You've probably seen this contraption before: Five small silver balls hang in a perfectly straight line by thin threads that attach them to two parallel horizontal bars, which are in turn attached to a base. They sit on office desks around the world.
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.
History of the Newton's Cradle
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.
But he didn't.
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.
Newton's Cradle Design and Construction
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.
Composition of Balls in a Newton's Cradle
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.
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 on the next page.
More Great Links
- Antonick, Gary. "Numberplay: How Does Newton's Cradle Work?" Dec. 6, 2010. (Jan. 10, 2012) http://wordplay.blogs.nytimes.com/2010/12/06/numberplay-newtons-cradle/
- Fowler, Michael. "Momentum, Work and Energy." Nov. 29, 2007. (Jan. 10, 2012) http://galileoandeinstein.physics.virginia.edu/lectures/momentum.html
- Goodstein, David L. "Mechanics." Encyclopedia Britannica. (Jan. 10, 2012) http://www.britannica.com/EBchecked/topic/371907/mechanics
- Hutzler, Stefan, Gary Delaney, et al. "Rocking Newton's Cradle." 5 August 2011. (Jan. 10, 2012) http://www.upscale.utoronto.ca/Practicals/Modules/FormalReport/AJP_Newtons_Cradle.pdf
- Kurtus, Ron. "Derivation of Principles of Newton's Cradle." May 30, 2010. (Jan. 10, 2012) http://www.school-for-champions.com/science/newtons_cradle_derivation.htm
- Simanek, Donald. "Newton's Cradle." May 13, 2003. (Jan. 10, 2012) http://www.lhup.edu/~dsimanek/scenario/cradle.htm
- Understanding Force. "The Law of Conservation of Momentum." (Jan. 10, 2012) http://www.understandingforce.com/momentum.html