|The Physics of Pool|
|Physics: An Overview|
Newton's First Law
Newton's Second Law
Newton's Third Law
Newton's Laws in Conclusion
Angular Deflection/Angle of Incidence
Physics: An Overview:|
It is a fact that it is the nature of all things to fall into chaos. It is one of the laws of physics, and seems paradoxical. But, it seems as if it is nature's scheme to create problems that force living things to think. Consider the game of pool. When you first break the rack, the odds are incredibly slim that the balls will line up in such a way that you can easily shoot your way to the end of the game in one run of 8 ball. It often seems as if "Murphy's Law" comes into play (if anything can go wrong, it will), and we are stuck in a tough position, regardless of how well we broke the rack. Learning to overcome this chaos is the prime goal of every pool player. Professional pool players have come to the realization that they need to understand the physics involved within in order to tame the game. Considerations taken regarding the laws of physics lead some players to believe that the obviously easy shot might not be the wisest one. Now, of course, to overcome this physical chaos, the first step is to overcome the mental chaos within (attempting to calculate angles of incidence, the exact momentum of the 3-ball as it ricochets off of the foot rail, etc.). Yet, it is still important for a pool player to understand the clockwork that is physics involved in the game.
On this page, that is our goal.
Quantities in physics can be signified using two different methods: scalar and vector. Scalar quantities have only magnitude, but no direction. Mass, for example, is a scalar quantity, since there is only a numerical value and no direction. Speed and distance are also scalar quantities. Vector quantities, however, are defined by both their magnitude and their direction. Various forces, as well as velocity, acceleration, and momentum, all are indicated by vector quantities. The majority of the values used in physics involve vector quantities.
To add vectors, you use a simple "tip-to-tail" method to find a resultant vector. To use this method, place the vectors at the same orientation with the same magnitude, and use triangular trigonometric functions to determine the third side and the angle. Since the vectors in the below diagram are at a 90 degree angle, the Pythagorean theorem can be used to determine the length of the third side:
32 + 42 = 52
Using sine laws and inverse sine, you determine the vertex angle between due east and the resultant as 53.13 degrees:
Newton's First Law:
Newton’s first law of motion states that a body remains at rest or in motion in a straight line at constant speed unless acted upon by an outside force. This law is very important in understanding the physics of billiards. When the ball is struck with the cue stick it, in theory, will stay at a constant speed until it makes contact with a wall or ball. However, due to the friction on the ball caused by the felt on the table, chalk on the cue stick, and air resistance, energy is lost in the system. The bumpers also give us a great example of how to apply Newton's first law of motion. The cue ball will follow a straight path until it hits a bumper. Once the cue hits the bumper, the law of reflection comes into play. As stated earlier, energy is also lost when a ball hits the bumper because of the "cushioning" effect.
Newton's Second Law:
Newton's second law of motion is important to a pool player in his decision with which speed he wishes to hit the ball. Newton's second law states that the net force on an object is equal to its mass times its aceeleration(F=ma). Since acceleration is a vector quantity, so also is force. The direction of the acceleration is also the direction of the applied force. One decision a pool player must face is how hard to play the shot. A shot with a high acceleration could play better off a rail, or could cause more balls to be set into motion through harder collisions. A player must also be careful not to cause too much acceleration. Too much acceleration can cause more balls to collide and can be dangerous to the stategy of the game.
Newton's Third Law:
Newton’s third law is also important in understanding the physics of billiards. This law states that if an object exerts a force on a second object, the second object exerts an equal force back on the first object. This law is involved in any collision between balls and walls. Because the weight and mass of the cue ball and object ball are the same, when the cue ball slides into the object ball at no angle, and with no top, bottom or side spin, the cue ball will stop dead, regardless of velocity. Also according to the third law the ball has a force of gravity pushing down on it which in turn the table acts as a force pushing upward on the ball.
Newton's Laws in Conclusion
Interactions among things or matters occur by forces. Forces cause motion and influence the basic kinematics of objects and they are important to study mechanics. Newton is the father of classical mechanics and has stated 3 laws dealing with forces: F = ma (m = mass, a = acceleration) For every force, there MUST also be an equal and opposite force All objects have inertia; that is, an object will remain at rest or in uniform motion unless acted upon by some outside force When solving problems involving force, a force diagram is drawn on the object which forces are acted upon. Then to compute the resulting acceleration, these forces are combined accordingly calculating the y-components and x-components of the vectors. Or to find the original force which resulted in acceleration of some object, the equation is simply manipulated to fit the situation.
This diagram shows forces that are in the box - ball system on a frictionless table. For the box, there is the weight which is equivalent to mass times g for gravitational acceleration. And then there are two normal forces which are denoted by the letter N. Normal forces are the forces perpendicular (normal) to the surface. These exist on any surface that are in contact with another. Nt is the normal force exerted by the table, and Ncis the normal force exerted by the ball. Same forces act on the ball except in different magnitudes. The coordinate axes for positive x and y directions are included in the diagram also. This is to give reference to directions for the force vectors. Any coordinate system suitable for the unique force diagrams may be used. But generally the positive directions for x and y are generally in the direction of the resulting accelerations.
When a collision occurs the balls begin to move from rest. This motion can be measured in terms of velocity. The linear momentum of an object is defined as the product of its inertial mass and its velocity. Momentum is a vector quantity which has the same direction as velocity. The relationship between them is described in this formula: p=mv. Momentum is symbolized by p, mass by m and velocity by v. An objects momentum can be changed due to a change of velocity. The mass of the objects does not change in the case of the collisions occurring within the game of billiards. This change in momentum is known as impulse which Fnet=change in momentum/change in time.
The laws applied to angular deflection, or angle of incidence, are key in plotting out a successful bank shot. Invariably (unless playing on a mangled pool table), the pool ball will deflect off of the rail at the same angle to the rail as it approached. For example, if the ball approaches at a 45 degree angle and deflects off of the rail, the ball will travel at an angle of 45 degrees to the rail towards the other side. This law can also be seen this way: The angles formed as the ball approaches and deflects form a mirror image of each other, as shown in the diagram below:
The line forming the sides of angles 1 and 2 is perpendicular to the pool table rail.
Head-on collisions are very predictable: when hitting a ball head-on, the ball with which the collision was made will travel in the same direction:
On an angular collision, the balls naturally would not collide in the same fashion. Assuming that the two balls are equal (which they should be, unless you have a dented-up set of pool balls), the balls should ALWAYS deflect at a 90 degree angle, as shown in the below diagram:
The net force applied to the balls causes a rotational velocity. The ball rotates around its center of mass. A change in the rotational speed can only occur when there is a net external interaction on the object. The cue ball is rotating, and since rotational momentum must be conserved the cue ball is still spinning at the same angular velocity as it was before the collision and the ball soon begins to roll. The use of english and spin changes the result. The friction of the cloth will put forward, never backward, roll on the balls. The leather cue tip is used to increase friction with the cue ball, which in turn provides control over the friction.
Torque is extremely important to a pool player. Torque is the rotational analog of force. When you apply a net torque, you change an object's angular momentum. Torque is measured by taking the radius and multiplying it by the force perpendicular to the radius (t = r x F). When measuring a hit from an angle, you need to use the equation torque = (radius x Force) x sin of the angle measured from the applied force and the radius. The larger the angle, the less amount of torque applied. Therefore, an angle of 90 degrees produces the maximum torque.
No matter where you look on the pool table, friction is always playing a major part in the game. First of all, the cloth on the pool table provides a massive amount of friction. The friction of the cloth will always put forward roll on the ball, regardless of the initial spin through english. The leather tip is used to increase friction with the cue ball, which in turn provides control over the friction of the cloth.
The laws of physics as they pertain to friction are definitely taken into consideration when taking a masse shot. When you first shoot a masse, the ball is given a non-forward spin. The friction of the cloth causes the ball to start spinning forward, therefore curving the ball in a different path and (hopefully) clearing the ball that you wish to get around.
Many players feel that using top or no english provides better control. True. Because top spin works with the natural friction of the cloth, it is easy to maintain control on the ball. Advanced players know how to use conflicting forces (laws of physics) to move the cue ball in magical ways. A novice should work with the laws, at least until they have learned to understand, or at least trust them.
Kinetic energy is used a lot in the game of pool. Kinetic energy can be found by taking one half of the mass times the velocity squared. Kinetic energy is very important to a pool player. He must control the energy of the ball by how hard his stroke is. A harder stroke would give it more velocity, giving it more kinetic energy. Because the kinetic energy is greater, it takes a ball longer distances to surcome to the forces of friction and come to a stop. This can be what a player desires, but a ball in motion too much can collide with other balls and produce unfavorable outcomes.
There is more physics involved on the billiards table than you would think. From the rotation to the friction to the angles of incidence and reflection, all of which are just a small part of the pool table and game. Friction on the billiards table is more complicated than one might have originally thought. “What actually occurs as the balls rolls is the creation and destruction of new bonds, however this is not directly what causes the balls to slow themselves down, as no kinetic energy is ever lost. However it is the vibrational energy that occurs during that process that causes the ball to lose energy in the form of heat and sound.(Ron Sheperd)”
Sliding and frictional forces are both independent quantities. A ball on a hard rubber surface would have a large sliding force and a small rolling resistance, whereas the opposite would occur on say a more slippery surface. This is actually why billiards tables are lined with felt because it is the most balanced of the two extremes. There are several different speeds of felt that are available with quicker and slower rolling speeds, varying with their individual coefficients of friction.
Another force involved in the game is the collision forces, between the balls, the cue stick and the ball, as well as the balls impacting the rails or pockets. Frictional forces act in a directional tangent to the surface of the balls at the point of contact between the balls. The linear forces accelerating the balls are directed by the balls centers, which is why you might see players line up the center of the ball with the direction they intend for it to go and that determines the angle for aim. The friction force vector that accelerates one ball is opposite of the force accelerating the other ball. The ratio of tangential and normal forces are constants that have to be determined using the coefficient of friction.
The reason players use chalk on the tip of their cue is that the chalk increases the coefficient of friction of the cue tip so more force can be used in the hitting of the cue ball. This allows for better control and accuracy when striking the pool balls. Spin is also often applied to pool balls in order to change their direction. Spin or “English” as it is sometimes called can be applied to whatever direction by hitting the ball off the center. The farther from the center the more spin there will be.
Billiards is very similar to chaos theories and quantum physics, where particles are enclosed and can bounce elastically. Physicists welcome billiards as a good model for chaos in a wide range of problems from thermodynamics to quantum mechanics. The laws of physics that are in play on the billiards table are everywhere around you. Scientists do a good deal of work with atoms bouncing elastically off of walls and each other in order to create spatial patterns.