduced, the amplitude of motion decays exponentially, eventually converging to the equilibrium point.
2. Calculation of Dynamic Tension Induced in the Pendulum Rod
The dynamic tension (Tension Force, F_T) induced in the rod when the pendulum is at an angle of θ is the result of the vector addition of the centrifugal force and the component of gravitational force.
F_T = m × g × cos(θ) + m × L × ω² [N]
Here, ω = dθ/dt is the angular velocity in rad/s. When the pendulum passes through the lowest vertical centerline (θ = 0), its speed reaches its peak, resulting in maximum dynamic tension. Conversely, at the peak of the swing (θ = θ_max), the angular velocity becomes ω = 0, leaving only the pure static load of m·g·cosθ acting on the rod.
3. Law of Conservation of Energy and Damping Loss Rate
The total mechanical energy E_total of a simple pendulum system is the sum of the following two components:
- Potential Energy (PE): Energy due to the height raised from the lowest reference point.
PE = m × g × L × (1 - cosθ) [J] - Kinetic Energy (KE): Energy due to the tangential linear velocity of the bob.
KE = 1/2 × m × (L·ω)² [J]
When air resistance is present, each swing generates damping friction loss as heat energy, causing the total energy to continuously decrease and dissipate.