Cradle pendulum

Cradle pendulum DEFAULT

Newton's Cradle

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Simulation of Newton's Cradle, a device consisting of several steel balls suspended from a cage. Uses the 2D Rigid Body Physics Engine.

Each ball acts like an individual pendulum. At rest all the balls are touching. The surprising thing is that when you lift one of the end balls and let it hit the resting balls, only the ball at the other end moves, and the initial ball stops at rest.

Click near an object to exert a rubber band force with your mouse. Click reset to set the balls at rest. Try changing gravity, elasticity (bounciness), and damping (friction).

Experiments to try:

  • Turn on "show energy". When is kinetic energy highest? When is potential energy highest?
  • Reduce elasticity (bounciness) to less than 1, for example 0.9
  • Make a small gap between each ball by increasing the offset to above zero, for example 0.1
  • Try using "simultaneous" collision handling method
  • Turn on "show forces" to see the contact forces and collision impacts.

There are "joints" attaching each pendulum to the background, two joints for each connection: one vertical, one horizontal. Each pendulum is a single rigid body, as opposed to being made of a flexible rope and steel ball in the real world toy.

The method of collision handling for this simulation is important. For the balls to be in resting contact with each other, only the "simultaneous" or "hybrid" methods of collision handling work correctly.

Also available are: open source code, documentation and a simple-compiled version which is more customizable.


AIXICWXI Newton's Cradle Balance Pendulum Ball Newton's Cradle Balance Balls Physics Science Decompression Pendulum Ornaments , Metal Pendulum Balls Made of Wooden Base & Steel Ball

AIXICWXI Newton's Cradle Balance Pendulum Ball Newton's Cradle Balance Balls Physics Science Decompression Pendulum Ornaments , Metal Pendulum Balls Made of Wooden Base & Steel Ball


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Product Details

  • 【High Quality】With steel frame and 5pcs high polished smooth steel balance balls, which is more durable and easy to transfer and store more energy when collide.Every steel ball has been precisely measured for uniform and equal weight. Our newton cradle use high quality nylon line which is strong enough to not be broken easily.The hard wooden base can keep the smooth operation of the newton's cradle balance balls.
  • 【Longer Swing Time】Balance Pendulum Ball could last for 15-20 seconds due to the resistance of the air and friction. You could find many ways to play, such as one ball on one end strikes adjacent sending the one on the opposite end into the air while the ones in the middle stay perfectly still, or two, or more... Waiting for you to discover more fun and stress-relieving ways to play.
  • 【Educational Tools】Newton's Cradle is also a perfectly educational gadget for schools. It can lead kids to understand the meaning of energy conservation. Let children received the baptism of science while playing. It not only shows a pendulum, but also shows the laws of conservation of Momentum and energy.
  • 【Applicating】 This Newton's Cradle is a fascinating decoration for desks. Not only could add beauty to the room, but also relieve stress when you playing. It is a cool and fun games for office man avoid anxious.The light of the silver balls shines in the study makes the whole room more stylish.
  • 【Perfect Gift】The Balance Balls Toy can be a fun gift to your friends, family and children. Everything is carefully packaged and secured in place to ensure the ropes get tangled on arrival.
Product Dimensions5.5 x 5.1 x 4.4 inches (14 x 13 x 11.2 cm)
Item Weight1.1 pounds (0.5 kg)
Country of OriginChina
Manufacturer recommended age6 years and up
Mfg Recommended age6 - 60 years


This classic desktop toy provides you many benefits and entertainments, is suitable for various locations such as house or office, and is an intriguing device that not only demonstrates pendulum motion, but also helps to show the laws of Conservation of Momentum and Energy. Whether you are studying physics knowledge or looking for a delicate gift, this balance ball is definitely what you .


Excellent educational science gadget. Not only demonstrates a pendulum, but also shows the laws of conservation of momentum and energy.
Science toy. When feeling tired of studying and working, you can stir balls to hear the clinking sound to ease.
Excellent educational & science gadget, own your very own piece of scientific history and explore significant values of physics.
Pull back one or more of the balance balls, release them and the same number of balls will swing up on the opposite side.
Great for home ornament, study or office desk table decor accessories, etc. Also good for birthday gift and kids toy.


Condition: 100% brand new
Material: Solid Wood + Stainless Steel
Color: Black+Sliver
Frame: Stainless Steel
Base: Solid Wood
Size: 14 x 13 x 11.3cm /5.5x5.1x4.4 inch(appr.)
Weight:500g /1.1Ib (Include Packing Box)
Occasions: Daily use, Education supplies, Psychotherapy, Birthday present, Business gift

Package Included:
1 x Balance Balls Toy


1.When you use this newton's cradle, please move it in a horizontal line, so the lines will not be tangled.
2.Please don't worry, if it was tangled, you could untangle it still, just need some patience.DIY your own cradle!
3.Please allow 1-2 cm error due to manual measurement and make sure you do not mind before ordering.
4.Please understand that colors may exist chromatic aberration as the different placement of pictures.

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Newton's cradle

Device that demonstrates conservation of momentum and energy via a series of swinging spheres

The Newton's cradle is a device that demonstrates the conservation of momentum and the conservation of energy with swinging spheres. When one sphere at the end is lifted and released, it strikes the stationary spheres, transmitting a force through the stationary spheres that pushes the last sphere upward. The last sphere swings back and strikes the nearly stationary spheres, repeating the effect in the opposite direction. The device is named after 17th-century English scientist Sir Isaac Newton and designed by French scientist Edme Mariotte. It is also known as Newton's pendulum, Newton's balls, Newton's rocker or executive ball clicker (since the device makes a click each time the balls collide, which they do repeatedly in a steady rhythm).[1][2]


When one of the end balls ("the first") is pulled sideways, the attached string makes it 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 does 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 impact of the first ball. When two (or three) balls are dropped, the two (or three) balls on the opposite side swing out. Some say 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.[3]

Physics explanation[edit]

Newton's cradle with two balls of equal weight and perfectly efficient elasticity. The left ball is pulled away and let go. Neglecting the energy losses, the left ball strikes the right ball, transferring all the velocity to the right ball. Because they are the same weight, the same velocity indicates all the momentum and energy are also transferred. The kinetic energy, as determined by the velocity, is converted to potential energy as it reaches the same height as the initial ball and the cycle repeats.
An idealized Newton's cradle with five balls when there are no energy losses and there is always a small separation between the balls, except for when a pair is colliding
Newton's cradle three-ball swing in a five-ball system. The central ball swings without any apparent interruption.

Newton's cradle can be modeled fairly accurately with simple mathematical equations with the assumption that the balls always collide in pairs. If one ball strikes four stationary balls that are already touching, these simple equations can not explain the resulting movements in all five balls, which are not due to friction losses. For example, in a real Newton's cradle the fourth 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[edit]

The conservation of momentum (mass × velocity) and kinetic energy (1/2 × mass × velocity2) 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 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 there is no need to account for heat and sound energy losses.

Steel does not compress much, but its elasticity is very efficient, so 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 effect seen in the cradle and gives an approximate solution to all its activities.

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 second ball striking the third ball) transfers all 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 second moving ball transferring its velocity to the first moving ball that just stopped, and the sequence repeats immediately and imperceptibly behind the first sequence, ejecting the fourth ball right behind the fifth ball with the same small separation that was between the two initial striking balls. If they are simply touching when they strike the third ball, precision requires the more complete solution below.

Other examples of this effect[edit]

The 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 as 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 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[edit]

For the simple solution to precisely predict the action, no pair in the midst of colliding may touch the third ball, because the presence of the third ball effectively makes the struck ball 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 to maximize the effect from pair-wise collisions.

More complete solution[edit]

A cradle that best follows the simple solution needs to have an initial separation between the balls that measures at least twice the amount that any one ball 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 two balls even when there is an initial separation. This solution simplifies to the simple solution when only two 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 gives 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 five equations, one for each ball—and five 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 five equations, producing zero.

Determining the velocities[4][5][6] 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 approximately follow Hooke's force law for springs, {\displaystyle F=k\cdot x}, but because the area of contact for a sphere increases as the force increases, colliding elastic balls follow Hertz's adjustment to Hooke's law, {\displaystyle F=k\cdot x^{1.5}}. This and Newton's law for motion ({\displaystyle 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 increases 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. Thirteen percent 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 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 are 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 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.[7] 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 vertical direction as the fourth ball. But in a real Newton's cradle, the fourth ball swings out as far as the fifth ball. 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[edit]

The forces in the Hertzian solution above were assumed to propagate in the balls immediately, which is not the case. Sudden changes in the force between the atoms of material build up to form a pressure wave. Pressure waves (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 five balls about ten times, although dispersing to less of a wavefront 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 backward 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 second ball. Then the reverberations reflect off the contact point between the second and third balls, but still converge at the last contact point, until the last ball is ejected—but it is less of a wavefront with each reflection.

Effect of different types of balls[edit]

Using different types of material does not change the action as long as the material is efficiently elastic. The size of the spheres does not change the results unless the increased weight exceeds the elastic limit of the material. If the solid balls are too large, energy is being lost as heat, because the elastic limit increases with the radius raised to the power 1.5, but the energy which had to be absorbed and released increases as the cube of the radius. 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.


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.[further explanation needed]


Large Newton's cradle at American Science and Surplus

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 physicistAbbé Mariotte in the 17th century.[1][8] 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.[citation needed] 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.[9] 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.[citation needed] Later a very successful chrome design for the Carnaby Street store Gear was created by the sculptor and future film director Richard Loncraine.[citation needed]

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.[10] 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 15-centimetre (6 in) chrome steel ball bearings, each weighing 15 kilograms (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 (see photo). Each ball was an inflatable exercise ball 66 cm (26 in) in diameter (encased in steel rings), and was supported from the ceiling using extremely strong magnets. It was dismantled in early August 2010 due to maintenance concerns.[citation needed]

In popular culture[edit]

Newton's cradle has been used more than 20 times in movies,[11] 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.[12] 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 Rosencrantz and Guildenstern Are Dead, and as a row of 1968 Eero Aarnio bubble chairs with scantily-clad women in them in Gamer.[13] In Storks, Hunter the CEO of Cornerstore has one not with balls, but with little birds. Newton's Cradle is an item in Nintendo's Animal Crossing where it is referred to as "executive toy".[14] In 2017, an episode of the Omnibuspodcast, featuring Jeopardy! champion Ken Jennings and musician John Roderick, focused on the history of Newton's Cradle.[15] Newton's cradle is also featured on the desk of Deputy White House Communications Director Sam Seaborn in The West Wing.

Rock band Jefferson Airplane used the cradle on the 1968 album Crown of Creation as a rhythm device to create polyrhythms on an instrumental track.

See also[edit]


  1. ^ ab"Newton's Cradle". Harvard Natural Sciences Lecture Demonstrations. Harvard University. 27 February 2019.
  2. ^Palermo, Elizabeth (28 August 2013). "How Does Newton's Cradle Work?". Live Science.
  3. ^Gauld, Colin F. (August 2006). "Newton's Cradle in Physics Education". Science & Education. 15 (6): 597–617. Bibcode:2006Sc&Ed..15..597G. doi:10.1007/s11191-005-4785-3. S2CID 121894726.
  4. ^Herrmann, F.; Seitz, M. (1982). "How does the ball-chain work?"(PDF). American Journal of Physics. 50. pp. 977–981. Bibcode:1982AmJPh..50..977H. doi:10.1119/1.12936.
  5. ^Lovett, D. R.; Moulding, K. M.; Anketell-Jones, S. (1988). "Collisions between elastic bodies: Newton's cradle". European Journal of Physics. 9 (4): 323. Bibcode:1988EJPh....9..323L. doi:10.1088/0143-0807/9/4/015.
  6. ^Hutzler, Stefan; Delaney, Gary; Weaire, Denis; MacLeod, Finn (2004). "Rocking Newton's Cradle"(PDF). American Journal of Physics. 72. pp. 1508–1516. Bibcode:2004AmJPh..72.1508H. doi:10.1119/1.1783898.C F Gauld (2006), Newton's cradle in physics education, Science & Education, 15, 597–617
  7. ^Hinch, E.J.; Saint-Jean, S. (1999). "The fragmentation of a line of balls by an impact"(PDF). Proc. R. Soc. Lond. A. 455. pp. 3201–3220.
  8. ^Wikisource:Catholic Encyclopedia (1913)/Edme Mariotte
  9. ^Schulz, Chris (17 January 2012). "How Newton's Cradles Work". HowStuffWorks. Retrieved 27 February 2019.
  10. ^"Newton's Crane Cradle (5 October 2011)" at IMDb
  11. ^Most Popular "Newton's Cradle" Titles – IMDb
  12. ^Concussion – Cinemaniac ReviewsArchived 11 February 2017 at the Wayback Machine
  13. ^13 Icons of Modern Design in Movies « Style Essentials
  14. ^Animal Crossing Official Game Guide by Nintendo Power. Nintendo.
  15. ^Omnibus: Newton's Cradle (Entry 835.1C1311)


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

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