NEWTON'S CRADLE

The cradle in motion.

Newton's cradle 5-ball system in 3D 2-ball swing

Newton's cradle, named after Sir Isaac Newton, is a device that demonstrates

conservation of momentum and energy via a series of swinging spheres. When one on

the end is lifted and released, the resulting force travels through the line and pushes the

last one upward. The device is also known as Newton's balls or "Executive Ball

Clicker".

Construction

A typical Newton's cradle consists of a series of identically sized metal balls suspended

in a metal frame so that they are just touching each other at rest. Each ball is attached to

the frame by two wires of equal length angled away from each other. This restricts the

pendulums' movements to the same plane.

Action

Newton's cradle 2-ball system. The left ball is pulled away and is let to fall; it strikes the

right ball and the left ball comes to nearly a dead stop. The right ball acquires most of

the velocity and almost instantly swings in an arc almost as high as the release height of

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the first ball. This shows that the right ball receives most of the energy and momentum

that was in the first ball.

If one ball is pulled away and is let to fall, it strikes the first ball in the series and comes

to nearly a dead stop. The ball on the opposite side acquires most of the velocity and

almost instantly swings in an arc almost as high as the release height of the Last ball.

This shows that the final ball receives most of the energy and momentum that was in the

first ball.

Newtons cradle 5-ball system. One ball is pulled away and is let to fall; it strikes the

first ball in the series and comes to nearly a dead stop. The ball on the opposite side

acquires most of the velocity and almost instantly swings in an arc almost as high as the

release height of the first ball. This shows that the final ball receives most of the energy

and momentum that was in the first ball.

The impact produces a shock wave that propagates through the intermediate balls. Any

efficiently elastic material such as steel will do this as long as the kinetic energy is

temporarily stored as potential energy in the compression of the material rather than

being lost as heat.

Intrigue is provided by starting more than one ball in motion. With two balls, exactly

two balls on the opposite side swing out and back.

Newton's cradle 3-ball swing in a 5-ball system. The central ball swings without any

apparent interruption.

More than half the balls can be set in motion. For example, three out of five balls will

result in the central ball swinging without any apparent interruption.

While the symmetry is satisfying, why does the initial ball (or balls) not bounce back

instead of imparting nearly all the momentum and energy to the last ball (or balls)? The

simple equations used for the conservation of kinetic energy and conservation of

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momentum can show this is a possible solution, but they cannot be used to predict the

final velocities when there are three or more balls in a cradle, because they provide only

two equations to find the three or more unknowns (velocities of the balls). They give an

infinite number of possible solutions if the system of balls is not examined in more

detail.

History

Christiaan Huygens used pendulums to study collisions. His work, De Motu Corporum

ex Percussione (On the Motion of Bodies by Collision) published posthumously in

1703, contains a version of Newton's first law and discusses the collision of suspended

bodies including two bodies of equal size with the motion of a moving body being

transferred to one at rest.

The principle demonstrated by the device, the law of impacts between bodies, was first

demonstrated by the French physicist Abbé Mariotte in the 17th century. Newton

acknowledged Mariotte's work, among that of others, in his Principia.

PHYSICS EXPLANATION

Newton's cradle can be modeled with simple physics and minor errors if it is incorrectly

assumed the balls always collide in pairs. If one ball strikes 4 stationary balls that are

already touching, the simplification is unable to explain the resulting movements in all 5

balls, which are not due to friction losses. For example, in a real Newton's cradle the 4th

has some movement and the first ball has a slight reverse movement. All the animations

in this article show idealized action (simple solution) that only occurs if the balls are not

touching initially and only collide in pairs.

Simple solution

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The conservation of momentum (mass × velocity) and kinetic energy (0.5 × mass ×

velocity^2) can be used to find the resulting velocities for two colliding elastic balls.

When all the balls weigh the same, the solution for a colliding pair is that the "moving"

ball stops relative to the "stationary" one, and the stationary one picks up all the other's

velocity (and therefore all the momentum and energy, assuming no friction, heat, or

sound energy losses). This effect from two identical elastic colliding spheres is the basis

of the cradle and gives an approximate solution to all its action without needing to use

math to solve the momentum and energy equations. For example, when two balls

separated by a very small distance are dropped and strike three stationary balls, the

action is as follows: The first ball to strike (the second ball in the cradle) transfers its

velocity to the third ball and stops. The third ball then transfers the velocity to the fourth

ball and stops, and then the fourth to the fifth ball. Right behind this sequence is the first

ball transferring its velocity to the second ball that had just been stopped, and the

sequence repeats immediately and imperceptibly behind the first sequence, ejecting the

fourth ball right behind the fifth ball with the same microscopic separation that was

between the two initial striking balls. If the 1st and 2nd balls had been firmly connected

at their adjoining surfaces, the initial strike would be the same as one ball having twice

the weight and this results in the last ball moving away much faster than the 4th ball, so

the initial separation is important.

When simple solution applies

In order for the simple solution to theoretically apply, no pair in the midst of colliding

can touch a third ball. This is because applying the two conservation equations to three

or more balls in a single collision results in many possible solutions.

Even when there is a small initial separation, a 3rd ball may become involved in the

collision if the initial separation is not large enough. This is because the 2nd ball starts

to move and can move into a 3rd ball before the 1st and 2nd balls' colliding surface has

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separated. When this occurs, the complete solution method described below must be

used. If the initial separations are large enough to prevent simultaneous collisions, the

complete solution simplifies to the case of independent collision pairs.

Small steel balls work well because they remain efficiently elastic with little heat loss

under strong strikes and do not compress much (up to about 30 µm in a small Newton's

cradle). The small, stiff compressions mean they occur rapidly, less than 200

microseconds, so steel balls are more likely to complete a collision before touching a

nearby 3rd ball. Steel increases the time during the cradle's operation that the simple

solution applies. Softer elastic balls require a larger separation in order to maximize the

effect from pair-wise collisions.

In a pair-wise collision, mass and initial velocity are the variables that are solved for in

the momentum and energy equations. For three or more simultaneously colliding elastic

balls, the relative compressibilities of the colliding surfaces are the additional variables

that determine the outcome. For example, five balls have four colliding points and

scaling (dividing) three of them by the fourth will give the three extra variables needed

(in addition to the two conservation equations) to solve for all five post-collision

velocities. But the compressions of the surfaces are interacting in a way that makes a

deterministic algebraic solution by this method very difficult. Instead of conservation of

momentum and energy, Newton's law and the compression of all four contact points is

used for a numerical solution as described below.

More complete solution

Determining the velocities for the case of one ball striking four initially-touching balls is

found by modeling the balls as weights with non-traditional springs on their colliding

surface. Steel is elastic and follows Hooke's force law for springs, F=k\cdot x, but

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because the area of contact for a sphere increases as the force increases, colliding elastic

balls will follow Hertz's adjustment to Hooke's law, \ F=k\cdot x^{1.5}. This and

Newton's law for motion (F=m\cdot a) are applied to each ball, giving five simple but

interdependent differential equations that are solved numerically.[7] When the fifth ball

begins accelerating, it is receiving momentum and energy from the third and fourth balls

through the spring action of their compressed surfaces. For identical elastic balls of any

type, 40% to 50% of the kinetic energy of the initial ball is stored in the ball surfaces as

potential energy for most of the collision process. 13% of the initial velocity is imparted

to the fourth ball (which can be seen as a 3.3 degree movement if the fifth ball moves

out 25 degrees) and there is a slight reverse velocity in the first three balls, −7% in the

first ball. This separates the balls, but they will come back together just before the fifth

ball returns making a determination of "touching" during subsequent collisions

complex. Stationary steel balls weighing 100 grams (with a strike speed of 1 m/s) need

to be separated by at least 10 µm if they are to be modeled as simple independent

collisions. The differential equations with the initial separations are needed if there is

less than 10 µm separation, a higher strike speed, or heavier balls.

The Hertzian differential equations predict that if two balls strike three, the fifth and

fourth balls will leave with velocities of 1.14 and 0.80 times the initial velocity.[9] This

is 2.03 times more kinetic energy in the fifth ball than the fourth ball, which means the

fifth ball should swing twice as high as the fourth ball. But in a real Newton's cradle the

fourth ball swings out as far as the fifth ball. In order to explain the difference between

theory and experiment, the two striking balls must have at least 20 µm separation (given

steel, 100 g, and 1 m/s). This shows that in the common case of steel balls, unnoticed

separations can be important and must be included in the Hertzian differential equations,

or the simple solution may give a more accurate result.

Description of Newton's Cradle

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Newton's Cradle consists of several metal balls (usually 5) suspended from a rack by

wires, such that they line up and are almost in contact when in a resting position. There

are two wires attached to each ball to keep the pendulum motion in one plane.

Pull up one ball

When an end ball (ball a) is pulled up and let go, it swings down as a pendulum and hits

the next ball. The energy and momentum from that ball is transmitted through the three

balls at rest to the fifth ball on the other end (ball e). That ball is propelled forward at

the same velocity as the first ball had, due to the force of the first collision.

This process continues as ball e reaches its peak and then swings down to hit the balls at

rest, propelling ball a forward and upward.

One Newton's Cradle ball raised and ready to swing

Note: Although the balls look like they are touching, they are really slightly apart (less

than the width of a human hair)

Two or more balls

If two or more balls are pulled up and let go at the same time, the collision will result in

the same number of balls will be propelled forward on the other end.

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Slowly slows down

The pendulum action will go back and forth until it gradually slows down due to losses

from friction and the elasticity of the balls.

Steel balls are usually used, because they deform very little upon collision and are

highly elastic—meaning only a small amount of energy is lost in the collision.

Requirements

There are several requirements on the contruction of Newton's Cradle, to make sure it

fuctions properly:

The balls in Newton's Cradle should be the same mass. Even a slight deviation will

change the derivation equations and result in slightly different results.

It doesn't matter how many balls are used, although the more you use, the greater the

chances for deviations.

The balls should be perfectly aligned. If some balls were not on a straight line, the

transfer of momentum and energy would also be misaligned, changing the outcome.

Spherical balls are used because their contact is approximately a point. Other shaped

objects could be used, but that increases the chances for misalignment.

Hard metal balls—such as made of hardened steel—are used to minimize losses in

energy due to elastic distortions.

Balls are hung with a pair of strings or wires in order to keep them in alignment and to

minimize losses due to friction.

Simulation

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The following simulation allows you to explore Newton's Cradle on your computer,

similar to using the real device.

Note that you must have the Flash player installed in your computer to use this

simulation.

Drag on one ball

Place you mouse pointer on an end ball, hold down the left mouse button, and drag the

ball, so it is at an angle. Then release the mouse button and let the ball swing free. You

will see that only one ball on the other end of the group swings up at about the same

speed as the ball you let go.

Notice that the balls start to slow down and will bounce less and less until they finally

stop. This is due to losses from friction and energy that is absorbed in the balls. The

effect is called damping of the periodic motion.

Drag multiple balls

You can drag two, three or four balls and let them go. The same number of balls you

release will be moved forward upon the collision with the moving balls. This verifies

the Law of the Conservation of Momentum, which states that the momentum (mass

times velocity) remains the same after a collision.

Summary

Newton's Cradle demonstrates laws of motion, including the Laws of Conservation of

Momentum and Energy. The simulation allows you to experiment swinging different

number of balls to verify the conservation of momentum.

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