By Emma Hermick

Introduction

Newton's Cradle is a device that uses pendulums to demonstrate the laws of Conservation of Momentum and Conservation of Energy. It's named Newton's Cradle because it demonstrates Sir Isaac Newton's 1687 work, Philosophiae Naturalis Principia. 17th century physicist Abbe Mariotte was the first to demonstrate the law of impact between bodies that governs the movement of Newton's Cradle. French ohy It consists of a series of identical balls, each attached by 2 strings to a frame. It was first manufactured in 1967 and is a popular desk toy now.

To use one of these toys, the first ball on one end is picked up and dropped. It transfers energy through the middle balls before causing the ball on the far end to swing up. When the ball comes back down, the action repeats. The middle balls don't move, while the two on the ends go up and down. The same thing will happen if two or three balls are picked up and dropped at the same time; the same number of balls on the other end will swing up and down.

By design, when the balls collide the strings that hold them up are vertical (assuming balls are only swung from one side). This means there are no horizontal forces from the string on the balls so linear momentum in the direction of swing must be conserved in the collision. Energy is also nearly conserved provided not too much noise and heat are produced.


Newtons_cradle_animation_book_2.gif

Description

Newton's Cradle consists of several metal balls (usually 5) suspended by two wires, so that they line up and are almost in contact when in a resting position.

One Ball

When an end ball (ball #1) is pulled up and let go, it swings down as a pendulum and hits the next ball. The energy and momentum from that ball is transmitted through the three balls at rest to the ball on the other end (ball #5). That ball is propelled forward at the same velocity as the first ball had, due to the force of the first collision.

This process continues as ball #5 reaches its peak and then swings down to hit the balls at rest, propelling ball #1 forward and upward.
newtons_cradle-one-ball.gif

Two or More Balls

If two or more balls are pulled up and let go, the collision will result in the same number of balls being propelled forward on the other end.

Slows Down

The action will go back and forth until it slowly slows down due to losses from friction and the elasticity of the balls.
Steel balls are usually used, because they deform very little upon collision and are highly elastic—meaning only a small amount of enegy is lost in the collision.

Explanation

The number of balls moving after impact equals the number of balls that are released by applying the Law of the Conservation of Energy and the Law of Conservation of Momentum.

Energy

The Law of the Conservation of Energy states that the total kinetic energy of a system with no external forces acting on it remains constant. That means that the kinetic energy of the moving ball or balls upon impact equals the kinetic energy of the balls leaving the other side of the row of balls. At those times, the force of gravity is not a factor.
  • KE = mv2/2 = MV2/2
where
  • KE is kinetic energy
  • m is the mass of the balls being released
  • v is the velocity on impact
  • M is the mass of the balls moved on the other end
  • V is the velocity of the second group of balls after impact

Momentum

The Law of Conservation of Momentum states that the total linear momentum (p) of a closed system is constant. That means that the momentum of the balls on impact equals the momentum of the second group of balls after impact:
p = mv = MV

Equations

Solve for v and square both sides of the equation:

  • v = MV/m
  • v2 = M2V2/m2

Substitute v2 in the energy equation mv2/2:

  • mv2/2 = mM2V2/2m2 = M2V2/2m

Since

  • mv2/2 = MV2/2
  • M2V2/2m = MV2/2

Thus:

  • M/m = 1 or M = m

This means the mass of the balls leaving equals the incoming mass. Since the balls are of equal mass, that means the same number of balls leave the series as those which impacted the group of balls.

Sources

http://www.school-for-champions.com/science/newtons_cradle.htm
http://physics.stackexchange.com/questions/3527/newtons-cradle
http://www.physics.org/article-interact.asp?id=32
http://www.sciencedaily.com/articles/n/newton's_cradle.htm