Newton's cradle



Newton's cradle, named after English 17th century scientist Isaac Newton, is a device that demonstrates conservation of momentum and energy using a series of swinging spheres. When one sphere at the end is lifted and released, it strikes the stationary spheres; a force is transmitted through the stationary spheres and pushes the last sphere upward. The device is also known as Newton's balls or Executive Ball Clicker.

Description
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.

Operation
When one of the end balls ("the first") is pulled sideways, the attached string causes it to follow an upward arc. When it is let go, it strikes the second ball and comes to nearly a dead stop. The ball on the opposite side acquires most of the velocity of the first ball and swings in an arc almost as high as the release height of the first ball. This shows that the last ball receives most of the energy and momentum of the first ball. The impact produces a compression 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. There are slight movements in all the balls after the initial strike but the last ball receives most of the initial energy from the drop of the first ball. When two (or three) balls are dropped, the two (or three) balls on the opposite side swing out. It is often claimed that this behavior demonstrates the conservation of momentum and kinetic energy in elastic collisions. However, if the colliding balls behave as described above with the same mass possessing the same velocity before and after the collisions then any function of mass and velocity is conserved in such an event.

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
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 perfectly elastic objects. These two equations are used to determine the resulting velocities of the two objects. For the case of two balls constrained to a straight path by the strings in the cradle, the velocities are a single number instead of a 3D vector for 3D space, so the math requires only two equations to solve for two unknowns. When the two objects weigh the same, the solution is very simple: the moving object stops relative to the stationary one and the stationary one picks up all the other's initial velocity. This assumes perfectly elastic objects, so we do not need to account for heat and sound energy losses.

Steel does not compress much, but its elasticity is very efficient which means it does not cause much waste heat. The simple effect from two same-weight efficiently elastic colliding objects constrained to a straight path is the basis of the interesting effect seen in the cradle and gives an approximate solution to all its action.

For a sequence of same-weight elastic objects constrained to a straight path, the effect continues to each successive object. For example, when two balls are dropped to strike three stationary balls in a cradle, there is an unnoticed but crucial small distance between the two dropped balls and the action is as follows: The first moving ball that strikes the first stationary ball (the 2nd ball striking the 3rd ball) transfers all its velocity to the 3rd ball and stops. The 3rd ball then transfers the velocity to the 4th ball and stops, and then the 4th to the 5th ball. Right behind this sequence is the 1st ball transferring its velocity to the 2nd ball that had just been stopped, and the sequence repeats immediately and imperceptibly behind the first sequence, ejecting the 4th ball right behind the 5th ball with the same small separation that was between the two initial striking balls. If they are simply touching when they strike the 3rd ball, the more complete solution described below is necessary in order to be precise.

Other examples of this effect
The interesting effect of the last ball ejecting with a velocity nearly equal to the first ball can be seen in sliding a coin on a table into a line of identical coins, as long as the striking coin and its twin targets are in a straight line. The effect can similarly be seen in billiard balls. The effect can also be seen when a sharp and strong pressure wave strikes a dense homogeneous material immersed in a less-dense medium. If the identical atoms, molecules, or larger-scale sub-volumes of the dense homogeneous material are at least partially elastically connected to each other by electrostatic forces, they can act like a sequence of colliding identical elastic balls. The surrounding atoms, molecules, or sub-volumes experiencing the pressure wave act to constrain each other similarly to how the string constrains the cradle's balls to a straight line. For example, lithotripsy shock waves can be sent through the skin and tissue without harm in order to burst kidney stones. The side of the stones opposite to the incoming pressure wave bursts, not the side receiving the initial strike.

When the simple solution applies
In order for the simple solution to precisely predict the action, no pair in the midst of colliding may touch a third ball because the presence of the 3rd ball effectively makes the ball being struck appear heavier. Applying the two conservation equations to solve the final velocities of three or more balls in a single collision results in many possible solutions, so these two principles are not enough to determine resulting action.

Even when there is a small initial separation, a third ball may become involved in the collision if the initial separation is not large enough. When this occurs, the complete solution method described below must be used.

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 third ball. Softer elastic balls require a larger separation in order to maximize the effect from pair-wise collisions.

More complete solution
A cradle that best follows the simple solution needs to have an initial separation between the balls that is at least twice the distance that the surface of balls compresses, but most do not. This section describes the action when the initial separation is not enough and in subsequent collisions that involve more than 2 balls even when there is an initial separation. This solution simplifies to the simple solution when only 2 balls touch during a collision. It applies to all perfectly elastic identical balls that have no energy losses due to friction and can be approximated by materials such as steel, glass, plastic, and rubber.

For two balls colliding, only the two equations for conservation of momentum and energy are needed to solve the two unknown resulting velocities. 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 to solve for all five post-collision velocities.

Newtonian, Lagrangian, Hamiltonian, and stationary action are the different ways of mathematically expressing classical mechanics. They describe the same physics but must be solved by different methods. All enforce the conservation of energy and momentum. Newton's law has been used in research papers. It is applied to each ball and the sum of forces is made equal to zero. So there are 5 equations, one for each ball, and 5 unknowns, one for each velocity. If the balls are identical, the absolute compressibility of the surfaces becomes irrelevant because it can be divided out of both sides of all 5 equations, into zero.

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 surfaces. Most materials like steel that are efficiently elastic will approximately follows Hooke's force law for springs, $$F=k\cdot x$$, but 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. 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 with initially touching balls the action is the same for the first strike, except the time to complete a collision will decrease in softer materials. 40% to 50% of the kinetic energy of the initial ball from a single-ball strike 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, the first ball having the largest at −7% of the initial velocity. This separates the balls, but they will come back together just before as the fifth ball returns. This is due to the pendulum phenomenon of different small angle disturbances having approximately the same time to return to the center. When balls are "touching" in subsequent collisions is complex, but still determinable by this method, especially if friction losses is included and the pendulum timing is calculated exactly instead of relying on the small angle approximation.

The differential equations with the initial separations are needed if there is less than 10 µm separation (less than width of a human hair) when using 100 gram steel balls with an initial 1 m/s strike speed.

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. This is 2.03 times more kinetic energy in the fifth ball than the fourth ball, which means the fifth ball would swing twice as high in the vertically direction 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 ≈10 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 gives a more accurate result.

Effect of pressure waves
The forces in the Hertzian solution above were assumed to be propagated in the balls immediately which is not the case. Sudden changes in the force between the atoms of a material builds up to form a pressure wave. Pressure waves (aka sound) in steel travel about 5 cm in 10 microseconds which is about 10 times faster than the time between the first ball striking and the last ball being ejected. The pressure waves reflect back and forth through all 5 balls about 10 times, although dispersing to less of a wave front with more reflections. This is fast enough for the Hertzian solution to not require a substantial modification to adjust for the delay in force propagation through the balls. In less-rigid but still very elastic balls such as rubber, the propagation speed is slower, but the duration of collisions is longer, so the Hertzian solution still applies. The error introduced by the limited speed of the force propagation biases the Hertzian solution towards the simple solution because the collisions are not affected as much by the inertia of the balls that are further away.

Identically-shaped balls help the pressure waves converge on the contact point of the last ball: at the initial strike point one pressure wave goes forward to the other balls while another goes backwards to reflect off the opposite side of the first ball, and then it follows the first wave, being exactly 1 ball-diameter behind. The two waves meet up at the last contact point because the first wave reflects off the opposite side of the last ball and it meets up at the last contact point with the second wave. Then they reverberate back and forth like this about 10 times until the first ball stops connecting with the 2nd ball. Then the reverberations will reflect off the contact point between the 2nd and 3rd balls, but still converging at the last contact point, until the last ball is ejected. But it's less of a wave front with each reflection.

Effect of different types of balls
Using different types of material will not change the action as long as the material is efficiently elastic. The size of the spheres will not change the results unless the increased weight exceeds the elastic limit of the material. If the solid balls are too large, energy will be lost as heat because the elastic limit increases as radius1.5 but the energy needed to be absorbed and released increases as radius3. Making the contact surfaces flatter can overcome this to an extent by distributing the compression to a larger amount of material but it can introduce an alignment problem. Steel is better than most materials because it allows the simple solution to apply more often in collisions after the first strike, its elastic range for storing energy remains good despite the higher energy caused by its weight, and the higher weight decreases the effect of air resistance.

Heat and friction losses
This discussion has neglected energy losses from heat generated in the balls from non-perfect elasticity, friction in the strings, friction from air resistance, and sound generated from the clank of the vibrating balls. The energy losses are the reason the balls eventually come to a stop, but they are not the primary or initial cause of the action to become more disorderly, away from the ideal action of only one ball moving at any instant. The increase in the non-ideal action is caused by collisions that involve more than two balls at a time, effectively making the struck ball appear heavier. The size of the steel balls is limited because the collisions may exceed the elastic limit of the steel, deforming it and causing heat losses.

Applications
The most common application is that of a desktop executive toy. Another use is as an educational physics demonstration, as an example of conservation of momentum and conservation of energy.

A similar principle, the propagation of waves in solids, was used in the Constantinesco Synchronization gear system for propeller / gun synchronizers on early fighter aircraft.

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 mass with the motion of the moving body being transferred to the 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.

There is much confusion over the origins of the modern Newton's cradle. Marius J. Morin has been credited as being the first to name and make this popular executive toy. However, in early 1967, an English actor, Simon Prebble, coined the name "Newton's cradle" (now used generically) for the wooden version manufactured by his company, Scientific Demonstrations Ltd. After some initial resistance from retailers, they were first sold by Harrods of London, thus creating the start of an enduring market for executive toys. Later a very successful chrome design for the 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 consisted of five one-ton concrete and steel rebar-filled buoys suspended from a steel truss. The buoys also had a steel plate inserted in between their two halves to act as a "contact point" for transferring the energy; this cradle device did not function well because concrete is not elastic so most of the energy was lost to a heat buildup in the concrete. A smaller scale version constructed by them consists of five 6 inch chrome steel ball bearings, each weighing 33 lb, and is nearly as efficient as a desktop model.

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

In popular culture
Newton's cradle has been used more than 20 times in movies, often as a trope on the desk of a lead villain such as Paul Newman's role in The Hudsucker Proxy, Magneto in X-Men, and the Kryptonians in Superman II. It was used to represent the unyielding position of the NFL towards head injuries in Concussion. It has also been used as a relaxing diversion on the desk of lead intelligent/anxious/sensitive characters such as Henry Winkler's role in Night Shift, Dustin Hoffman's role in Straw Dogs, and Gwyneth Paltrow's role in Iron Man 2. It was featured more prominently as a series of clay pots in Rosenkrantz and Guildenstern are Dead, and as a row of 1968 Eero Aarnio bubble chairs with scantily-clad women in them in Gamer. In Storks, Hunter the CEO of Cornerstore has one not with balls, but with little birds.

Literature

 * B. Brogliato: Nonsmooth Mechanics. Models, Dynamics and Control, Springer, 2nd Edition, 1999.
 * B. Brogliato: Nonsmooth Mechanics. Models, Dynamics and Control, Springer, 2nd Edition, 1999.