Newtons cradle что это
Обновлено: 30.06.2024
If you pull a ball up and out and then release it, it falls back and collides with the others with a loud click. Then, instead of all four remaining balls swinging out, only the ball on the opposite end jumps forward, leaving its comrades behind, hanging still. That ball slows to a stop and then falls back, and all five are briefly reunited before the first ball is pushed away from the group again.
This is a Newton's cradle, also called a Newton's rocker or a ball clicker. It was so-named in 1967 by English actor Simon Prebble, in honor of his countryman and revolutionary physicist Isaac Newton.
Despite its seemingly simple design, the Newton's cradle and its swinging, clicking balls isn't just an ordinary desk toy. It is, in fact, an elegant demonstration of some of the most fundamental laws of physics and mechanics.
The toy illustrates the three main physics principles at work: conservation of energy, conservation of momentum and friction. In this article, we'll look at those principles, at elastic and inelastic collisions, and kinetic and potential energy. We'll also examine the work of such great thinkers as Rene Descartes, Christiaan Huygens and Isaac Newton himself.
Given that Isaac Newton was one of the early founders of modern physics and mechanics, it makes perfect sense that he would invent something like the cradle, which so simply and elegantly demonstrates some of the basic laws of motion he helped describe.
Despite its name, the Newton's cradle isn't an invention of Isaac Newton, and in fact the science behind the device predated Newton's career in physics. John Wallis, Christopher Wren and Christiaan Huygens all presented papers to the Royal Society in 1662, describing the theoretical principles that are at work in the Newton's cradle. It was Huygens in particular who noted the conservation of momentum and of kinetic energy [source: Hutzler, etal]. Huygens did not use the term "kinetic energy," however, as the phrase wouldn't be coined for nearly another century; he instead referred to "a quantity proportional to mass and velocity squared' [source: Hutzler, et al.].
Conservation of momentum had first been suggested by French philosopher Rene Descartes (1596 - 1650), but he wasn't able to solve the problem completely -- his formulation was momentum equals mass times speed (p=mv). While this worked in some situations, it didn't work in the case of collisions between objects [source: Fowler].
It was Huygens who suggested changing "speed" to "velocity" in the formula, which solved the problem. Unlike speed, velocity implies a direction of motion, so the momentum of two objects of the same size traveling the same velocity in opposite directions would be equal to zero.
Even though he didn't develop the science behind the cradle, Newton gets name credit for two main reasons. First, the law of conservation of momentum can be derived from his second law of motion (force equals mass times acceleration, or F=ma). Ironically, Newton's laws of motion were published in 1687, 25 years after Huygens provided the law of conservation of momentum. Second, Newton had a greater overall impact on the world of physics and therefore more fame than did Huygens.
While there can be many aesthetic modifications, a normal Newton's cradle has a very simple setup: Several balls are hung in a line from two crossbars that are parallel to the line of the balls. These crossbars are mounted to a heavy base for stability.
On small cradles, the balls are hung from the crossbars by light wire, with the balls at the point of an inverted triangle. This ensures that the balls can only swing in one plane, parallel to the crossbars. If the ball could move on any other plane, it would impart less energy to the other balls in the impact or miss them altogether, and the device wouldn't work as well, if at all.
All the balls are, ideally, exactly the same size, weight, mass and density. Different-sized balls would still work, but would make the demonstration of the physical principles much less clear. The cradle is meant to show the conservation of energy and momentum, both of which involve mass. The impact of one ball will move another ball of the same mass the same distance at the same speed. In other words, it'll do the same amount of work on the second ball as gravity did on the first one. A larger ball requires more energy to move the same distance -- so while the cradle will still work, it makes it more difficult to see the equivalence.
As long as the balls are all the same size and density, they can be as big or as small as you like. The balls must be perfectly aligned at the center to make the cradle work the best. If the balls hit each other at some other point, energy and momentum is lost by being sent in a different direction. There's usually an odd number of balls, five and seven being the most common, though any number will work.
So now that we've covered how the balls are set up, let's look at what they're made of and why.
In a Newton's Cradle, ideal balls are made out of a material that is very elastic and of uniform density. Elasticity is the measure of a material's ability to deform and then return to its original shape without losing energy; very elastic materials lose little energy, inelastic materials lose more energy. A Newton's cradle will move for longer with balls made of a more elastic material. A good rule of thumb is that the better something bounces, the higher its elasticity.
Stainless steel is a common material for Newton's cradle balls because it's both highly elastic and relatively cheap. Other elastic metals like titanium would also work well, but are rather expensive.
It may not look like the balls in the cradle deform very much on impact. That's true -- they don't. A stainless steel ball may only compress by a few microns when it's hit by another ball, but the cradle still functions because steel rebounds without losing much energy.
The density of the balls should be the same to ensure that energy is transferred through them with as little interference as possible. Changing the density of a material will change the way energy is transferred through it. Consider the transmission of vibration through air and through steel; because steel is much denser than air, the vibration will carry farther through steel than it will through air, given that the same amount of energy is applied in the beginning. So, if a Newton's cradle ball is, for example, more dense on one side than the other, the energy it transfers out the less-dense side might be different from the energy it received on the more-dense side, with the difference lost to friction.
Other types of balls commonly used in Newton's cradles, particularly ones meant more for demonstration than display, are billiard balls and bowling balls, both of which are made of various types of very hard resins.
Amorphous metals are a new kind of highly elastic alloy. During manufacturing, molten metal is cooled very quickly so it solidifies with its molecules in random alignment, rather than in crystals like normal metals. This makes them stronger than crystalline metals, because there are no ready-made shear points. Amorphous metals would work very well in Newton's cradles, but they're currently very expensive to manufacture.
Conservation of Energy
The law of conservation of energy states that energy -- the ability to do work -- can't be created or destroyed. Energy can, however, change forms, which the Newton's Cradle takes advantage of -- particularly the conversion of potential energy to kinetic energy and vice versa. Potential energy is energy objects have stored either by virtue of gravity or of their elasticity. Kinetic energy is energy objects have by being in motion.
Let's number the balls one through five. When all five are at rest, each has zero potential energy because they cannot move down any further and zero kinetic energy because they aren't moving. When the first ball is lifted up and out, its kinetic energy remains zero, but its potential energy is greater, because gravity can make it fall. After the ball is released, its potential energy is converted into kinetic energy during its fall because of the work gravity does on it.
When the ball has reached its lowest point, its potential energy is zero, and its kinetic energy is greater. Because energy can't be destroyed, the ball's greatest potential energy is equal to its greatest kinetic energy. When Ball One hits Ball Two, it stops immediately, its kinetic and potential energy back to zero again. But the energy must go somewhere -- into Ball Two.
Ball One's energy is transferred into Ball Two as potential energy as it compresses under the force of the impact. As Ball Two returns to its original shape, it converts its potential energy into kinetic energy again, transferring that energy into Ball Three by compressing it. The ball essentially functions as a spring.
This transfer of energy continues on down the line until it reaches Ball Five, the last in the line. When it returns to its original shape, it doesn't have another ball in line to compress. Instead, its kinetic energy pushes on Ball Four, and so Ball Five swings out. Because of the conservation of energy, Ball Five will have the same amount of kinetic energy as Ball One, and so will swing out with the same speed that Ball One had when it hit.
One falling ball imparts enough energy to move one other ball the same distance it fell at the same velocity it fell. Similarly, two balls impart enough energy to move two balls, and so on.
But why doesn't the ball just bounce back the way it came? Why does the motion continue on in only one direction? That's where momentum comes into play.
Conservation of Momentum
Momentum is the force of objects in motion; everything that moves has momentum equal to its mass multiplied by its velocity. Like energy, momentum is conserved. It's important to note that momentum is a vector quantity, meaning that the direction of the force is part of its definition; it's not enough to say an object has momentum, you have to say in which direction that momentum is acting.
When Ball One hits Ball Two, it's traveling in a specific direction -- let's say east to west. This means that its momentum is moving west as well. Any change in direction of the motion would be a change in the momentum, which cannot happen without the influence of an outside force. That is why Ball One doesn't simply bounce off Ball Two -- the momentum carries the energy through all the balls in a westward direction.
But wait. The ball comes to a brief but definite stop at the top of its arc; if momentum requires motion, how is it conserved? It seems like the cradle is breaking an unbreakable law. The reason it's not, though, is that the law of conservation only works in a closed system, which is one that is free from any external force -- and the Newton's cradle is not a closed system. As Ball Five swings out away from the rest of the balls, it also swings up. As it does so, it's affected by the force of gravity, which works to slow the ball down.
A more accurate analogy of a closed system is pool balls: On impact, the first ball stops and the second continues in a straight line, as Newton's cradle balls would if they weren't tethered. (In practical terms, a closed system is impossible, because gravity and friction will always be factors. In this example, gravity is irrelevant, because it's acting perpendicular to the motion of the balls, and so does not affect their speed or direction of motion.)
The horizontal line of balls at rest functions as a closed system, free from any influence of any force other than gravity. It's here, in the small time between the first ball's impact and the end ball's swinging out, that momentum is conserved.
When the ball reaches its peak, it's back to having only potential energy, and its kinetic energy and momentum are reduced to zero. Gravity then begins pulling the ball downward, starting the cycle again.
Elastic Collisions and Friction
There are two final things at play here, and the first is the elastic collision. An elastic collision occurs when two objects run into each other, and the combined kinetic energy of the objects is the same before and after the collision. Imagine for a moment a Newton's cradle with only two balls. If Ball One had 10 joules of energy and it hit Ball Two in an elastic collision, Ball Two would swing away with 10 joules. The balls in a Newton's cradle hit each other in a series of elastic collisions, transferring the energy of Ball One through the line on to Ball Five, losing no energy along the way.
At least, that's how it would work in an "ideal" Newton's cradle, which is to say, one in an environment where only energy, momentum and gravity are acting on the balls, all the collisions are perfectly elastic, and the construction of the cradle is perfect. In that situation, the balls would continue to swing forever.
But it's impossible to have an ideal Newton's cradle, because one force will always conspire to slow things to a stop: friction. Friction robs the system of energy, slowly bringing the balls to a standstill.
Though a small amount of friction comes from air resistance, the main source is from within the balls themselves. So what you see in a Newton's cradle aren't really elastic collisions but rather inelastic collisions, in which the kinetic energy after the collision is less than the kinetic energy beforehand. This happens because the balls themselves are not perfectly elastic -- they can't escape the effect of friction. But due to the conservation of energy, the total amount of energy stays the same. As the balls are compressed and return to their original shape, the friction between the molecules inside the ball converts the kinetic energy into heat. The balls also vibrate, which dissipates energy into the air and creates the clicking sound that is the signature of the Newton's cradle.
Imperfections in the construction of the cradle also slow the balls. If the balls aren't perfectly aligned or aren't exactly the same density, that will change the amount of energy it takes to move a given ball. These deviations from the ideal Newton's cradle slow down the swinging of the balls on either end, and eventually result in all the balls swinging together, in unison.
For more details on Newton's cradles, physics, metals and other related subjects, take a look at the links that follow.
Contents
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.
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. The simplification overestimates the kinetic energy in the 5th ball by 2.2%. 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 x velocity) and kinetic energy (0.5 x mass x velocity^2) can be used to find the resulting velocities for 2 colliding elastic balls (see elastic collision). For 3 or more balls, the velocities can be calculated in the same way if the collisions are a sequence of separate collisions between pairs. In Newton's cradle, all the balls weigh the same, so 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). If both are "moving", you can pick one to be your "stationary" frame of reference. This simple and interesting 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 2 balls separated by a very small distance are dropped and strike 3 stationary balls the action is as follows: The 1st ball to strike (the 2nd ball in the cradle) transfers 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 1st sequence, ejecting the 4th ball right behind the 5th ball with the same microscopic separation that was between the two initial striking balls. If the two initial balls had been microscopically welded together, the initial strike would be the same as one ball having twice the weight and this results in only the last ball moving away much faster than the others in both theory and practice, so the initial separation is important.
When the simple solution applies, the balls more efficiently transfer the velocity from one ball to the next, maintaining the interesting effect. So contrary to intuition, the effects are more noticeable when the balls are not touching and therefore more closely follow independent collisions.
When simple solution applies
In order for the simple solution to theoretically apply, no pair in the midst of colliding can touch a 3rd ball. This is because applying the two conservation equations to 3 or more balls in a single collision results in many possible solutions.
"Touching" in this discussion means when a ball is still compressed on one side during a collision, it begins compression on the other side from the next collision. So "touching" may include small initial separations, which will need the complete Hertzian solution described below. If the separations are large enough to prevent simultaneous collisions, the Hertzian differential equations simplify to the case of independent collision pairs.
Small steel balls work well because they remain efficiently elastic (less heat loss) under strong strikes and hardly compress (up to about 30 microns 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. So 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 interesting effect from pair-wise collisions. For example, when 2 balls strike, there needs to be about 1/2 mm separation for rubber balls much in order to get the 4th and 5th balls to eject with nearly the same velocity, but only half the width of a hair for steel balls.
The extra variables needed to determine the solution for 3 or more simultaneously colliding elastic balls are the relative compressibilities of the colliding surfaces. For example, 5 balls have 4 colliding points and scaling (dividing) 3 of the compressibilities by the 4th will give the 3 extra variables needed (in addition to the two conservation equations) to solve for all 5 post-collision velocities. The compressions of the surfaces are interacting in a way that makes a deterministic algebraic solution difficult to find. Numerical step-wise solutions to the differential equations have been used.
More complete solution
Determining the velocities for the case of 1 ball striking 4 "touching" balls is found by modeling the balls as weights with non-traditional springs on their colliding surface. Steel is elastic and follows Hook's force law for springs, F=k*x, but because the area of contact for a sphere increases as the force increases, colliding elastic balls will follow Hertz's adjustment to Hook's law, F=k*x^1.5. This and Newton's law for motion (F=m*a) are applied to each ball, giving 5 simple but interdependent ("touching") differential equations that are solved numerically. [ 9 ] When the 5th ball begins accelerating, it is receiving momentum and energy from the 3rd and 4th 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 4th ball (which can be seen as a 3.3 degree movement if the 5th ball moves out 25 degrees) and there is a slight reverse velocity in the first 3 balls, -7% in the first ball. This separates the balls, but they will come back together just before the 5th 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 microns 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 micron separation, a higher strike speed, or heavier balls. [ 10 ]
The Hertzian differential equations predict that if 2 balls strike 3, the 5th and 4th balls will leave with velocities of 1.14 and 0.80 times the initial velocity. [ 11 ] This is 2.03 times more kinetic energy in the 5th ball than the 4th ball, which means the 5th ball should swing twice as high as the 4th ball. But in a real Newton's cradle the 4th ball swings out as far as the 5th ball. In order to explain the difference between theory and experiment, the 2 striking balls must have at least 20 microns 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 come out more accurate.
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. [ 12 ]
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. These energy losses are why 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.
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. [ 13 ] [ 14 ] Newton acknowledged Mariotte's work, among that of others, in his Principia.
Invention and design
There is much confusion over the origins of the modern 'Newton's cradle'. An unknown designer in Canada has been credited, without any substantiating evidence, 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 his iconic wooden version manufactured by his company, Scientific Demonstrations Ltd . After some initial resistance from uncomprehending 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 consists of five 1-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 recently [ when? ] taken down due to its need for frequent maintenance.
Newton's cradle
Newton's cradle — noun A sophisticated toy consisting of five metal balls hanging in a frame, caused to hit against one another at speeds which vary • • • Main Entry: ↑newton … Useful english dictionary
Newton's cradle — The cradle in motion … Wikipedia
Cradle — may refer to: Mechanical devices: Bassinet, a small bed, often on rockers, in which babies and small children sleep Ship cradle, supports a ship that is dry docked Cradle (grain), in agriculture is a device based upon a scythe to cleanly reap and … Wikipedia
Newton (Blake) — Newton (1795 1805) 460 x 600 mm. Collection Tate Britain Newton is a monotype by the English poet, painter and printmaker William Blake first completed in 1795,[1] but reworked and reprinted in 1805 … Wikipedia
Isaac Newton — Sir Isaac Newton … Wikipedia
Pendule de newton — Le pendule de Newton est un pendule particulier se composant de cinq billes et permettant d illustrer les théories de conservation de la quantité de mouvement et de l énergie. Sommaire 1 Description 2 Expériences … Wikipédia en Français
Newtons cradle — Schwingendes Kugelstoß oder Newton Pendel Ein Kugelstoßpendel (auch Kugelpendel, Newtonpendel oder Newton Wiege) ist eine Anordnung von hintereinander beidseitig aufgehängten Kugeln gleicher Masse und Pendellänge. Wenn man die am weitesten rechts … Deutsch Wikipedia
MythBusters (2011 season) — Country of origin Australia United States Broadcast Original channel Discovery Channel Original run … Wikipedia
Applications
The most common application is that of a desktop executive toy. A less common use is as an educational aid, where a tutor explains what is happening or challenges students to do so.
Newton's cradle
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 an executive ball clicker, [ 1 ] Newton's balls, [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] Newton's pendulum, [ 7 ] or Newtonian Demonstrator. [ 8 ]
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 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.
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 doesn't the initial ball (or balls) 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 momentum can show this is a possible solution, but they can't be used to predict the final velocities when there are 3 or more balls in a cradle because they provide only 2 equations to find the 3 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.
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