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Derivation of Principles of Newton's Cradle
by Ron Kurtus (27 July 2009)
Newton's Cradle device provides a physical demonstration of the Laws of Conservation of Momentum and Energy. It 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. There is a mini-quiz near the end of the lesson.
Useful tools: Metric-English Conversion | Scientific Calculator.
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 in a given direction 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 x number of balls of mass m to strike the stationary balls. By the Conservation of Momentum:
(1) p = xmv = 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 = ½xmv2 = ½MV2
Find velocity
Take equation (1) and solve for m:
m = MV/xv
Substitute in equation (2):
½xmv2 = ½MV2
½xv2MV/xv = 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 = xmv = MV
v = MV/xm
Square v:
v2 = M2V2/x2m2
Substitute in equation (2):
½xmv2 = ½MV2
½xmM2V2/x2m2 = ½MV2
M/xm = 1
M = xm
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
The following resources provide information on this subject:
Websites
Conservation of Momentum - Mathematical explanation from the University of Winnipeg, Canada
Books
Top-rated
books on Physical Science
Mini-quiz to check your understanding
If you got all three correct, you are on your way to becoming a Champion in Physical Science. If you had problems, you had better look over the material again.
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Derivation of Principles of Newton's Cradle