⏱️ Kinetic Energy of a Pendulum Calculator
Calculate kinetic energy of a pendulum
Angle from vertical at which to calculate KE
Maximum angle from vertical (release angle)
How to Use This Calculator
Enter Mass
Input the mass (m) of the pendulum bob in kilograms (kg). This is the mass of the object swinging at the end of the pendulum. Mass must be greater than zero.
Enter Length
Input the length (L) of the pendulum in meters (m). This is the distance from the pivot point to the center of mass of the bob. Length must be greater than zero.
Enter Current Angle
Input the current angle (θ) in degrees from the vertical position. This is the angle at which you want to calculate the kinetic energy. Angles should be between -90° and 90°. At the bottom (θ = 0°), kinetic energy is maximum.
Enter Maximum Angle
Input the maximum angle (θ_max) in degrees. This is the angle from which the pendulum was released (amplitude). This determines the total energy of the system. At maximum angle, kinetic energy is zero and potential energy is maximum.
Enter Gravity (Optional)
Input the gravitational acceleration (g) in m/s². Default is 9.81 m/s² (Earth's surface). Use different values for other planets or locations. Gravity must be greater than zero.
Calculate and Review
Click the "Calculate" button to compute the kinetic energy, velocity, potential energy, and total energy. The calculator uses energy conservation: total energy = kinetic energy + potential energy, and this total remains constant throughout the motion.
Formula
KE = mgh_max - mgh_current
h = L(1 - cos(θ))
v = √(2gh)
Where:
• KE = Kinetic energy (J)
• m = Mass (kg)
• g = Gravitational acceleration (m/s²)
• h = Height difference (m)
• L = Pendulum length (m)
• θ = Angle from vertical (radians)
• v = Velocity (m/s)
Example 1: Pendulum at Bottom
A pendulum with mass 1 kg, length 1 m is released from 45° and swings to the bottom (0°). Calculate the kinetic energy at the bottom.
Given:
• Mass (m) = 1 kg
• Length (L) = 1 m
• Current angle (θ) = 0°
• Maximum angle (θ_max) = 45°
Solution:
h_max = L(1 - cos(45°)) = 1 × (1 - 0.707) = 0.293 m
h_current = L(1 - cos(0°)) = 0 m
KE = mgh_max - mgh_current = 1 × 9.81 × 0.293
KE ≈ 2.87 J
v = √(2 × 9.81 × 0.293) ≈ 2.40 m/s
Example 2: Pendulum at Mid-Point
Same pendulum at 30° angle. Calculate kinetic energy.
Given:
• Mass (m) = 1 kg, Length (L) = 1 m
• Current angle (θ) = 30°
• Maximum angle (θ_max) = 45°
Solution:
h_max = 0.293 m (from previous)
h_current = L(1 - cos(30°)) = 1 × (1 - 0.866) = 0.134 m
KE = mg(h_max - h_current) = 1 × 9.81 × (0.293 - 0.134)
KE ≈ 1.56 J
Frequently Asked Questions
How is kinetic energy calculated for a pendulum?
Kinetic energy is calculated using energy conservation. The total energy (potential + kinetic) remains constant. At maximum angle, all energy is potential. As the pendulum swings down, potential energy converts to kinetic energy. The formula is KE = mgh_max - mgh_current, where h = L(1 - cos(θ)) is the height above the lowest point.
When is kinetic energy maximum for a pendulum?
Kinetic energy is maximum when the pendulum is at the bottom of its swing (θ = 0°), where potential energy is minimum. At this point, the pendulum has the highest velocity. At the maximum angle (amplitude), kinetic energy is zero and potential energy is maximum. The energy continuously converts between potential and kinetic as the pendulum swings.
Why does the height formula use h = L(1 - cos(θ))?
This formula comes from the geometry of circular motion. When a pendulum is at angle θ, the vertical height difference from the bottom is h = L - Lcos(θ) = L(1 - cos(θ)). At θ = 0° (bottom), h = 0. At θ = 90°, h = L. This height difference determines the potential energy and thus the kinetic energy available.
What happens to kinetic energy as the pendulum swings?
As the pendulum swings from maximum angle to bottom: kinetic energy increases from 0 to maximum. From bottom back to maximum angle: kinetic energy decreases from maximum back to 0. This is because energy converts between potential (high at extremes) and kinetic (high at bottom). The total energy remains constant (conservation of energy).
Can I use this for a simple pendulum only?
This calculator uses the simple pendulum model, which assumes: the string is massless and rigid, the bob is a point mass, and small-angle approximation is valid. For large angles or physical pendulums (extended mass), the formulas become more complex. However, this calculator works well for most practical pendulum applications.
Where is pendulum kinetic energy used in real-world applications?
Understanding pendulum kinetic energy is important in: pendulum clocks (energy transfer mechanisms), seismometers (measuring earthquake motion), metronomes (timing devices), amusement park rides (swinging motions), physics education (demonstrating energy conservation), and analyzing any oscillating system under gravity.
About Kinetic Energy of a Pendulum Calculator
The kinetic energy of a pendulum calculator computes the kinetic energy at any point in a pendulum's swing using energy conservation principles. The formula uses KE = mgh_max - mgh_current, where height is calculated from h = L(1 - cos(θ)).
This calculator is essential for students studying energy conservation, engineers analyzing oscillating systems, and anyone understanding pendulum dynamics. It demonstrates how potential energy converts to kinetic energy as a pendulum swings, with total energy remaining constant throughout the motion.