Kinetic Energy of a Pendulum Calculator
Enter the pendulum bob mass, string length, release angle, and current swing angle to find the kinetic energy at that position. The calculator applies conservation of mechanical energy so you also get the velocity, height drop, potential energy, and the period of the swing. All values update as you type.
Formula
Worked example
A 1 kg bob on a 1 m string released from 30 deg: h0 = 1 x (1 - cos 30 deg) = 0.1340 m. Total energy = 1 x 9.807 x 0.1340 = 1.314 J. At the bottom (0 deg) KE = 1.314 J and v = sqrt(2 x 1.314 / 1) = 1.620 m/s. Period = 2pi x sqrt(1 / 9.807) = 2.006 s.
How a pendulum exchanges energy
A simple pendulum converts energy back and forth between two forms throughout each swing. At the release point the bob is stationary, so all the mechanical energy is gravitational potential energy (PE = mgh, where h is the height above the bottom of the arc). As the bob swings down that PE converts to kinetic energy. At the very bottom h = 0, so all energy is kinetic and the bob moves at its fastest. On the way back up the process reverses: KE converts back to PE until the bob momentarily stops at the far side at the same height it started from (ignoring air resistance and pivot friction). Total mechanical energy stays constant throughout: KE + PE = E_total = mgh0.
The height formula and why angle matters
The height of the bob above the lowest point is not simply proportional to the angle. Because the bob travels along an arc, the height is h = L(1 - cos(theta)), where L is the string length and theta is the angle from vertical. At small angles (below about 15 deg) this is roughly h = L * theta^2 / 2 in radians, but at larger angles the cosine relationship becomes significant. For example, at 30 deg the bob rises to 13.4% of the string length, while at 60 deg it rises to 50% of the string length. This nonlinearity also means the period formula T = 2 pi sqrt(L/g) is an approximation valid only for small angles. The actual period is longer for larger release angles.
Velocity and the bottom-of-arc speed
From KE = (1/2)mv^2 and KE = mg(h0 - h), velocity at any point is v = sqrt(2g(h0 - h)). The maximum speed occurs at the bottom where h = 0: v_max = sqrt(2gh0). Notice that mass cancels out: a heavier bob and a lighter bob released from the same angle reach the bottom at exactly the same speed (Galileo's insight). The speed depends only on the height drop and gravitational acceleration, not on mass. Mass does affect the energy stored, E = mgh0, which matters for applications like wrecking balls and pendulum impact tests where force is needed.
Period and the independence from mass
The period T = 2 pi sqrt(L/g) depends only on string length and gravitational acceleration, not on mass or amplitude (for small angles). This is why pendulums were used in clocks for centuries: the period stays nearly constant as the swing gradually decays due to friction. A 1 m pendulum on Earth has a period of about 2.006 s. The Moon's lower gravity (1.62 m/s^2) would stretch that to about 4.94 s. If you need accuracy at large angles, add correction terms: T = 2 pi sqrt(L/g) * (1 + theta^2/16 + ...) where theta is in radians.
Pendulum energy at common angles
| Position | Angle (% of release) | KE fraction | PE fraction |
|---|---|---|---|
| Bottom of arc | 0% | 100% | 0% |
| 25% of release angle | 25% | ~97% | ~3% |
| 50% of release angle | 50% | ~87% | ~13% |
| 70% of release angle | 70% | ~72% | ~28% |
| Release point | 100% | 0% | 100% |
Fraction of total mechanical energy that is kinetic at various swing angles (relative to release angle). At 0 deg the bob is at the bottom; at 100% of the release angle it is back at the start.
Frequently asked questions
Where is kinetic energy maximum in a pendulum?
Kinetic energy is maximum at the very bottom of the swing (theta = 0 deg), where the height above the lowest point is zero. At that instant all of the total mechanical energy is kinetic and the bob moves at its fastest speed. This is the mirror image of the release point, where KE = 0 and all energy is potential.
Does mass affect the kinetic energy of a pendulum?
Yes, mass scales the total energy and therefore the kinetic energy: E_total = mgh0, so doubling the mass doubles both the stored PE and the KE at every point along the arc. However, mass does not affect the velocity. Because velocity comes from v = sqrt(2g * delta_h) and mass cancels out, two bobs of different mass released from the same angle always hit the bottom at the same speed.
What is the kinetic energy at the release point?
Zero. At the instant you release the bob from rest it has no velocity, so KE = (1/2)mv^2 = 0. All the mechanical energy is gravitational potential energy at that moment. As the bob swings down, potential energy converts to kinetic energy, reaching a maximum at the bottom.
How does gravity affect the kinetic energy?
Stronger gravity stores more potential energy for a given height, which means more kinetic energy at the bottom. The total energy scales as g: E = mgh0 = mgL(1 - cos theta0). On the Moon (g = 1.62 m/s^2) the same pendulum released from the same angle would have about 1/6 the kinetic energy it has on Earth, and would swing more slowly.
Is the period formula accurate for large angles?
Only approximately. T = 2pi * sqrt(L/g) is derived using the small-angle approximation sin(theta) = theta, which holds well for release angles below about 15 deg. Above 15 deg the true period is noticeably longer. At 30 deg the error is about 1.7%; at 60 deg it exceeds 7%. For precise work at large angles use the complete elliptic integral of the first kind or a numerical integration.
What happens to energy in a real pendulum?
A real pendulum loses energy gradually to air resistance and pivot friction, so the swing amplitude decays over time. This is called damping. The total mechanical energy is no longer constant: KE + PE decreases each cycle. The rate of decay depends on the damping coefficient. This calculator assumes an ideal, undamped pendulum with no energy loss.
Can I use this calculator for a physical (compound) pendulum?
This calculator models a simple pendulum, which assumes all mass is concentrated at the bob and the string or rod is massless. A physical (compound) pendulum has distributed mass along its length. For that case, replace L with L_eff = I / (md), where I is the moment of inertia about the pivot and d is the distance from the pivot to the centre of mass. The energy formulas (PE = mgh, KE = E - PE) still hold, but the effective length changes the period and the height calculation.