## Where’s The Cue Ball Going? – Snookered by The Laws of Physics

BBC – World Snooker Championship 2016

With the World Snooker Championship having just wrapped up at the Crucible Theater in Sheffield, Ronnie O’Sullivan saw himself equaling Stephen Hendry’s record number of seven titles after two weeks of grueling matches. While the commentators praised the players for executing spectacular shots, a physics student may find themselves attempting to recall lessons on impacts and collisions, and perhaps wondering if professional snooker players and physicists are more alike than we are different.

## Newtonian Cue-Action

Snooker players often rely on striking the cue ball at an angle to the target ball in order to gain position for their next shot. Taking a shot straight on is undesirable as it makes it very difficult to change the direction in which the cue ball travels after it strikes the object ball. Mathematically speaking however, this type of shot is described very simply by Newton’s experimental law, which states that the ratio of the final and initial relative speeds of two bodies in collision is constant. This ratio is known as the coefficient of restitution and is denoted by the letter e. The value of reveals interesting properties of the collision. Namely,

• = 1 : Kinetic energy is conserved and the collision is classified as being “elastic”.
• 0 < e < 1 : Kinetic energy is not conserved and the collision is said to be “inelastic”.
• = 0 : The maximum amount of kinetic energy is lost in the collision.
• > 1 : Kinetic energy is gained in the collision.

## Introducing Spin

In the case of a snooker match, the player can manipulate the outcome of the collision by applying spin to the cue ball. This is achieved by striking the ball above or below the center of the ball. The diagram below gives a detailed visualization of the various ways spin can be applied to the cue ball.

Applying top spin is known as a stun shot and causes the cue ball to run forwards after striking the object ball. This is due to the cue ball having a greater forwards angular momentum which carries the cue ball forwards after the collision. Alternatively, a screw shot can be played, which places angular momentum in the direction opposite to the motion of the cue ball and causes the ball to retreat backwards in the direction of the collision. Each of these shots are useful for gaining position on the next shot of the match.

True professionals make use of side spin in conjunction with top or back spin to really take control of the cue ball and produce shots which may seem impossible to the uninitiated.

Ronnie O’ Sullivan demonstrating extreme accuracy and cue ball control using side spin.

It may come as a harsh realisation to some that there is low correlation between knowledge of collisions and snooker ability. While it appears there is a certain element of luck involved in snooker, one certainty is that the laws of physics are at play and unlikely to find themselves snookered.

## Particle Spin is not a measure of how much a particle spins around?

When looking at Physics, one of the more scarier and misunderstood concepts is that of particle spin. I too didn’t get the general idea of the concept until it was derived in front of me, but more importantly shown its effects, most notably the Zeeman effect. Before beginning to delving into the Zeeman effect, a general recap will be done so that everybody is up to speed and nobody is left behind. It should be noted that the model used to explain the atom is the Bohr model which isn’t 100% correct but gives a good baseline understanding of what occurs in the Zeeman effect.

Figure 1 – An image of the Bohr model as seen on academickids.com, where E is the energy of the photon, h is the Planch’s constant and f is the frequency of the photon.[1]

Bohr Model: The definition of a neutral atom is that it doesn’t have an overall charge. This means the protons (positively charged particles in the centre of the atom, known as the nucleus) and the electrons (the negatively charged particles) that orbits it are the same in quantity. There are also neutral particles that act as glue for the protons known as neutrons. The increased radius away from the nucleus shows a bigger orbit and therefore a bigger energy of hat electron. The distance aren’t equal in size but is shown for clarity sake. An electron can absorb or emit a certain quantity (a quanta) of electromagnetic radiation energy, known as a photon, which allows the electron to move from one energy level (quantised/discrete number n) to the other. This emitted photon will have a certain frequency (E=hf) or wavelength (c=λf where c is the speed of light and λ is the wavelength) as seen in figure 2.

Figure 2 – An illustration showing the range of electromagnetic radiation and how it is the same thing but of different frequencies/wavelengths, as seen on britannica.com.[2]

Getting a grasp with the infamous spin: The (simple) definition of momentum is a mass moving with a certain velocity(speed with a certain direction)/speed. This can be thought of as a simple linear movement, as walking down straight street or a rotational one, which is about spinning in a circle. The total angular momentum is broken up into two bits and can be thought of through planets. The first part is the orbital momentum which is the momentum due to the planet (particle) as the planet goes around a center point (the sun in figure 3’s case or the nucleus in the electrons case). The other is rotational momentum which is the planet itself spinning, as earth does everyday.

Figure 3 – an illustration of the earth orbiting the sun, as its rotating symbolised by the red arrow, as seen on researchgate.net.[3]

This can also be said for particles like electrons, there is the orbital like momentum and the ‘spin’ momentum. Its much better to think of it as a the ‘mathsy’ counterpart that would take place of the spinning momentum of a planet. In case you were wondering why we know its not the particle actually spinning, calculations were preformed. To get the same effects from particle spin as from it rotating, the particle would have to spin faster then the speed of light, which is impossible. It should be noted that electrons have positive spins and negative spins, so if it was a planet, it would be like the planet was spinning clockwise and anticlockwise.

Bohr Model: If we make the situation a bit more difficult and imagine figure 1 is now not a circle but a sphere with different onion layers around the nucleus (figure 3) then an ‘issue’ occurs, symmetry begins to develop. From the singular quantum number n needed to describe what the energy of an electron is, now more then one are needed due to the fact that we are now working with a 3-D problem (so not just n, but now it is more like n it is n x and n y and n z which signify the three main directions possible in a 3-D problem, intuitively can be thought of as the three lines seen in every room corner). It is found that different arrangements of the quantum numbers can lead to the same electron energy, which in quantum physics is called degenerate levels.

Figure 3 – An illustration on how certain certain energy levels would look like in a Bohr-like model as seen on istockphoto.com.[4]

Zeeman effect: Due to the spin from electrons being either positive or negative, this can be used to separate/lift the degenerate states from the electrons in the presence of a magnetic field. In figure 4 the normal degenerate state can be seen as a normal red ring, one wavelength from the laser. When a magnetic field is applied then they separate from each other because of the magnetic field, giving them different wavelengths and moving them away from each other in the fancy optical setup that was used, giving figure 5.

Figures 4 and 5 – figure 4 (left) is the normal laser without magnetic field, figure 5 (right) is the laser when a magnetic field is applied

References:

1. Bohr model – Academic Kids. (2022). Retrieved 9 May 2022, from https://academickids.com/encyclopedia/index.php/Bohr_model
2. electromagnetic spectrum | Definition, Diagram, & Uses. (2022). Retrieved 9 May 2022, from https://www.britannica.com/science/electromagnetic-spectrum
3. Padgett, M. (2022). Retrieved 9 May 2022, from https://www.researchgate.net/figure/The-motion-of-the-Earth-which-spins-on-its-axis-as-it-orbits-the-Sun-is-analogous-to-that_fig2_301577086
4. Sphere layers – iStock. (2022). Retrieved 9 May 2022, from https://www.istockphoto.com/photos/sphere-layers