## So how do these laws apply to newton's cradle ?

As stated before, the physics behind Newton’s Cradle and just about any process in the universe can be extremely excessive. To simplify our task of analyzing our system, we will look at a few different scenarios of a simple motion: One ball being displaced and released. This is the simplest performance of this device, but it makes our analysis much easier. Also, the rules and laws used to predict an outcome can be used to predict the outcome of any motions of Newton’s Balls. On other thing to note is that we will be neglecting forces such as drag and friction (with the exception of a few minor comments). In reality, these play an important role in the motion, but we will only consider a “perfect system.” The motion being considered is the first demonstration in the video on the “Construction” link.

## Stage 1 :

The first scenario is when the system is just sitting on a table top, and none of the balls are moving. The velocity is equal to “0” in this case, so there is no momentum or kinetic energy. Furthermore, we have set our datum for potential energy on the same axis on which the balls lie when they are at rest (as this will be the case for all scenarios). Because they indeed are at rest, the height in our energy equation is “0” and thus potential energy is “0.” Therefore both the momentum and the energy equations are:  0 = 0. Not very exciting…  This does not mean that no forces are acting on the balls. Take a look at the free body diagram to the right. The force due to gravity is balanced by the vertical components of the string. The ball is in equilibrium. The horizontal components are also equal; and therefore, the ball does not move side to side. Still, not very exciting…

## stage 2 :

Let’s take a look when one of the balls is displaced from its equilibrium position by an external force (human hand). Refer to the photo on the left. Again, it is a free body diagram and also a snapshot of the ball immediately after it is released. Now the forces shown are not balancing each other out, and there is acceleration. This acceleration leads to our velocity we will talk about. The vertical component of the force of gravity clearly outweighs that of the combined tensions in the strings. As the ball progresses through its path, the forces will balance to keep it on that path.

But what about our momentum and energy equations?

MOMENTUM: There is external force (gravity) acting on our body, so equations pertaining to this will not help us much. At the moment, momentum is changing.

ENERGY: Initially, there is no velocity, so initial KE is “0.” PE on the other hand is governed by (mgh)- all of which are nonzero quantities.

SUMMARY: Momentum is not yet considered. Initial energy is PE + KE, but velocity is “0” so KE is “0.” Initial PE though is simply (mgh).

## stage 3 :

As the balls velocity increases, the momentum and KE increase. One thing to note is that the PE is decreasing because the height (h) from our datum is decreasing while (m) and (g) are remaining constant. PE is, in a sense, being converted to KE. Now the ball is at the position it was in before anyone even touched the apparatus. It is literally at the moment of impact with the rest of the balls. This time however, the ball possesses momentum and energy; it is at its maximum velocity. These are not two different energies that the ball has at the same time. The only energy is that talked about in the energy equation. Momentum is simply a tool to predict effects of impacts, and it becomes very handy at this moment.

MOMENTUM: We want to know what will happen immediately after impact. We know that the initial momentum is going to equal the final, and the momentum is at a maximum (velocity is at a maximum). The equation tells us that the product of the mass and the velocity of the single ball is equal to the product of the mass and velocity of whatever is moving after the collision. Looking at the device, this leaves two options: when the ball strikes, the rest of the balls will move away at a much slower velocity or one ball of equal mass will move away at an equal velocity. We know that the latter is true.

ENERGY: As the ball accelerates towards the site of impact, its PE is being converted to KE. By the time the ball reaches the datum (impact site is on this), the final PE is “0” (h = 0) and the final KE is at its max. If we put these results into our equation, we find that the initial PE is equal to the final KE. By doing some measurements, we can calculate the original PE and therefore find the final KE and the velocity. This velocity is the same as the velocity for the momentum.

SUMMARY: The ball was at rest with PE stored. After the release, an imbalance of forces gave it acceleration on a curvilinear path. It gains KE while losing PE; and at the bottom of its path, it has a velocity that can be used to determine momentum. Momentum then takes over for our calculations of the next stage.