OSCILLATIONS & ENERGY

Pendulum Laboratory

Release a swing. Trace the forces. Follow the energy.

Pull. Release. Observe.

Ready

Forces Velocity & acceleration a components Vacuum reference

Speed

Bob radius is drawn to scale with a 4 px minimum. The view fits each length. Trail: last 2 s. Dashed gray: optional vacuum reference.

Drag the bob along its arc; release to run from rest. Keyboard: ← / → changes the angle; Home centers it; Enter or Space releases. Limit ±80°.

Force arrows share a scale. Velocity and acceleration use separate scales; acceleration components share the total acceleration scale. Zero vectors have no arrow.

ANGLE

ANGULAR VELOCITY

SPEED

DRAG MAGNITUDE

ELAPSED TIME

T₀ = 2π√(L/g) is the small-angle, undamped approximation. Actual periods use two equilibrium crossings in the same direction. Step = 0.05 s.

Motion & energy

Last 30 s · time (s) →

Angle θ (°) · dashed gray = optional vacuum reference

Energy (J)

K · kineticU · potentialE · mechanical (dashed)Q · dissipated

Follow the energy

Kinetic K

Potential U

Dissipated Q

Bars share the initial energy scale. Potential energy is zero at the lowest point. Drag removes energy through negative work.

K = ½mL²ω²
U = mgL(1 − cos θ)
Q = ∫ −Fdrag v dt
K + U + Q ≈ E₀

Why?

What restores the bob?

The tangential component of gravity is −mg sin θ. It points toward equilibrium. Tension points inward and constrains the circular path; it does no work.

Why is speed greatest at the bottom?

In vacuum, K + U is constant. At the bottom U = 0, so kinetic energy and speed are greatest. At a turning point, speed is zero.

Can acceleration remain at the bottom?

Yes. Even when tangential gravity is zero, moving along a circle needs inward radial acceleration v²/L. With air resistance, drag also slows the bob tangentially at the bottom.

How do length and gravity affect period?

T₀ = 2π√(L/g). Four times the length gives twice the small-angle period. Four times the gravity halves it. Gravity selection never changes air density.

Why does mass not change the ideal period?

The gravitational force and inertia both scale with mass, so m cancels from θ″ = −(g/L) sin θ. This holds at large angles too when there is no drag.

Why do large swings take longer?

At large angles, sin θ is smaller than θ in radians. The restoring acceleration is weaker than the small-angle linear model predicts, so the undamped period increases with amplitude.

Why does air reduce amplitude?

Fdrag = −½ρCdA v|v| always opposes motion. Its power Fdrag v is nonpositive, reducing mechanical energy and the height of successive swings.

Why do speed and radius increase drag?

Drag magnitude is proportional to v² and area A = πr². At the same air density, doubling speed or radius multiplies drag by four.

Why does a heavier bob lose less speed?

For the same radius and speed, drag force is unchanged, but drag deceleration is |Fdrag|/m. Doubling mass halves this deceleration.

Educational approximation: constant sphere Cd = 0.47. Real drag varies with flow conditions. The bob is a point mass for dynamics; radius sets drag area and appearance. No generic damping is added.

TRY A CHALLENGE

A two-second swing.

On Earth, with a fixed 5° release and no air resistance, adjust length to achieve a measured period of 2.0 seconds.

Aim for 2.00 ± 0.05 s. Run until a complete cycle is measured.

Show a hint

Rearrange the approximation: L ≈ g(T/2π)². Too fast? Lengthen the pendulum. Too slow? Shorten it.