Newton's cradle is a classic science gadget that, in addition to being a desk decoration, demonstrates to us two fundamental conservation laws in physics. It usually has four or more steel spheres hanging from the end of the thin cable. When one sphere is lifted and released, it collides with the other stationary spheres. In a collision, the force is transmitted through the stationary spheres to the last sphere and the last sphere in the row is pushed up. Now the same movement is repeated in the opposite direction. An oscillating motion occurs, which is why this gadget is also called Newton's pendulum.
These swinging steel spheres in Newton's cradle demonstrate to us two important principles in physics. These principles are the law of conservation of energy and the law of conservation of momentum.
Conservation of mechanical energy
According to the law of conservation of mechanical energy, the total mechanical energy of a closed system is constant. Mechanical energy consists of two components - kinetic energy and potential energy. Newton's cradle demonstrates how the potential energy of a sphere turns into kinetic energy and vice versa. This means that one type of energy becomes another, but does not disappear. When we lift the sphere, we increase its potential energy because it is now higher than the other spheres relative to the ground.
If we raise the sphere to some height above the ground, it receives additional gravitational potential energy (Ep). This energy is proportional to the mass (m) of the ball, the height of the ball from the ground (h), and the acceleration to the gravity (g). Now that we release the sphere, it naturally wants to move to a point in the gravitational field, where it has less potential energy. The sphere starts to gather speed and move downwards, and the potential energy of the sphere begins to convert into kinetic energy (Ek).
Kinetic energy is related to the movement of the sphere and is equal to half of the sphere's mass (m/2) multiplied by the velocity of the sphere squared. Just before colliding with another sphere, all the initial potential energy that was given to the sphere, has now been converted into kinetic energy.
The falling sphere carries momentum and it collides with the next sphere in the row, which has the same mass. The first sphere stops and the second sphere starts to move, giving its energy to the third sphere, and so on. When the momentum is carried to the last sphere in the row, it swings upward at an initial speed equal to the final speed of the first sphere before the collision with the second sphere. As the sphere swings upwards, it loses its speed and thus its kinetic energy. Kinetic energy is now being converted into potential energy again. When the sphere stops in the highest position, the kinetic energy is equal to zero and the potential energy is maximum.
As the spheres oscillate back and forth, then the sum of the potential and kinetic energy remains constant, because the potential energy of the sphere is converted into kinetic energy and vice versa. The law of conservation of mechanical energy tells us, that the total mechanical energy (ME) of the system does not increase or decrease, but remains constant.
The total mechanical energy remains constant, but the oscillation finally stops, because the mechanical energy is also converted into other forms of energy. If the spheres oscillate and collide, energy is also lost as heat and sound, so the oscillation of the spheres eventually stops.
Conservation of momentum
Another fundamental conservation law, that Newton's cradle demonstrates to us, is the law of conservation of momentum. According to the law of conservation of momentum, the total momentum of an isolated system (oscillating spheres in our example) is conserved. Momentum (p) is a property of the body, that the body has by virtue of its mass and motion and is equal to the product of the mass (m) and speed (v) of that body. Momentum is a vector quantity, which means that it has both magnitude and direction.
All spheres in Newton's cradle have the same mass and collisions between spheres can ideally be considered elastic. This means, that the kinetic energy of the sphere does not convert into other forms of energy, like heat energy or sound energy in a collision. However, as we already read, kinetic energy is gradually converted into thermal energy and sound energy, and the spheres stop oscillating at some point. That is, we need to idealize a bit and consider collisions between spheres to be almost elastic.
Let us first look at a system consisting of two spheres. Both spheres have the same mass, but the first one is stationary, while the other move at some speed towards the first sphere. The first sphere moving at some speed carries momentum, but the second sphere is stationary and its momentum is equal to zero.
When a head-on elastic collision occurs, the first sphere remains stationary, and the second sphere moves forward at the same speed at which the first sphere moved before the collision. Thus the total momentum of the system before and after the collision remained the same.
In Newton's cradle, we have a system of five spheres. All spheres have equal mass and after an elastic collision, the first sphere stops and transfers its momentum to the next sphere in the row, which in turn to the next and so on. Finally, the last sphere in the row begins to move at the same speed as the first sphere moved before the collision with the second sphere. Hence it carries the same momentum as the first sphere before the collision and the total momentum of the system remained unchanged.
If we initially lift two, three, or four spheres, then after the collisions, the same number of spheres moves up the other side of the line, and the total momentum of the system before and after the collisions remains unchanged. For example, let's take a closer look at what happens when we lift two spheres. We take into account, that there is a very small cap between the spheres so that spheres are almost in contact with each other.
1) Sphere 2 and sphere 3 collide first
2) Two collisions occur simultaneously - between 1 and 2 and between 3 and 4
3) Two collisions occur simultaneously again- between 2 and 3 and between 4 and 5
4) Sphere 3 collides with sphere 4
Spheres 5 and 4 move up from the other end of the line and the total momentum of the system remains unchanged.