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, it strikes the stationary spheres; a force is transmitted through the stationary spheres and pushes the last one upward. The device is also known as Newton's balls or "Executive Ball Clicker".[1][2][3][4]

## 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 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. 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.

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 momentum of the first ball disappears and reappears as the momentum of the last ball as it rises into the air. This shows that the momentum of the system remains constant.

With two balls dropped, exactly two balls on the opposite side swing out and back. With three balls dropped, three balls will swing back and forth, with the central ball appearing to swing without interruption.

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

## 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.[5] [6] 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

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 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 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.[8]

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.

Gravity and the pendulum action influence the middle balls to return near the center positions at nearly the same time in 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.[10]

#### Heat and friction losses

This discussion has assumed there are no heat losses from the balls' striking each other or friction losses from air resistance and the strings. However in the real world, these energy losses are the reason the balls eventually come to a stop. The higher weight of steel reduces the relative effect of air resistance. 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.

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

## Invention and design

Large Newton's cradle at American Science and Surplus

The experimental use of pendular devices, to demonstrate the law of impacts between bodies, was first described by Mariotte in the 17th century.

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. A smaller scale version constructed by them consists of five 6" chrome steel ball bearings, each weighing 33 pounds, and is nearly as efficient as a desktop model.

The cradle device with the largest diameter collision balls on public display, was on display for more than a year in Milwaukee, Wisconsin at retail store American Science and Surplus. Each ball was an inflatable exercise ball 26" 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.

## References

1. ^ "Sciencedemonstrations.fas.harvard.eu". Sciencedemonstrations.fas.harvard.edu. Retrieved 3 November 2011.
2. ^ "Hendrix2.uoregon.edu". Hendrix2.uoregon.edu. Retrieved 3 November 2011.
3. ^ "claymore.engineer.gvsu.edu". claymore.engineer.gvsu.edu. Retrieved 3 November 2011.
4. ^ "Demo.pa.mus.edu". Demo.pa.msu.edu. Retrieved 3 November 2011.
5. ^ "Harvard website page on Newton's Cradle". Retrieved 2007-10-07.
6. ^ "Catholic Encyclopedia: Edme Mariotte". Retrieved 2007-10-07.
7. ^ Herrmann, F.; Seitz, M. (1982). "How does the ball-chain work?". American Journal of Physics 50. pp. 977–981.
8. ^ Lovett, D. R.; Moulding, K. M.; Anketell-Jones, S. (1988). "Collisions between elastic bodies: Newton's cradle". European Journal of Physics 9 (4): 323. doi:10.1088/0143-0807/9/4/015. edit
9. ^ Hinch, E.J.; Saint-Jean, S. (1999). "The fragmentation of a line of balls by an impact". Proc. R. Soc. Lond. A 455. pp. 3201–3220.
10. ^ Hutzler, Stefan; Delaney, Gary; Weaire, Denis; MacLeod, Finn (2004). "Rocking Newton's Cradle". American Journal of Physics 72. pp. 1508–1516.

## Literature

• Herrmann, F. (1981). "Simple explanation of a well-known collision experiment". American Journal of Physics 49 (8): 761. doi:10.1119/1.12407. edit
• B. Brogliato: Nonsmooth Mechanics. Models, Dynamics and Control, Springer, 2nd Edition, 1999.