NEWTON’S CRADLE: AN EXTENDED EXAMPLE OF ENERGY IN SIMPLE HARMONIC MOTION

Vivian Tsang Physics 101 Section 203

Assumptions:

Newton’s Cradle is quite complex so for the purposes of our Learning Objective, we will view the five balls as point masses and view the balls as a single oscillating mass and an example of Simple Harmonic Motion

We will set the reference point as the point that balls two, three, four and five start at (shown in the next image)

Assume that friction forces are negligible and the oscillation occurs indefinitely

Newton’s Cradle: the basics

The law of conservation of energy: energy cannot be created or destroyed.

It can change forms such as from kinetic to potential which the Newton’s Cradle utilizes

Potential energy is energy objects have stored due to gravity or elasticity

 Kinetic energy is energy objects have due to motion

1 2 3 4 5

To begin, imagine ball one is held in place by an imaginary hand:Kinetic Energy equals zero Potential Energy equals the total energy of the system

Reference point

-A 0 A

Step By Step Explanation I

To initiate the motion, we lift up the first ball giving the ball potential energy and defining it equal to the total energy of the system since nothing is ‘moving’ at this point

When this ball is released, its potential energy is converted into kinetic energy because of the work of gravity

At the lowest point, just before ball one hits ball two, all its potential energy has been converted into kinetic energy and thus, the kinetic energy is now equal to the total energy in the system

1 2 3

4

5

Although the balls look like they’re not ‘moving’, a conversion of kinetic to potential energy is taking place

[Although taken care of in our assumptions, in case you were wondering what happens when ball one collides with ball two…

The kinetic energy in ball one is transferred to ball two as potential energy as ball two is ‘compressed’ when hit by ball one

When ball two decompresses, the potential energy it previously stored is transferred into kinetic energy used to ‘compress’ ball three

And so the process continues until ball five]

As ball five decompresses, its potential energy is transferred into kinetic energy

Because ball five has no ball next in line, its kinetic energy swings it out with the same speed that ball one had when it was released

1 2 3

4

5

Step By Step Explanation II

Kinetic Energy equals zero Potential Energy equals the total energy of the system This is the same energy distribution as when the Newton's cradle’s motion first began

Graphing the Motion: A Plot of Kinetic and Potential Energy as a Function of Displacement

When potential energy equals zero, total energy equals the kinetic energy the ball has and the ball is at its lowest point (at the reference point)

When kinetic energy equals zero, total energy equals the potential energy the ball has and the ball is at its highest point above the reference point

Conclusion:

Note that: although potential energy (U) and kinetic energy (K) are not constant individually through the oscillation of the ball, their sum is always equal to the total energy (E)

Therefore, at any point in the oscillation, the total energy in the system is conserved