Newton's cradle is a device, usually made of metal, that demonstrates the principles of conservation of momentum and conservation of energy in physics with swinging spheres.
When one sphere at the end is lifted and released, it strikes the stationary spheres, compressing them and thereby transmitting a pressure wave through the stationary spheres, which creates a force that pushes the last sphere upward. The last sphere swings back and strikes the stationary spheres, repeating the effect in the opposite direction. The cradle thus demonstrates conservation of momentum and energy.
The device is named after 17th-century English scientist Sir Isaac Newton and was 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).
Operation
When one of the balls at the end ("the first") is pulled sideways, the attached string constrains it along an upward arc. When released it strikes the second ball and comes nearly, but not entirely, to a dead stop. The succeeding ball acquires most of the velocity of the first ball and propagates the diminished momentum. Eventually the last ball, having received a successively diminished portion of the first's energy and momentum, begins the process anew in the opposite direction. Each impact produces a sonic wave that propagates through the medium of the intermediate balls. (Any efficiently elastic material such as steel suffices, as long as the kinetic energy is temporarily stored as potential energy in the compression of the material rather than being lost as heat. This is similar to bouncing one coin of a line of touching coins by striking it with another coin, and which happens even if the first struck coin is constrained by pressing on its center such that it cannot move.) In each phase of the process, efficient mechanical energy is lost; Newton's cradle is not a perpetual motion machine. This would stand true even in the absence of air resistance, as in a vacuum.
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. Thus, this first-level explanation is a true, but incomplete, description of the motion.
Physics explanation
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 The conservation of momentum and kinetic energy 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 have the same mass, 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-mass 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-mass 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 of its momentum to the third ball and stops. The third ball then transfers the momentum to the fourth ball and stops, and then the fourth to the fifth ball.
Right behind this sequence, the second moving ball is transferring its momentum 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 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. As a medical 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. In the Indian game carrom, a striker stops after hitting a stationary playing piece, transferring all of its momentum into the piece that was hit.
When the simple solution applies 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 more massive. 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
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 velocitiesC F Gauld (2006), Newton's cradle in physics education, Science & Education, 15, 597–617 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, , 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, . This and Newton's law for motion () are applied to each ball, giving five simple but interdependent differential equations that can be 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. Forty to fifty percent 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. Of the initial velocity, 13% 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.