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Explanation of Derivation of Principles of Newton's Cradle by Ron Kurtus - Succeed in Understanding Physics. Also refer to physical science, Isaac Newton, Simon Prebble, Conservation of Momentum and Energy, mass, velocity, friction, inertia, pendulum, School for Champions. Copyright © Restrictions
Derivation of Principles of Newton's Cradle
by Ron Kurtus (revised 30 may 2010)
Newton's Cradle provides a physical demonstration of the Laws of Conservation of Momentum and Energy. It shows how swinging one or more balls toward the remaining balls will result in the same number of balls being projected from the other end of the device.
The device usually consists of five metal balls of the same size and mass. They are suspended on wires and aligned such that they are in a row. If a number of balls are swung to strike the remaining stationary balls on one end of the row an equal number of balls will move on the other end.
A simple derivation proves this mathematically. There are physical situations where there is deviation from the principles.
(See Newton's Cradle for an explanation and animation of the device.)
Questions you may have include:
- What are the Laws of Conservation?
- What is the derivation?
- What are some deviations?
This lesson will answer those questions.
Useful tool: Metric-English Conversion
Point masses
Suppose the balls were equal point masses. When one point mass struck the second point mass, the momentum and energy would be instantaneously transferred from the first to the second, continuing to the last point mass, which would then move away from the group. The same would occur if two or more points struck the series.
However, instantaneous motion would require infinite acceleration and zero mass.
Spring collision
When a moving small metal spring collides with a perfectly hard wall, its CM slows until the spring reaches a compression according to its mass and initial velocity. Then it slowly increases its velocity in the opposite direction, as it restores its original shape.
If the spring strikes a perfectly hard object of the same mass, the spring will start to compress and its CM will slow down, while the other object slowly accelerates. ??
If the spring strikes another spring of the same mass, spring #1 will start to compress and its CM will slow down, while the srping #2 also compresses and its CM slowly accelerates.
The compression can be thought of a sound wave starting from the CM of srping 1 and moving to the CM of spring 2. Once it reaches CM 2, the springs separate. The CM 1 is located not where they were in contact but at a point d = vt, where t is the time it took the shock wave to reach other other CM
Two ball collision
When a moving ball collides with a stationary ball, a comprssion wave occurs in each ball.
Ideal configuration
One problem inthe actual Newton's cradle is that
A moving ball acts as if its mass was concentrated at its geometric center.
The ideal configuration for Newton's Cradle is that the balls are equal point masses. By default, they would be perfectly elastic.
When one point mass struck the second point mass, the momentum and energy would be instantaneously transferred from the first to the second, continuing to the last point mass, which would then move away from the group. The same would occur if two or more points struck the series.
Laws of Conservation
Newton's Cradle is based on the Laws of Conservation of Momentum and Energy.
Conservation of Momentum
The Law of Conservation of Momentum states that in a closed system, the momentum in a given direction is constant.
Momentum is designated as:
p = mv
where
- p is the momentum
- m is the mass
- v is the velocity (speed in a given direction)
- mv is m times v
Conservation of Energy
The Law of Conservation of Energy states that in a closed system, the energy is constant.
Energy is designated as:
KE = ½mv2
where
- KE is the kinetic (moving) energy
- v2 is the velocity squared or v times v
In other words, if the balls at one end strike the row with a given energy, the energy will be transferred to the balls at the other end.
Derivation
Suppose you swing n number of balls of mass m to strike the stationary balls. By the Conservation of Momentum:
(1) p = nmv = MV
where
- M is the total mass of the balls moved on the other end
- V is the velocity of the balls moved on the other end
Likewise, due to the Conservation of Energy:
(2) KE = ½nmv2 = ½MV2
Find velocity
Take equation (1) and solve for m:
m = MV/nv
Substitute in equation (2):
½nmv2 = ½MV2
½nv2MV/nv = ½MV2
v = V
In other words, the balls on the other end of the row will move out at velocity v.
Find mass
Take equation (1) and solve for v:
p = nmv = MV
v = MV/nm
Square v:
v2 = M2V2/n2m2
Substitute in equation (2):
½nmv2 = ½MV2
½nmM2V2/n2m2 = ½MV2
M/nm = 1
M = nm
The mass of the balls moved will be the same as the mass of the initial balls.
Outcome
In other words, since all the balls have the same mass, if you swung two balls at a given velocity and struck the row, two balls on the other end would move outward at the same velocity. If you swung four balls, four would move from the other end.
Requirements and deviations
The balls in Newton's Cradle should be the same mass. Even a slight deviation will change the derivation equations and result in slightly different results.
It doesn't matter how many balls are used, although the more you use, the greater the chances for deviations.
The balls should be perfectly aligned. If some balls were not on a straight line, the transfer of momentum and energy would also be misaligned, changing the outcome.
Spherical balls are used because their contact is approximately a point. Other shaped objects could be used, but that increases the chances for misalignment.
Hard metal balls—such as made of hardened steel—are used to minimize losses in energy due to elastic distortions.
Balls are hung with string or wire that will keep them in alignment and minimize losses due to friction.
Summary
Newton's Cradle provides a physical demonstration of the Laws of Conservation of Momentum and Energy. If a number of balls are swung to strike the remaining stationary balls on one end of the row an equal number of balls will move on the other end. A simple derivation proves this mathematically. Certain requirements on the balls assure the Laws are met.
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Resources and references
The following resources provide information on this subject:
Websites
Newton's Cradle - by Donald Simanek, Lock Haven University - Thourough analysis
Conservation of Momentum - Mathematical explanation from the University of Winnipeg, Canada
Books
Top-rated
books on Laws of Motion
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