⚙️ Physics · Momentum · Energy
📅 March 2026 ⏱ ~6 min read Ages 8–14

Pendulum & Newton's Cradle

Pull one ball back on a Newton's Cradle and release it. One ball flies out the other side — not two slow ones, not one fast and one slow. Why? The answer involves two conservation laws working together, and it's one of the most satisfying demonstrations in all of physics.

The Pendulum

A pendulum is just a weight (called a bob) on a string that swings back and forth. Galileo Galilei noticed in 1602 — supposedly while watching a lamp swinging in a cathedral — that the swing takes the same amount of time regardless of how far you pull it back (for small angles, at least).

This property — called isochronism — made pendulums the basis of accurate clocks for over 200 years.

⏱ Fun Fact
Christiaan Huygens built the first pendulum clock in 1656. It was accurate to about 10 seconds per day — a massive improvement over the best existing clocks, which drifted by 15 minutes per day!

As the pendulum swings, energy constantly converts between two forms:

The total energy (PE + KE) stays constant — it just keeps converting back and forth. This is the conservation of energy.

Period and Length

The time for one complete swing (back and forth) is called the period T. It depends only on the length L of the pendulum and gravity g:

T = 2π × √( L / g )

On Earth, g = 9.8 m/s². A 1-metre pendulum has a period of about 2 seconds — one second each way.

Pendulum length → Period (on Earth)

  • 10 cm (about a ruler length): T ≈ 0.63 s
  • 25 cm (a quarter metre): T ≈ 1.0 s
  • 1 m (a tall pendulum clock): T ≈ 2.0 s
  • 67 m (Foucault Pendulum, Panthéon, Paris): T ≈ 16.4 s

The mass of the bob doesn't matter at all! A heavy steel ball and a light wooden ball on the same length string will swing in perfect sync. Galileo proved this by pushing two pendulums with different-weight bobs and watching them stay together.

🏠 Try It at Home

Tie a washer to a 25 cm piece of string. Hold it and let it swing. Count 10 swings — it should take about 10 seconds. Now try with a heavier washer. Same timing!

Conservation of Momentum

Momentum (p) is how much "moving force" an object has. It equals mass times velocity:

p = m × v

The Law of Conservation of Momentum says: in any collision, the total momentum before equals the total momentum after. Momentum cannot be created or destroyed — only transferred.

When ball A (mass m, velocity v) hits stationary ball B (mass m):

Before: p = m×v + m×0 = mv
After: p = m×v₁ + m×v₂ must still equal mv

Conservation of Kinetic Energy

Newton's Cradle uses hard steel balls. Collisions between hard balls are elastic — the balls don't squish or heat up, so kinetic energy is also conserved (in addition to momentum):

Conservation of Momentum

m·v = m·v₁ + m·v₂

Total momentum (mv) before = total momentum after

Conservation of KE

½mv² = ½mv₁² + ½mv₂²

Total kinetic energy before = total kinetic energy after

These are two equations. When one ball of mass m hits another at speed v, you have two unknowns (v₁ and v₂) and two equations. The unique solution for equal masses is: v₁ = 0, v₂ = v.

The first ball stops dead. The second ball moves away at the original speed. There is no other solution that satisfies both laws simultaneously.

Why Exactly One Ball Flies Out?

Newton's Cradle has 5 equal-mass steel balls. When you pull back 1 ball and release it, it travels at speed v and hits the row.

The collision must satisfy both conservation of momentum and conservation of kinetic energy. For 5 equal-mass balls, the only solution is: the one ball at the end flies out at speed v, and all the middle balls remain stationary.

🔵🔵🔵🔵⬡→
1 ball in: 1 ball exits at the same speed. The momentum (mv) and kinetic energy (½mv²) are transferred through the chain. The three middle balls briefly compress and act as an intermediary.
🔵🔵🔵⬡⬡→
2 balls in: 2 balls exit at the same speed. Both laws are satisfied with 2mv going in and 2mv coming out.
🔵🔵⬡⬡⬡→
3 balls in: 3 balls exit. Same logic applies.
🤔 Why Not 2 Slower Balls?
Try it mathematically. If 1 ball hits at v, and 2 balls exit at v/2 each:

Momentum check: m×v/2 + m×v/2 = mv ✅ (OK!)
KE check: ½m(v/2)² + ½m(v/2)² = ¼mv² ≠ ½mv² ❌ (Wrong!)

Half the kinetic energy went missing. That's not allowed in an elastic collision. The "2 slow balls" answer is forbidden by physics!

What About Two Balls?

Real Newton's Cradles aren't perfectly elastic — a tiny bit of energy is always lost as heat and sound with each click. Over time, the swings get smaller and smaller until the cradle comes to rest. This is damping.

For a truly elastic Newton's Cradle, the balls would swing forever — just like a frictionless pendulum. In the real world, nothing is perfectly elastic, but steel comes remarkably close.

You can try it with a virtual pendulum simulation to see the idealised, undamped version:

🕰️ Open Pendulum →