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Brown University
Physics Department
Physics 0050/0070
Newton’s Cradle
Newton’s Cradle
Equipment:
Science First, Newtonian Demonstrator (40-135)
Introduction:
Newton’s Cradle, aptly named after Sir Isaac
Newton, is designed to show the results of Newton’s
Third
Law of Motion through conservation of both energy and
momentum. Newton’s Third Law states, “For every
action,
there is an equal and opposite reaction.” The Law of
Conservation of Energy asserts, “Energy is neither
created, nor destroyed, but can be changed from one form to another.” The Law of Conservation
of Momentum similarly states, “Momentum is neither created, nor destroyed.” The balls of this
Newtonian Demonstrator, when displaced a given height and released, show these concepts as
the same displacement is mirrored by the balls on the other end. In an ideal case, this would
continue indefinitely. However, friction from the air, string and balls will eventually bring the
balls to rest.
Directions for use:
1. Place the Newtonian Demonstrator on a smooth level surface away from any magnetic
fields. Ideally, orient the balls parallel to the east-west plane to avoid interference from
the Earth’s magnetic field. Earth’s magnetic field ranges from 30 to 60 microteslas (0.30.6 gauss).
2. To release balls, raise from the row about 5 inches and release as smoothly as possible. It
is possible to release any number of balls from either or both sides of the row.
3. Allow the balls to come to rest. Repeat to observe desired combinations.
Discussion:
Newton’s Cradle, if you neglect the effect of friction, shows elastic collisions, in which
energy and momentum are both conserved. Momentum, P, is defined as an objects mass, m,
times its velocity, v.
P = mv
In an elastic collision, the total momentum before the collision takes place must equal the
total momentum after the collision takes place.
Pinitial = Pfinal
This yields:
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Brown University
Physics Department
Physics 0050/0070
Newton’s Cradle
m1v1 + m2v2 = m1v1 + m2v2
initial
final
final
initial
Figure 1: Elastic collision of two objects
In an elastic collision, energy is also conserved. Potential energy, PE, is the energy stored
within a system. In this case, the potential energy comes from gravity, g, when the ball of mass,
m, is lifted a certain height, h.
PE = mgh
Kinetic energy, KE, is the energy of a moving system. A system of mass, m, when
moving has a velocity, v, which gives a kinetic energy of:
KE = 12 mv 2
Conservation of energy states that no energy is gained or lost, and that the total energy, E,
remains the same.
Einitial = E final
PEinitial + KE initial = PE final + KE final
It is necessary to treat each ball as an individual mass when calculating momentum and
energy. If you displace one ball and treat the remaining four balls as a single mass, you would
find that the four balls would all be displaced but in proportion to the original ball. However, if
you treat each ball individually, the transfer of momentum can be seen between each ball, and
this agrees with the observed motion of the ball on the end being displaced the same as the
original incident ball.
This is a simplified explanation of the observed phenomenon. Because there are 5 balls,
giving 5 unknown velocities, and only 2 equations of conservation of energy and momentum,
there are an infinite number of solutions. However, only very specific motions are observed.
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Brown University
Physics Department
Physics 0050/0070
Newton’s Cradle
Figure 2: Some possible starting positions
There are several alternative models. The more complete models take into account the
finite size and elastic properties of the balls and treat them as five masses coupled by springs.
Another approach to explaining the phenomena is to assume the balls at rest are not initially in
contact, but have small gaps between them. This allows for consideration of successive two body
collisions.
There are several ways that the Newtonian Demonstrator can be observed more closely
and quantized. A video camera and VideoPoint software on a computer may be used to record
the motion of the demonstrator and will allow for frame by frame review of the motion. Another
possible method would to be to use a Pasco Motion Sensor to track the horizontal motion of the
balls. The motion sensor could be used to observe the friction that eventually brings the balls to a
stop. The Newtonian Demonstrator is an elegant demonstration of many of the basic laws of
physics.
References:
Gavenda, JD, and JR Edgington. "Newton's Cradle and Scientific Experiment." The Physics
Teacher 35 (1997): 411-17. The Physics Teacher. 20 July 2009
<http://scitation.aip.org/tpt/>.
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Brown University
Physics Department
Physics 0050/0070
Newton’s Cradle
Hutzler, Stefan, Gary Delaney, Denis Weaire, and Finn MacLeod. "Rocking Newton's Cradle."
American Journal of Physics 72 (2004): 1508-516. American Journal of Physics. 20 July
2009 <http://scitation.aip.org/ajp/>.
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