The cradle in motion.

Newton's cradle 5-ball system in 3D

2-ball swing

Newton's cradleFrom Wikipedia, the free encyclopedia

Newton's cradle, named after Sir Isaac Newton, is a device that demonstratesconservation of momentum and energy via a series of swinging spheres. Whenone on the end is lifted and released, the resulting force travels through the lineand pushes the last one upward. The device is also known as Newton's balls or

"Executive Ball Clicker".[1][2][3][4]

Contents

1 Construction2 Action3 History4 Physics explanation

4.1 Simple solution4.1.1 When simple solution applies4.1.2 More complete solution4.1.3 Heat and friction losses

5 Applications6 Invention and design7 See also8 References9 Literature10 External links

Construction

A typical Newton's cradle consists of a series of identically sized metal balls suspended in a metal frame so that theyare just touching each other at rest. Each ball is attached to the frame by two wires of equal length angled away fromeach other. This restricts the pendulums' movements to the same plane.

Action

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 ballon the opposite side acquires most of the velocity and almost instantly swings in an arc almost as high as the releaseheight of the Last ball. This shows that the final ball receives most of the energy and momentum that was in the firstball.

The impact produces a shock wave that propagates through the intermediate balls. Any efficiently elastic materialsuch as steel will do this as long as the kinetic energy is temporarily stored as potential energy in the compression ofthe 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 sideswing out and back.

More than half the balls can be set in motion. For example, three out of five balls will result in the central ballswinging without any apparent interruption.

While the symmetry is satisfying, why does the initial ball (or balls) not bounce back instead of imparting nearly all themomentum and energy to the last ball (or balls)? The simple equations used for the conservation of kinetic energy andconservation of momentum can show this is a possible solution, but they cannot be used to predict the final velocitieswhen there are three or more balls in a cradle, because they provide only two equations to find the three or moreunknowns (velocities of the balls). They give an infinite number of possible solutions if the system of balls is not

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

the first ball. This shows that the right

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.

examined in more detail.

History

Christiaan Huygens used pendulums to study collisions. His work, De MotuCorporum ex Percussione (On the Motion of Bodies by Collision) publishedposthumously in 1703, contains a version of Newton's first law and discussesthe collision of suspended bodies including two bodies of equal size with themotion of a moving body being transferred to one at rest.

Physics explanation

Newton's cradle can be modeled with simple physics and minor errors if it isincorrectly assumed the balls always collide in pairs. If one ball strikes 4stationary balls that are already touching, the simplification is unable to explainthe resulting movements in all 5 balls, which are not due to friction losses. Thesimplification overestimates the kinetic energy in the 5th ball by 2.2%. All theanimations in this article show idealized action (simple solution) that onlyoccurs if the balls are not touching initially and only collide in pairs.

Simple solution

The conservation of momentum (mass × velocity) and kinetic energy(0.5 × mass × velocity^2) can be used to find the resulting velocities for twocolliding elastic balls. For three or more balls, the velocities can be calculatedin the same way if the collisions are a sequence of separate collisions betweenpairs. In Newton's cradle, all the balls weigh the same, so the solution for acolliding 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 themomentum and energy). If both are moving, one is picked to be the"stationary" frame of reference. This effect from two identical elastic collidingspheres is the basis of the cradle and gives an approximate solution to all itsaction without needing to use math to solve the momentum and energyequations. For example, when two balls separated by a very small distance aredropped and strike three stationary balls, the action is as follows: The first ballto strike (the second ball in the cradle) transfers its velocity to the third balland 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 balltransferring its velocity to the second ball that had just been stopped, and thesequence repeats immediately and imperceptibly behind the first sequence,ejecting the fourth ball right behind the fifth ball with the same microscopicseparation that was between the two initial striking balls. If the two initial ballshad been microscopically welded together, the initial strike would be the sameas one ball having twice the weight and this results in only the last ball movingaway much faster than the others in both theory and practice, so the initialseparation is important.

When the simple solution applies, the balls more efficiently transfer the velocity from one ball to the next, maintainingthe effect. So the effects are more noticeable when the balls are not touching and therefore more closely followindependent collisions.

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 isbecause applying the two conservation equations to three or more balls in a single collision results in many possiblesolutions.

Newton's cradle 3-ball swing in a

5-ball system. The central ball swings

without any apparent interruption.

"Touching" in this discussion means when a ball is still compressed on one sideduring a collision, it begins compression on the other side from the nextcollision. So "touching" may include small initial separations, which will needthe complete Hertzian solution described below. If the separations are largeenough to prevent simultaneous collisions, the Hertzian differential equationssimplify to the case of independent collision pairs.

Small steel balls work well because they remain efficiently elastic (less heatloss) under strong strikes and hardly compress (up to about 30 µm in a smallNewton's cradle). The small, stiff compressions mean they occur rapidly (lessthan 200 microseconds), so steel balls are more likely to complete a collisionbefore touching a nearby 3rd ball. Steel increases the time during the cradle'soperation that the simple solution applies. Softer elastic balls require a largerseparation in order to maximize the effect from pair-wise collisions. Forexample, when two balls strike, there needs to be about 0.5 mm separation forrubber balls in order to get the fourth and fifth balls to eject with nearly thesame velocity, but only half the width of a hair for steel balls.

The extra variables needed to determine the solution for three or more simultaneously colliding elastic balls are therelative compressibilities of the colliding surfaces. For example, five balls have four colliding points and scaling(dividing) three of the compressibilities by the fourth will give the three extra variables needed (in addition to the twoconservation equations) to solve for all five post-collision velocities. The compressions of the surfaces are interactingin a way that makes a deterministic algebraic solution difficult to find. Numerical step-wise solutions to the differentialequations have been used.

As air resistance and string friction slow the "ejected" ball(s) down, the other balls may come back together andcollide and separate before the faster ball(s) return, thereby allowing the simple solution to apply on subsequent strikeseven if the first strike did not.

More complete solution

Determining the velocities for the case of one ball striking four "touching" balls is found by modeling the balls asweights with non-traditional springs on their colliding surface. Steel is elastic and follows Hooke's force law forsprings, , but because the area of contact for a sphere increases as the force increases, colliding elasticballs will follow Hertz's adjustment to Hooke's law, . This and Newton's law for motion ( )are applied to each ball, giving five simple but interdependent ("touching") differential equations that are solved

numerically.[5] When the fifth ball begins accelerating, it is receiving momentum and energy from the third and fourthballs through the spring action of their compressed surfaces. For identical elastic balls of any type, 40% to 50% of thekinetic 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 movesout 25 degrees) and there is a slight reverse velocity in the first three balls, −7% in the first ball. This separates theballs, but they will come back together just before the fifth ball returns making a determination of "touching" duringsubsequent collisions complex. Stationary steel balls weighing 100 grams (with a strike speed of 1 m/s) need to beseparated 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.[6]

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.[7] This is 2.03 times more kinetic energy in the fifth ball than thefourth ball, which means the fifth ball should swing twice as high as the fourth ball. But in a real Newton's cradle thefourth ball swings out as far as the fifth ball. In order to explain the difference between theory and experiment, thetwo striking balls must have at least 20 µm separation (given steel, 100 g, and 1 m/s). This shows that in the commoncase 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.

Gravity and the pendulum action influence the middle balls to return near the center positions at nearly the same timein subsequent collisions. This and heat and friction losses are influences that can be included in the Hertzian equations

to make them more general and for subsequent collisions.[8]

Heat and friction losses

This discussion has assumed there are no heat losses from the balls' striking each other or friction losses from airresistance and the strings. However in the real world, these energy losses are the reason the balls eventually come to astop. The higher weight of steel reduces the relative effect of air resistance. The size of the steel balls is limitedbecause the collisions may exceed the elastic limit of the steel, deforming it and causing heat losses.

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.[9] [10] Newton acknowledged Mariotte's work, among that of others, inhis Principia.

Applications

The most common application is that of a desktop executive toy. Another use is as an educational physicsdemonstration, as an example of conservation of momentum.

Invention and design

There is much confusion over the origins of the modern Newton's cradle. Marius J. Morin has been credited asbeing the first to name and make this popular executive toy. However, in early 1967, an English actor, SimonPrebble, coined the name "Newton's cradle" (now used generically) for the wooden version manufactured by hiscompany, Scientific Demonstrations Ltd. After some initial resistance from retailers, they were first sold by Harrods ofLondon, thus creating the start of an enduring market for executive toys. Later a very successful chrome design forthe Carnaby Street store Gear was created by the sculptor and future film director Richard Loncraine.

The largest cradle device in the world was designed by Mythbusters and consists of five one-ton concrete and steelrebar-filled buoys suspended from a steel truss. The buoys also had a steel plate inserted in between their two halvesto act as a "contact point" for transferring the energy; this cradle device did not function well. A smaller scale versionconstructed by them consists of five 6" chrome steel ball bearings, each weighing 33 pounds, and is nearly as efficientas a desktop model.

The cradle device with the largest diameter collision balls on public display, was on display for more than a year inMilwaukee, Wisconsin at retail store American Science and Surplus. Each ball was an inflatable exercise ball 26" indiameter (enclosed in cage of steel rings), and was supported from the ceiling using extremely strong magnets. It wasdismantled in early August of 2010 due to maintenance concerns.