Calculate a pendulum bob's kinetic energy from speed or from swing angles, including the energy at any point during the motion.
Advanced options
How to use our Kinetic Energy of a Pendulum Calculator
- Choose What do you know? to match your problem: Mass and speed, Mass, length, and release angle, or Energy at a current angle.
- Enter Bob mass (kg). Use the mass of the bob itself, not the string or rod.
- If you chose Mass and speed, enter Speed (m/s). The calculator then uses that speed directly to find Kinetic energy.
- If you chose a geometry mode, enter Pendulum length (m), Release angle, and Gravitational acceleration (m/s^2). In Energy at a current angle mode, also enter Current angle.
- Open Advanced options if needed and set Angle input unit before typing angles, then choose Display format and Decimal places if you want custom result formatting.
- Click Calculate. Read Kinetic energy first, then use Speed at chosen position, Vertical drop from release, Potential energy decrease from release, and Maximum kinetic energy at bottom for extra context.
- Sanity-check the result: in geometry modes, Potential energy decrease from release should match Kinetic energy for the ideal model, and Speed at chosen position should be largest at the bottom.
Definitions
Bob mass: The mass of the pendulum's moving weight. In the kinetic energy formula, kinetic energy equals 1/2 times mass times speed squared.[2]
Speed at chosen position: How fast the bob is moving at the point you selected. Higher speed means higher kinetic energy for the same mass.[2]
Release angle: The starting angle from straight down where the bob begins from rest. Straight down is the lowest position of the swing.
Current angle: The angle from straight down at the moment you want to analyze. In the current-angle mode, it should not be farther from straight down than the release angle.
Vertical drop from release: How far the bob falls vertically between the release point and the chosen position. That drop provides the height change used in gravitational potential energy.
Potential energy decrease from release: The amount of gravitational potential energy lost as the bob moves downward. For an ideal loss-free motion, a drop of height h changes energy by mgh and that energy becomes kinetic energy.[3]
Simple pendulum: An ideal model where the bob acts like a point mass on a massless string or rod, so energy trades mainly between gravitational potential energy and kinetic energy.[4]
Gravitational acceleration: The strength of gravity used in the calculation. The default value 9.80665 m/s^2 is the standard acceleration of gravity.[1]
Common mistakes and quick fixes
Mistake: Entering Bob mass (kg) in grams instead of kilograms.
Fix: Convert to kilograms first. For example, 500 g should be entered as 0.5 in Bob mass (kg).
Mistake: Typing angles in radians while Angle input unit is still set for degrees.
Fix: Set Angle input unit before entering Release angle or Current angle so both angles are interpreted the same way.
Mistake: Using the full string length plus bob diameter for Pendulum length (m).
Fix: Enter the distance from the pivot to the bob's center for Pendulum length (m).
Mistake: Entering a negative value for Speed (m/s) in Mass and speed mode.
Fix: Use speed as a nonnegative size only. If direction matters in your class, direction is not part of Kinetic energy here.
Mistake: Making Current angle larger than Release angle for a bob released from rest.
Fix: In Energy at a current angle mode, Current angle must not have a larger absolute value than Release angle, or the position is not feasible for that release.
Mistake: Assuming Maximum kinetic energy at bottom should appear in Mass and speed mode.
Fix: That output is for geometry-based setups, where the calculator can compare your chosen position with the bottom of the swing.
Limitations & Key Assumptions / Boundary Conditions
- The geometry modes assume an ideal simple pendulum released from rest, with no friction, air drag, or energy loss.
- Bob mass (kg) must be greater than 0, and Gravitational acceleration (m/s^2) must be greater than 0.
- Pendulum length (m) must be greater than 0 in geometry-based modes.
- In Mass and speed mode, Speed (m/s) cannot be negative.
- In Energy at a current angle mode, the absolute value of Current angle must not exceed the absolute value of Release angle for a release-from-rest swing.
- The simple model is most practical for ordinary swings with a concentrated bob and a light string or rod. Real pendulums with a heavy rod, a large bob size, or strong damping can differ from the ideal result.
- Very large angles can still be computed with the energy formulas, but real motion may differ more from the simple ideal setup.
- Small rounding effects can produce a tiny negative energy near zero in edge cases; those values are treated as zero rather than as physically negative kinetic energy.
Methodology
Core idea
This calculator uses either the direct kinetic energy formula or energy conservation for an ideal pendulum. Kinetic energy is the energy of motion, and for a moving mass it is given by 1/2 times mass times speed squared.[2] When a bob falls through a vertical height, the lost gravitational potential energy mgh becomes kinetic energy if losses are ignored.[2]
Formulas used
KE = 0.5 * m * v^2
h(theta) = L * (1 - cos(theta))
KE_max = m * g * L * (1 - cos(theta0))
KE(theta) = m * g * L * (cos(theta) - cos(theta0))
v = sqrt(2 * KE / m)
How each mode works
Mass and speed: The calculator reads Bob mass (kg) and Speed (m/s), then computes Kinetic energy with KE = 0.5 * m * v^2. It also reports the same speed as Speed at chosen position.
Mass, length, and release angle: The bob is assumed to start from rest at the release angle and swing to the bottom. The vertical drop from release is found from the pendulum geometry, then the available kinetic energy at the bottom is m * g * vertical drop. For a pendulum, that bottom energy can also be written as m * g * L * (1 - cos(theta0)).
Energy at a current angle: The calculator compares the height at Release angle with the height at Current angle. The difference in height is the Vertical drop from release, and the matching kinetic energy is the lost potential energy. This gives KE(theta) = m * g * L * (cos(theta) - cos(theta0)).
Mini example
Suppose Bob mass (kg) = 2, Pendulum length (m) = 1.5, Release angle = 30 deg, Current angle = 15 deg, and Gravitational acceleration (m/s^2) = 9.81.
vertical drop = 1.5 * (cos(15 deg) - cos(30 deg)) = 0.1498506939 m
KE = 2 * 9.81 * 0.1498506939 = 2.9400706207 J
v = sqrt(2 * 2.9400706207 / 2) = 1.7146634129 m/s
So the bob has about 2.94 J of kinetic energy at 15 deg, and it is moving about 1.715 m/s. At the bottom, the maximum kinetic energy for the same release would be larger.
Display and angle handling
If Angle input unit is set to degrees, the calculator converts degrees to radians before using cosine. If it is set to radians, the angle values are used as radians directly. Auto display uses regular decimals for ordinary values and switches to scientific notation only when values are extremely small or large.
Assumptions behind the results
The default gravity value is 9.80665 m/s^2, the standard acceleration of gravity.[1] The energy-trading idea used here matches the ideal pendulum picture where potential energy and kinetic energy exchange during motion.[4] Real systems can differ because of air drag, pivot friction, a heavy rod or string, or a bob that is not well modeled as a point mass.