Cycle II Regents Physics

John Dewey High School

Mr. Klimetz

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.

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)

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.

Single Sphere, Raised at One End ________________________________________________________________

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Single Sphere, Raised at Opposite End ____________________________________________________________

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Two Spheres, Raised at One End ________________________________________________________________

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Two Spheres, Raised at Opposite End _____________________________________________________________

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Three Spheres, Raised at One End _______________________________________________________________

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Three Spheres, Raised at Opposite End ___________________________________________________________

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Four Spheres, Raised at One End ________________________________________________________________

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Four Spheres, Raised at Opposite End ____________________________________________________________

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Single Sphere Raised at Both Ends _______________________________________________________________

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Two Spheres Raised at Both Ends _______________________________________________________________

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Two Spheres Raised at One End and Three Spheres Raised at Opposite End

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Four Spheres Raised at One End and One Sphere Raised at Opposite End

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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?