Pendulum Kinetic Energy Calculator

Pendulum Energy at an Angle

A pendulum bob stores gravitational potential energy when displaced from its lowest point. As it swings downward, potential energy converts to kinetic energy. At the bottom, kinetic energy is maximum and potential energy is minimum.

Kinetic Energy Formula

$$KE = mgL(1 - \cos\theta)$$

Mass is $m$, length is $L$, gravity is $g$, and $\theta$ is the angle from vertical. The height gain above the lowest point is $h = L(1 - \cos\theta)$, so $KE = mgh$.

Energy Conservation

Ignoring friction and air resistance, total mechanical energy stays constant. At the release angle, kinetic energy is zero and potential energy is $mgL(1 - \cos\theta_0)$. At any lower angle, the kinetic energy equals the potential energy lost since release.

Frequently Asked Questions

Why use (1 - cos θ) for height?

At angle θ, the bob is higher than at the bottom by $L - L\cos\theta = L(1 - \cos\theta)$.

What is kinetic energy at the bottom?

At the lowest point, θ = 0 and KE equals the potential energy at the release angle (if released from rest).

Does length affect energy?

Yes. A longer pendulum raised to the same angle stores more energy because height gain is larger.

What angle gives maximum energy?

Maximum stored energy occurs at the highest release angle. At 180 degrees the bob is directly above the pivot (unstable equilibrium).

Are friction losses included?

No. This is the ideal energy from height alone. Real pendulums lose a small amount to air drag and pivot friction.

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Pendulum Kinetic Energy Calculator

Pendulum Energy at an Angle

Kinetic Energy Formula

Energy Conservation

Frequently Asked Questions

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