Kinetic Energy of a Pendulum Calculator
Use energy conservation to get a swinging pendulum's kinetic energy, speed and tension at any angle, with a KE and PE chart.
⏱️ What is the Kinetic Energy of a Pendulum?
The kinetic energy of a pendulum is the energy of motion of the swinging bob at a particular moment, KE = ½ m v². It is not constant: it is zero at the two ends of the swing, where the bob stops for an instant, and it is largest at the lowest point, where the bob moves fastest. Because the only forces that do work are gravity (the string pulls at right angles to the motion), the sum of kinetic and gravitational potential energy stays constant for an ideal pendulum. That conservation of mechanical energy lets you find the kinetic energy at any angle without solving the equation of motion: KE = m g L (cos θ − cos θ0), where θ0 is the release angle and θ the angle of interest, both measured from the vertical. This calculator evaluates the formula and shows the full picture, including speed, potential energy and string tension.
The idea behind the formula is simple geometry. When the string makes an angle θ with the vertical, the bob is h = L (1 − cos θ) above its lowest point, so its potential energy, measured from that point, is m g L (1 − cos θ). At the release angle the bob is momentarily at rest, so all of the energy is potential: E = m g L (1 − cos θ0). Anywhere else the kinetic energy is whatever is left over, E minus the potential energy, which simplifies to the formula above. At the bottom, θ = 0 and the kinetic energy equals the total, giving the maximum speed vmax = √(2 g L (1 − cos θ0)). The mass cancels from the speed, which is why a heavy and a light bob on the same string move equally fast.
These results are useful well beyond textbook exercises. Engineers use them for wrecking-ball and impact tests, clock designers use them to understand how much energy a driving mechanism must replace each swing, and teachers use them to show energy conversion. The curve of kinetic and potential energy against angle in the chart makes the exchange visible: the two curves mirror each other and always add up to the dashed total line. If you want the time of a swing instead of its energy, the physical pendulum calculator gives the period of an extended body, and the general formulas are on the kinetic energy calculator and the potential energy calculator.
Some limits to keep in mind: the model ignores air drag and friction at the pivot, which slowly drain the energy, and it treats the string as massless and the bob as a point. It also does not need the small-angle approximation, so it stays accurate for large swings. If you release the pendulum with a push instead of from rest, the starting kinetic energy is not zero and this simple release-angle model no longer applies.
📐 Formula
📖 How to Use This Calculator
Steps
💡 Example Calculations
Example 1 — A Lab Pendulum Part Way Through the Swing
A 0.5 kg bob on a 1.2 m string, released at 40°, observed at 15° on Earth
Example 2 — The Speed at the Bottom
A 2 kg bob on a 2 m string released at 60°, at the lowest point (0°)
Example 3 — The Same Swing on the Moon
A 1 kg bob on a 1 m string, released at 30°, observed at 10° with g = 1.62 m/s²
Example 4 — A Large Release Angle
A 1 kg bob on a 1 m string released horizontally (90°), observed at 45° on Earth
❓ Frequently Asked Questions
🔗 Related Calculators
How do you find the kinetic energy of a pendulum?
Use conservation of energy: KE = m g L (cos theta - cos theta0), where theta is the current angle and theta0 the release angle, both from the vertical. For 0.5 kg, 1.2 m, released at 40 degrees, KE at 15 degrees is 1.176 J.
What is the maximum speed of a pendulum?
The speed is greatest at the lowest point: v max = sqrt(2 g L (1 - cos theta0)). A 2 m pendulum released at 60 degrees reaches 4.43 m/s on Earth. It does not depend on the mass.
Where is the kinetic energy of a pendulum highest?
At the bottom of the swing, where the potential energy is zero and all of the total energy is kinetic. At the two extremes the bob stops for an instant and the kinetic energy is zero.
Does the mass of the bob affect the speed?
No. Mass appears on both sides of the energy equation and cancels, so v = sqrt(2 g L (cos theta - cos theta0)). A heavier bob has more energy, not more speed.
How do you find the potential energy of a pendulum?
Measure the height above the lowest point, h = L (1 - cos theta), and use PE = m g h. The potential energy is zero at the bottom and equals the total energy at the release angle.
Is the total energy of a pendulum constant?
In the ideal case, yes. Kinetic and potential energy trade places, and their sum stays at m g L (1 - cos theta0). Air drag and friction at the pivot slowly remove energy in a real pendulum.
What is the tension in the string at a given angle?
T = m g (3 cos theta - 2 cos theta0). At the bottom this becomes m g (3 - 2 cos theta0), which is larger than the weight. At the release angle it falls to m g cos theta0.
Does the length of the pendulum change the kinetic energy?
Yes. The energies scale with L through the height drop L (1 - cos theta). The speed scales with the square root of L, so a longer pendulum released from the same angle is faster at the bottom.
Does this work for large angles?
Yes. Energy conservation is exact for any release angle up to 180 degrees, so this calculator is valid for large swings where the small-angle period formula fails.
How does gravity change the kinetic energy?
The energies scale linearly with g and the speeds with its square root. The same swing on the Moon, with g = 1.62 m/s², has about one sixth of the Earth energy.