Energy Conservation

This is Energy Conservation in a Pendulum Simulation.

In this simulation, we demonstrates the principle of energy conservation in a pendulum system.

As the pendulum swings:

• At the highest points (θ = ±θ₀), the pendulum has maximum potential energy and zero kinetic energy.
• At the lowest point (θ = 0), the pendulum has maximum kinetic energy and minimum potential energy.
• In the absence of friction/drag, the total mechanical energy (PE + KE) remains constant.
• The period T of a pendulum (time for one complete swing) is T = 2π√(L/g).

How to Use:

Use the control buttons to play, pause, step backward, step forward, or reset the simulation.

Observe how the energy transforms between potential and kinetic while the total energy remains constant.

Physics Behind It

The motion of a simple pendulum follows the equation:

θ(t) = θ₀·cos(ωt)

Variables in the equation:

• θ is Angular displacement at time t (in radians).
• θ₀ is Initial angular displacement (amplitude).
• ω = √(g/L), ω is Angular frequency (rad/s).
• g is Acceleration due to gravity (9.8 m/s² on Earth).
• L is Length of the pendulum (in meters).
• T is the time period of a pendulum (time for one complete swing in seconds), T = 2π√(L/g).

Potential Energy (PE)

PE = mgh

Where:

• m is Mass of the pendulum bob (in kg).
• g is Acceleration due to gravity (9.8 m/s²).
• h is Height of the bob above its lowest point (in meters).
• h = L(1 - cosθ) where L is pendulum length and θ is angular displacement.

Kinetic Energy (KE)

KE = 1/2 * mv²

Where:

• m is Mass of the pendulum bob (in kg).
• v is Tangential velocity of the bob (in m/s).
• v = ωL√(θ₀² - θ²) where ω is angular frequency, L is length.

Real-World Considerations

In real pendulums, energy is gradually lost due to:

• Air resistance.
• Friction at the pivot point.
• Energy transfer to support structure.
• This causes the amplitude of oscillation to gradually decrease over time.

Applications

Pendulums are used in:

• Clocks and timekeeping devices.
• Seismometers for detecting earthquakes.
• Scientific demonstrations of periodic motion.
• Energy transformation studies.