The Physics Of Newton’s Cradle

Cal Berwick and George McCain

Voss Honors Physics C

February 28, 2023

Abstract

Introduction

In class, we have spent a considerable amount of time on both collisions and pendulums. Perhaps the most popular example of a demonstration that applies both concepts simultaneously is the Newton's Cradle, a popular desk toy. The cradle is constructed as such: three to five balls hang independently, suspended by two strings connected to a metal frame. The balls are typically made of steel, or some other hard material which transfers energy well. The balls hang close enough to each other to where they touch, but they still hang straight from their respective strings.

In a perfect world, for every ball you add to the initial set (which is lifted and then released to start the mechanism), an equal number of balls swing out the other side, while all the other balls stay still. In theory, total momentum and total energy are both conserved, and this assumption allows us to simulate what we expect to happen under elastic conditions. In the real world, however, the cradle sways unexpectedly, energy is lost, and as such the balls reach lower velocities and heights than the ‘perfect’ Physics calculation suggest. There are 4 formulas that we used for our calculations:

We also derived a formula that relates the initial velocity of the first ball (or clump of balls) in a Newton’s cradle collision with the final velocity of the last ball, assuming that kinetic energy and momentum are conserved.

Procedure

For each of our experiments, we used a Newton's cradle with steel balls 1.99 cm in diameter. For the purposes of the analysis section, we refer to each of the balls as Ball 1, Ball 2, and Ball 3 (left to right). The Newton's Cradles we acquired used a length of low-grade fishing line, secured to either end of two horizontal bars and attached to the ball via a small metal loop. Each ball hung between the these bars such that the fishing line formed a V shape. For measurement, we used an iPhone 14 recording at 60 frames per second at 1080p. We also used two vertical stanchions with horizontal attachments, typically used for previous pendulum lab work, to suspend a meter stick above the Newton's cradle. The meter stick was at a height at which it was visible between the supporting structure of the Newton's Cradle and slotted inside of the V shape created by the fishing line. This way, we avoided any kind of parallax effect created by the meter stick being too far from the experiment itself. We recorded ten trials, but only two were relevant in our final analysis: 1 ball vs 2 balls, 2 balls vs 1 ball. For every trial, we recorded the cradle for at least one minute and thirty seconds, to ensure that we didn’t miss any odd behaviors by cutting the demonstration short.

For trials 1 & 2, the setup was fairly simple. We simply had to remove two of the balls from the cradle with scissors, such that only three of the steel balls remained. We began recording and proceeded to wait for any of the swaying of the balls to cease. Then, we took hold of the ball(s) (one in the case of Trial 1 and two in the case of Trial 2), and let them go from about half from the top of the strings to where they hung previously.

After we had collected all of the video, we used Vernier to individually track the positions of each of the balls, using the object feature to create different systems for each of the balls. Once we had collected all the data from the X and Y positions of each of the balls for each of the trials, we imported the data into an excel sheet. From there, we calculated, for each of the balls in each of the trials, the X and Y velocities, angular (XY) Velocity, Potential Energy, and Kinetic Energy. Then, using the combined data, we calculated the total Energy, total Potential Energy, total Kinetic Energy, and Total Momentum.

Trial 1 (1 ball vs. 2 balls)

• In Trial 1, we released one ball from the left side to collide with two other stationary balls. Based on the relationships we established assuming that the collisions are elastic, we know that the final velocity should be equal to $sqrt(2gh)$.

• We dropped Ball 1 from a height of 0.055 meters, which means we anticipate the final velocity to be $sqrt(2g*0.055)= 1.039 m/s$.

Considering Momentum

The below momentum over time graph shows the momentum of each ball over time in a simple 1v2 collision arrangement. As Ball 1 accelerates towards Balls 2 & 3, it gains momentum until it collides with Ball 2, at which point its momentum drops off sharply, lingering around zero. The momentum of Ball 2 increases slightly during the collision but then returns to zero; as Ball 2 collides with Ball 3, the momentum of Ball 3 spikes dramatically until it reaches the pre-collision momentum of Ball 1. Ball 3 then gradually loses momentum until it reaches its maximum height where it reverses direction and swings back the other way with a negative velocity, hence why its momentum goes below zero (because $p=m*v$) as it falls back towards Ball 2. The momentum of Ball 2 changes as it is slightly jostled around during the collision, but again it immediately loses all of that momentum as it collides with Ball 1, and the sequence repeats itself.

Momentum Over Time Graph

What about Total Momentum?

The total momentum over time graph is not constant. In fact, total momentum follows a sinusoidal trend, beginning at zero before Ball 1 is released, reaching its peak at the bottom of its swing (t=0.2) when Ball 1 reaches its highest velocity and collides with Ball 2, and then decreasing steadily before hitting zero when Ball 3 reaches its maximum height.

What’s going on here?

At first, we were surprised to see that the graph of Total Momentum fluctuates up and down in waves rather than being a flat line. We know that the total momentum of a system of objects that are not acted on by outside forces is conserved.

But Newton’s cradle is acted on by an outside force— gravity. As the balls swing and their height changes, the force of gravity in the tangential direction also changes. Gravity adds momentum to the system when either of the outside balls falls towards the middle ball. After momentum is transferred through the middle ball and sends the other outside ball swinging, gravity decelerates the outside ball, taking momentum away from the system, until the ball reaches its maximum height and all of its velocity is lost. Then, the force of gravity accelerates the outside ball back towards the middle ball, adding momentum to the system. Because there is an outside force (gravity) accelerating and decelerating the balls and allowing them to swing, it makes sense that the graph shows that momentum is not conserved. In fact, momentum is conserved, but gravity periodically modifies the total momentum of the system, which produces the above effect in the graph. Interesting!

But how exactly does gravity ‘modify’ the momentum of the system?

We can calculate the force of gravity in the tangential direction experienced by each of the balls over time, and then plot the total force alongside the total momentum to see visually how gravity is modifying the system’s momentum.

At any given point, the force of gravity in the tangential direction for each of the balls equals $mgsinθ$. Though $g$ and $m$ are constants, measuring $θ$ would be very difficult. Luckily, we know $l$ (the length of the string) and $r$ (the radius of each ball). At rest, the distance from the center of mass of Ball 1 to the COM of Ball 2 equals two radii, or $d$ (the diameter of each ball). Because we know where Ball 1 starts from (initial position) and where it ends up ($PosBall2 - d$), we can derive a formula for $mgsinθ$ for Ball 1.

We followed the same process with Ball 2 & Ball 3 to derive their respective formulas for $mgsinθ$. Because we established COM of Ball 2 as x = 0, we don’t need to account for the difference of $2r$ as we do for both of the outside balls. Using these formulas, we can calculate $mgsinθ$ over time for each of the balls and then add them to find “Total Force” over time. Plotting this value on a graph with Total Momentum Over Time demonstrates the connection between force and momentum.

Total Force vs. Total Momentum Over Time

The above graph shows the relationship between the total amount of ‘force’ (force of gravity in the tangential direction) experienced by the 3 balls over time, and the total momentum over time. When Total Force is positive, Total Momentum is increasing. When Total Force is flat, Total Momentum does not change. When Total Force is negative, Total Momentum is decreasing. When Total Force equals zero, it corresponds to a maximum or a minimum for Total Momentum. This demonstrates a key insight: that force is equal to the derivative of momentum.

Trial 2 (2 balls vs. 1 ball)

Our next question was whether this relationship between Total Force and Total Momentum would appear if we dropped two balls into one as opposed to one ball into two. In our second trial, we sought to shed some light on this question, while looking for any other interesting insights that might be found in our results.

Even though the arrangement is different from Trial 1, the formulas for the force of gravity in the tangential experienced by each ball are the same. Because Ball 1 and Ball 2 start together and remain together as they swing towards Ball 3, Ball 2 is always $2r$ ahead of Ball 1. Similarly, after the initial collision, Ball 1 nearly comes to rest and the other two balls move together, with Ball 3 always $2r$ ahead of Ball 2. The $mgsinθ$ formulas for Trial 1 still apply for Trial 2 because they account for this difference of $d$. Therefore:

• Ball 1 $mgsinθ = mg * -((x+d)/l)$

• Ball 2 $mgsinθ = mg * -(x/l)$

• Ball 3 $mgsinθ = mg * -((x-d)/l)$

Using these formulas, we calculate “Total Force” in Excel and plot it alongside Total Momentum.

When Total Force is negative, Total Momentum is decreasing. When Total Force is flat, Total Momentum does not change. When Total Force is negative, Total Momentum is decreasing. When Total Force = 0, Total Momentum has a maximum or a minimum. Clearly, the relationship between total force and total momentum is the same, even when we swing two balls into one instead of vice versa. This confirms our finding that force is equal to the derivative of momentum.

Considering Energy

Using the velocities we calculated in Excel, we graphed the KE over time for each of the three balls, and it helped us recognize something fascinating and unexpected about the way the balls collide.

As you can see in the PE over time graph, Ball 2 follows a very clean, consistent sinusoidal curve. As it swings back and forth, it reaches its peak PE and proceeds back toward the other balls for the collision. Balls 1 and 3 seem to trade places. During the collision, whichever ball had previously been traveling with Ball 2 does work to transfer kinetic energy (and momentum) through Ball 2 and into the other outside ball, resulting in a sinusoidal curve which matches that of Ball 2.

In the KE over time graph, we observe a similar shape to PE over time, although mirrored. Balls 1 and 3 seem to trade kinetic energy, with the KE of Ball 2 following the same trend as whichever of the outside balls has KE (they swing together, after all). However, at each peak in kinetic energy, where the collision occurs, there is consistently a dip in Ball 2’s KE. Interestingly, our theoretical predictions (elastic conditions) do not show this result for the KE of Ball 2. Also, there is a moment at each collision (peaks of KE) where the KE of Ball 1 and Ball 3 are not on the x-axis, equal to zero. That is, all three balls are moving at some point during the collision.

So, why is this happening?

We reviewed the Trial 2 video to help us make sense of what’s going on here, and we noticed something curious about the spacings between the balls. As they swing, Balls 1 & 2 do not move in perfect tandem, but a small space forms between them.

In a 2v1 situation, aa very small space forms between Ball 1 and 2 as they swing towards Ball 3. When Ball 2 hits Ball 3, it's a regular collision. That is, nearly all of the Kinetic Energy of Ball 2 is transferred to Ball 3. At that point, Ball 2 is stationary. Then Ball 1 collides with Ball 2, accelerating it and doing work to transfer its KE to Ball 2. During a collision, it takes a certain amount of time for each of the balls to accelerate and decelerate. Because of the minutely small space between Ball 1 & Ball 2, it also takes time for Ball 1 to cover the distance between the two so they can collide.

In cases where the time it takes Ball 2 to decelerate to 0 m/s is greater than the time it takes Ball 1 to cover the short distance, Ball 2 will have enough time to reach 0 kinetic energy. With this understanding, it makes sense that we would expect to see a momentary dip in KE for Ball 2 around the point of collision, which otherwise follows a fairly uniform sinusoidal curve.

In cases where the time it takes Ball 2 to decelerate to 0 m/s is less than the time it takes Ball 1 to cover the short distance, Ball 2 will not have enough time to reach 0 kinetic energy. However, it will still be slowing down and colliding with Ball 3. We will still see a dip in the Ball 2 KE graph, although not quite as pronounced. At this point, Ball 2 and Ball 3 are still touching, and so the kinetic energy of Ball 1 is evenly transferred to Balls 2 and 3, such that they depart the collision with the same distance between them as balls 1 and 2 pre-collision.

Sources of Uncertainty & How to Improve Them

In our experiment, we used a standard iPhone, cheap Newton’s Cradles, and our own hands to record data. To say the least, there are many ways in which we could have improved upon recording and setting up our experiment.

We filmed at 60 frames per second. In the past, we have been able to divide this uncertainty out over many periods, iterations, etc. However, because of the variable nature of our experiment, this was not an option. Because of the inconsistent shutter speed of the camera, the uncertainty is kept to 1/60 of a second, or about 0.017 second. Filming with more frames would have been nice because we could more clearly see what’s going on during collisions that appear instantaneous, occurring in less than one frame. All footage was also captured in 1080p. Because we used 1080p, we can account for an uncertainty of +/- 0.003 meters. If we wanted to improve this experiment for the future, we would use a camera with the ability to shoot video at a high frame rate while maintaining the ability to use Vernier.

Obviously, in a perfect world with perfect materials, energy is transferred completely, and the strings stay completely taught and the balls completely stationary. For any future experiments of this nature, possibly using a harder steel or higher grade of fishing line would reduce the already insignificant uncertainty that these factors have.

Furthermore, we used various things which may have effected the motion of the balls. The fishing line, for instance, may have stretch with the centripetal acceleration of the balls. The table we used may not have been perfectly level, and unfortunately it can quite be shaky. There may have also been some “squishing” of the steel balls during the collisions, causing them to bounce. This level of uncertainty, however, would have little to no effect on the balls. As with almost all experiments, expanding the proportions of the demonstration would help to reduce uncertainty. For example, making the string longer makes the periods longer, so that uncertainty makes up a smaller percentage of our time measurement. To make everything bigger, we’d need to construct a massive Newton’s Cradle. To this end, we can look to MythBusters for inspiration.

Final Thoughts

All in all, this experiment was full of discovery. Although we initially set out to produce a multitude of data for various types of collisions, we found ourselves enraptured by the discoveries of our most simple experiments. After we delved into the results from these two trials—specifically in exploring why the total momentum over time graph shows that total momentum is not conserved—we realized we needed to transform our experiment so that we could produce a more specific, higher quality project that focuses on what actually interested us (and what would interest readers) instead of a broad, uninteresting analysis that includes more complex collisions but glosses over the Physics at play.

With more time, we would have liked to investigate 5-ball pendulums and to mess around with how binding the balls together with tape changes the way momentum and energy are transferred between them, but we still arrived at many worthwhile insights, even if we initially bit off more than we could chew. From this, we learned the importance of starting small and not getting ahead of oneself in scientific research, and that there are some very fascinating Physics behind Newton’s Cradle, despite how visually simple the mechanism appears.