Objective. A thorough investigation of the design, operation, and functioning of a "Newton's Cradle" apparatus will effectively lead to a broader and clearer understanding of Newton's Third Law, the Law of Conservation of Momentum, the Law of Conservation of Energy, and the behavior of objects experiencing elastic as well as inelastic collisions.
Introduction
History of the "Cradle". The apparatus familiarly and metaphorically known as "Newton's Cradle": is a modern-day outgrowth of a similar apparatus developed and employed by Newton during the late 17th century. Newton's device consisted of a pair of elastic (nondeformable) pendulums of constant dimension, mass, and length that were brought into direct contact via collision when each was in the base-level (vertical) position. Since much was known about the physics of pendulums even in Galileo's (and hence Newton's) time, particularly the ubiquitous and egalitarian force of gravity, direct control of period on length, and negligible frictional resistance, such a device was well-suited to experimentation on and demonstration of dynamic relationships. However, a Newton's Cradle is by no means a simple device. In fact, lively debate and curiosity over the physical behavior of the apparatus invariably springs up when it appears in the physics classroom.
Use and Purpose of the "Cradle". Primarily, apart from being an office desktop toy, Newton's Cradle (or Newtonian Demonstrator to which it has been alternately referred) illustrates Newton's Third Law. That is, forces always occur in pairs and that for every applied force there is an equal but oppositely directed force for a body to be in equilibrium. This is illustrated by simply raising a single pendulum to modest height on one side of the apparatus, releasing it, and observing the response of the four remaining pendulums (at rest) when the collision occurs. The apparatus is also used to demonstrate the Law of Conservation of Linear Momentum.That is, when two (or more) elastic bodies collide, the sum of the momenta of the system of pendulums before collision equals the sum of the momenta of the system of pendulums after collision. This occurs regardless of direction and how many pendulums are raised or remain in the at rest position. Similarly, the elastic nature of the collisions of the pendulums provides an exceptionally clear demonstration of the Law of Conservation of Energy. An elastic collision between two (or more) bodies is one in which the sum of the kinetic energies of the bodies (at the instant) before collision is equal to the sum of the kinetic energies of the bodies (at the instant) after collision. That is, in an elastic collision, whatever kinetic energy is lost by one body is gained by the other body. Additionally, the cyclic conversion of gravitational potential to mechanical kinetic back to gravitational potential energy is the most characteristic feature of the cradle as the pendulums rise and fall in a simple harmonic fashion. This behavior, which is identical to that of a ballistic pendulum, provides a clear demonstration of the features which characterize inelastic collisions in which the mechanical kinetic energy of the pendulums becomes converted into gravitational potential energy during a portion of the swing cycle and is therefore not conserved.
Architectural Features and Structural Design of "Newton's Cradle". The architecture of a "cradle" is simple yet rigorously exacting in dimension, mass, and orientation of each of the structural elements (see figures below).
m1
m2
m3
m4
m5
L1,2,3,4,5
Newton's
Cradle
pivot level
attaching
lines
dA
dB
dC
dD
contact line
base level
L3
L2
L1
L4
L5
Equal Masses: m1 = m2 = m3 = m4 = m5
Equal Lengths: L1 = L2 = L3 = L4 = L5
Equal Distances: dA = dB = dC = dD
Anatomy
of
Newton's Cradle
The device consists of a sturdy four post-and-crossbeam chrome metal frame from which 5 polished steel spheres of equal mass (m1 = m2 = m3 = m4 = m5) suspended from two equal lengths of high tensile strength, low mass nylon fishing line in turn attached to the horizontal crossbeam called the pivot level are affixed.The V-shape of the attaching lines, clearly visible in the end view of the apparatus as well as the lateral swaying character of the pendulums have undoubtedly led to the coining of the term "cradle" for this device. Each pair of attaching lines are exactly the same length, possessing the same dimension in the vertical (L1 = L2 = L3 = L4 = L5). Consequently, the angles between the pairs of strings are equal, and the height of each ball above the horizontal is equal. The lateral distance between adjacent string pairs is exactly equal to the diameter of each of the spheres (dA = dB = dC = dD). Therefore each sphere is in point-contact with adjacent spheres, the points of contact in turn all in alignment. The horizontal line of contact, called the contact line, is the line along which our system of spheres interact. That is, the system is one of linear motion and interaction. Since the lengths of of the attaching lines are equal, the times of descent of the spheres from a raised position above the base level are equal, regardless of the height of the start positions of the spheres. Therefore, interactions between the spheres occur when the center of each is in the contact line, and the attaching lines are all vertical, despite having experienced rotational motion during descent. One must also keep in mind the fact that our "system of interaction" to which we have referred consists of five separate, unattached, discrete spheres of equal mass and elastic character. Each behaves as a separate and discrete elastic body, distinct and apart from the others, as there is no physical connection between any and all spheres. Also, regard the collisions as instantaneous, and with no loss of energy, neither to friction nor to heating.
End View
The Physics of Elastic and Inelastic Collisions.
Inelastic Collisions. Most ordinary collisions are inelastic. The colliding bodies deform each other, generate heat, or remove kinetic energy from the system in other ways. As a result, the total final kinetic energy is less than the total initial kinetic energy. That is, the total kinetic energy of the system of colliding objects is not conserved. However, the Law of Conservation of Momentum continues to hold for inelastic as well as elastic collisions.
Elastic Collisions. There are a number of collisions like those between two steel spheres such as the ones in our cradle or between two billiard balls that are very nearly elastic. In nuclear physics in particular, some of the collisions that we observe between the tiny subatomic and atomic particles of matter are elastic and thus lend themselves to fairly simple analysis. Much basic information about these particles comes from the study of their elastic collisions. Among the simplest of these is the "head-on" collision of the type seen in this apparatus.
Head-On Collisions Between Two Particles. In solving elastic collision problems, we generally combine the use of two powerful principles: the conservation of kinetic energy and the law of momentum conservation. Let us illustrate the procedure for a head-on collision between an elastic sphere of mass m, having an initial velocity v1, and a second elastic sphere of mass M that is initially at rest. Let v2 be the velocity of the first sphere after collision and w2 be the velocity of the second sphere after collision. Because this is a head-on collision, all the motion after collision will take place in the line of the original motion of the first sphere. The total kinetic energy before the collision is simply that of the first sphere, 1/2mv1exp2. The total kinetic energy after the collision is the sum of the kinetic energy of the first sphere, 1/2mv2exp2, and the kinetic energy of the second sphere, 1/2Mw2exp2. Since in an elastic collision the kinetic energy before the collision is equal to the kinetic energy after the collision,
1/2mv1exp2 = 1/2mv2exp2 + 1/2Mw2exp2
(1)
From the law of conservation of momentum, the total momentum before collision is equal to the total momentum after the collision. Before the collision, the total momentum is that of the first sphere, mv1. After the collision, the total momentum is the sum of the new momentum of the first sphere, mv2 and the total momentum of the second sphere, Mw2. Thus we have the equation
mv1 = mv2 + Mw2
(2)
Now we know the masses of the two spheres and the initial velocity v1. The unknown quantities are v2 and w2. The velocities of the two spheres after collision. Since we have two equations (1) and (2), relating these velocities, we can solve for them. Therefore
v2 = [(m - M)/(m + M)]v1
(3)
and
w2 = [(2m)/(m + M)]v1
(4)
Elastic Collisions Between Equal Masses. Equations (3) and (4) give the velocities of the two spheres after the collision. In the special case in which both masses are equal, the result is of particular interest. For this case, substituting m = M in (3) we get w2 = v1. This means that the first sphere comes to a complete stop after the collision, while the second sphere moves off with exactly the same velocity, v1, that the first sphere had before the collision.
Exercises
Based on the previous tutorial as well as your knowledge of Newton's Third Law, the Laws of Conservation of Momentum and Energy, and the behavior of elastic bodies in collision, predict the response of the Newton's Cradle apparatus in each of the following situations in writing on the line beneath the heading marked Prediction. Then, test your prediction by performing the experiment on the apparatus itself, and recording your observations. Then compare and contrast your predictions with your observations.
Part A
ScenarioPredictionObservation
Single Sphere, Raised at One End ________________________________________________________________
1.A neutron moving at 1.5 x 10exp7 m/s has a head-on elastic collision with a oxygen nucleus at rest. Assuming that the mass of the oxygen nucleus is 16 times that of the neutron, find (a) the velocity of the oxygen nucleus after the collision, and (b) the velocity of the neutron after the collision.
2.Two gas molecules having equal masses have an elastic head-on collision. At the moment of collision the velocity of the first molecule is 2.0 x 10exp3 m/s while the second molecule is at rest. What are the velocities of the respective molecules after the collision?
The device consists of a sturdy four post-and-crossbeam chrome metal frame from which 5 polished steel spheres of equal mass (m1 = m2 = m3 = m4 = m5) suspended from two equal lengths of high tensile strength, low mass nylon fishing line in turn attached to the horizontal crossbeam called the pivot level are affixed.The V-shape of the attaching lines, clearly visible in the end view of the apparatus as well as the lateral swaying character of the pendulums have undoubtedly led to the coining of the term "cradle" for this device. Each pair of attaching lines are exactly the same length, possessing the same dimension in the vertical (L1 = L2 = L3 = L4 = L5). Consequently, the angles between the pairs of strings are equal, and the height of each ball above the horizontal is equal. The lateral distance between adjacent string pairs is exactly equal to the diameter of each of the spheres (dA = dB = dC = dD). Therefore each sphere is in point-contact with adjacent spheres, the points of contact in turn all in alignment. The horizontal line of contact, called the contact line, is the line along which our system of spheres interact. That is, the system is one of linear motion and interaction. Since the lengths of of the attaching lines are equal, the times of descent of the spheres from a raised position above the base level are equal, regardless of the height of the start positions of the spheres. Therefore, interactions between the spheres occur when the center of each is in the contact line, and the attaching lines are all vertical, despite having experienced rotational motion during descent. One must also keep in mind the fact that our "system of interaction" to which we have referred consists of five separate, unattached, discrete spheres of equal mass and elastic character. Each behaves as a separate and discrete elastic body, distinct and apart from the others, as there is no physical connection between any and all spheres. Also, regard the collisions as instantaneous, and with no loss of energy, neither to friction nor to heating.