Calvin Lareau
^Newton's Cradle Joke
Embedded Files
Abstract Video!!
Abstract Video!!
IMG_4666.movClick "open in a new tab." If you have trouble viewing the video, try downloading it.
Christiaan Huygens
History
History
The term “Newton’s Cradle” is actually accredited to an English actor named Simon Prebble, who began manufacturing the first desktop versions of the apparatus in 1967 out of wood. A sculptor, Richard Loncraine, quickly entered the competition, developing a chrome version of the cradle much more similar to those seen today, and by the early 1970s, it was a staple on the desks of business executives. However, the investigation of the physics behind the contraption significantly predate these events. As early as 1662, Christiaan Huygens, a Dutch physicist and mathematician, was outlining the theoretical principles present in the cradle, such as kinetic energy and conservation of momentum. Huygens even used a series of pendulums, not dissimilar to those I discuss today, to demonstrate these concepts. While Isaac Newton obviously researched and greatly further the understanding of these ideas, it's worth noting that he did not publish his laws of motion until 25 years later 1687, especially considering he gets all the credit.
Issac Newton
Relevant Background Theory
Relevant Background Theory
Energy:
There are two types of energy pertinent to the discussion of the cradle: gravitational potential energy and kinetic energy. Gravitational potential energy is the energy stored within an object due to its position within a gravitational field, and is given by the equation U = mgh, where m is mass in kilograms, h is height in meters, and g is the gravitational constant (9.8 on Earth). Kinetic energy is the energy an object has due to its motion, expressed by the equation K = ½ mv2, with v being velocity.
Momentum:
Objects in motion possess not only kinetic energy, but another quantity of motion called momentum, which is equal to mv, the mass of the object times its velocity. The momentum principle states that a change in an object's momentum is caused by a force applied to that object over some duration of time, which is called an impulse. The magnitude of the impulse is equal to the change in momentum.
Figure 1: The basic energy model for an isolated system
Conservation Laws:
Finally, the laws of conservation of momentum and conservation of energy state that within an isolated system, the total amount of both of these quantities remain the same. A system can be any specific group/collection of objects or particles which one defines; an isolated system is one in which there are no external forces or the external forces are balanced and add to zero.
Figure 2: Side view of measurement apparatus with metal marble cradle.
Figure 3: Top-down view of glass marble cradle
Methods and Materials
Methods and Materials
24 popsicle/wooden sticks
string
4 glass marbles
4 metal marbles
Hot glue
16 clips
I used simple materials to allow others to replicate my testing at home. For the frame, I made two identical squares by hot gluing four craft/popsicle sticks together at the corners. I then connected these squares horizontally at each corner using another four sticks, creating a cube. For each pendulum, I cut a length of string of 0.4 m, marked it in the middle, and affixed a marble to this point. Finally, I used two binder clips to attach the pendulums to either side of the frame and suspend them in the middle. I adjusted their positions such that they rested against each other, were symmetrical, and were level in height. I repeated this process twice, once with glass marbles of a mass of 5.7 grams, and once with metal marbles, which were massed at 28 grams.
Testing was conducted identically for the two pendulums. Rulers were positioned vertically along both sides of the cradle. For each run, the first ball was lifted to a height of 0.9 m above the resting position or origin. The ball was released the ball and a slow motion recording was captured to take measurements with. I measured the height the 4th ball achieved, measuring from the bottom of the ball. Due to technology limitations, velocity was examined indirectly by measuring the duration of the swing using the video. 3 runs were performed for each type of marble.
Theoretical Work
Theoretical Work
Restricting the system to the pendulums and the earth as a gravitational body and assuming no external forces influence the balls’ motion, the Newton's cradle is an isolated system where energy and momentum are conserved. When the 1st ball is lifted from the origin it obtains gravitational potential energy due to its height; as it falls, gravitational potential energy is converted to kinetic energy. Due to conservation of energy, the ball’s kinetic energy at the bottom of its swing (origin/height zero) will be equal to the gravitational potential energy it had at the top.
Concept Through Equations: The first line states total energy at top and bottom are equal, from conservation of energy. Velocity is zero at the top of the swing, and at the bottom of the swing the ball reaches a height of zero.
Solving for v, an equation for the 1st ball's velocity at the bottom (see right) can be found. This equation shows that the velocity is independent of the mass, and it gives an expected velocity of 4.2 m/s for a height of 0.9 m.
From its velocity and mass, the 1st ball will have momentum. When the 1st ball collides with the 2nd, assuming it stops completely, the 1st ball receives an impulse equal but opposite to its momentum. For momentum to be conserved, the 2nd ball must receive an impulse equal to the momentum the first ball had. The collision is repeated until the 4th and final ball receives the impulse. Thus, the 4th ball's momentum and, given uniform mass, velocity are equal to those which the 1st ball had (see right equations).
Therefore, the 4th ball's kinetic energy will be equal to that of the 1st at the bottom of its swing. By conservation of energy, as the 4th ball swings up it should reach the same height the 1st was dropped from: 0.9 m.
Concept Through Equations
Figure 4: Data Table containing individual and average values for drop height (m), height regained (m), and duration of swing (s), as well as calculations for (%) average height lost (m).
Video 2 : Slow motion video of a run for the metal marbles.
IMG-4359.MOVResults
Results
*There is an uncertainty of about +/-0.02 m across individual height measurements due to motion blur and other method limitations.
For the glass marbles, the 4th marble reached a height of 0.8 m on average, which represents an average loss of 0.10 m. This equates to an 11% difference from our original height and expected value of 0.90 m. Proportionally, this constitutes an 11% loss in gravitational potential energy. The 4th metal marble reached a height of 0.82 m on average, losing 0.08 m on average. This equates to an 8.9% difference from the expected value, and, proportionally, an 8.9% loss in gravitational potential energy. The percent difference between the average regained heights for the two trials was calculated as just 0.02%.
The average duration of the swing for the glass marbles was 2.83 second, compared to 2.93 seconds for the metal marbles. The percent difference between these average durations is 3.5%
Discussion
Discussion
The 11% and 8.9% losses in gravitational potential energy by the end of the 4th balls' swings indicate that my Newton's cradles are not isolated systems, as energy and/or momentum are not conserved. External forces from the environment must be influencing their motion. These forces likely include friction between the spheres, as well as air resistance. Additionally, the frames suspending the pendulums may not be perfectly stable; swaying might impact the direction of the impulses such that they are not in line with the balls' path causing momentum to be lost. The low percent differences of 0.02% for the average heights and 3.5% for the average swing durations between the metal and glass marble trials support the theoretical work which showed that mass does not impact the balls' velocity or energy transfer.
Conclusion
Conclusion
IMG_4668.movSources:
https://ssl.fastdir.com/~fastdir/space/sotlschool/T46/NewtonsCradlePDF.pdf
biography.com/scientist/isaac-newton
Page updated
Report abuse