“The Physics Ball Sports” Marc Walker Department of Physics, University of Warwick, Coventry, England. December, 1999. Abstract All ball sports are governed by the physics associated with aerodynamics and fluid flow. Here, we shall investigate Bernoulli's theorem and the Magnus effect, and then apply them to four sports - baseball, golf, football and cricket. We find that aerodynamics has a huge role in defining all of these sports. All of them see curvature of the path of the ball whilst in flight. This is due to the Magnus effect, and to the boundary layer surrounding the ball. We find that dimples on a golf ball help to increase distance, whilst the seam on a cricket ball can enhance swing. We also see the effect of sidespin on the ball, with spin bowling in cricket, and the curveball in baseball, which curves around 18 inches in flight. We also find that atmospheric conditions play a huge role in the distance of flight of a batted baseball, by increasing or reducing the air resistance, or drag, experienced by the ball. This essay is intended as an introduction to some of the basic physics which we can see in some of the world’s most popular sports. If you have any comments, please email me. 1. Introduction In this essay, we will consider the physics involved in four sports - football, golf, cricket and baseball. Firstly, we will consider the principles behind the aerodynamics of spheres. Then, we will apply these theories to the four sports previously mentioned. We will also consider the collision between baseball and bat. 1.1 Bernoulli's Principle Bernoulli's Principle states that regions of higher velocity in a fluid correspond to regions of lower pressure in that fluid. This is a direct result from Newton's Second Law of Motion: d 2 x F = m (1) dt 2 Hence, it can be clearly seen that if a constant force, F, is applied to a mass, m, the mass will accelerate. If the mass is accelerating, then its kinetic energy, Ek, is increasing by equation 2. 2 1 dx Ek = m (2) 2 dt This is the work done by the force, as it moves through a distance x. The work done is also defined as the product of the magnitude of the force and the distance over which it acts on the mass m, i.e. d 2 x Work Done = F.x = mx (3) dt 2 Now consider fluid motion. A line within the fluid to which the velocity vector is tangential is called a streamline. If the flow is steady, the fluid always moves along the streamlines. Atoms of the fluid are changing their velocities, but the velocity of the fluid at each point in space remains the same over time. Consider a particular element of a fluid over a period of time. v' v A' A Streamline Figure I - Stationary Fluid Flow When the fluid volume element is at A, it has a velocity v. If the fluid is stationary, all volume elements must have a velocity v as they pass A. This is also true of position A', where all elements have velocity v'. Therefore, we can consider the velocity distribution to be a function of position rather than time. To study the motion of a fluid, we also need to consider pressure. Pressure (P) is defined as the normal force (F) per unit area (A) exerted on a surface. Hence, F P = (4) A Pressure is expressed in units of newtons per square metre (Nm-2), or Pascals. The motion of the volume element is determined not only by the pressure exerted by the rest of the fluid, but also by gravity and other forces. We first determine the resultant of the external forces (gravity, electric, etc.), on the volume of fluid due to the pressure. Consider a fluid moving along a cylindrical pipe of variable cross-section A (see Figure II, below). Figure II - Forces on a fluid element due to the pressure. The x-axis is made coincident with the axis of the pipe. Consider an element of thickness dx, cross-sectional area A, and therefore volume Adx. Let p and p' be the values of pressure on the left and right sides of the volume element. Therefore, by equation 4, the force differential, df, is df = F − F ' = PA − P ' A = − ( P '− P ) A (5) However, p'-p is the pressure differential across the element. The element is of thickness dx. So, if p'-p = dp, then dp df = − (dp ) A = − ( Adx ) (6) dx Since Adx is the volume of the fluid, we conclude that the force per unit volume along the x-axis on the fluid due to the pressure is dp F = − (7) dx This result shows that the force acts in the direction of decreasing pressure. This is analogous to a potential field. In addition to the pressure, there may be other external forces, such as gravity or an external electric field, acting on the fluid inside the volume element. Let Ep be the corresponding potential per unit volume due to other external forces. Then the total potential energy due to pressure and other effects is p + Ep. We assume that the fluid is incompressible. For an incompressible fluid, the internal energy does not change, so we can ignore it when we consider the changes of energy of the volume element. The conservation of energy applied to an incompressible fluid, states that if all the forces acting on an incompressible fluid are conservative, and we follow the motion of a small volume of fluid, we find that the total energy per unit volume remains constant. Thus, the sum of kinetic energy, potential energy and pressure is a constant. i.e. 1 2 ρv + P + E = const. (8) 2 p where ρ is the density of the fluid. This result is known as Bernoulli's Theorem. 1.2 Magnus Effect and the Boundary Layer The Magnus Effect describes the flow of air around a rotating sphere. The basic concept was formulated in the 19th century, and expanded by Prandtl in the early 1900s. The commonly accepted explanation is that a spinning object creates a whirlpool of rotating air about itself. On the side where the motion of the whirlpool is in the same direction as that of the windstream to which the object is exposed, the velocity will be enhanced. On the opposite side, where the motions are opposed, the velocity will be decreased. According to Bernoulli's principle, this creates a pressure differential and a force perpendicular to the velocity vector of the sphere. This is the Magnus force. In reality, the phenomenon is a little more complex. It involves a thin layer of fluid, very close to the solid wall of the sphere. This is known as the boundary layer. Because every fluid possesses a viscosity, there is a shearing force between adjacent regions of the fluid (in this case air), which move at different speeds. This shearing force between layers decreases as we move further away from the sphere, until the sphere has a negligible effect on the airflow. When we reach this point, the air moves with free stream velocity. The faster moving layers drag the slower-moving layers along. Between the edge of the sphere and the region of free stream velocity, the shearing force caused by the sphere is of great importance. This is the previously mentioned boundary layer. The boundary layer is of the order of a few millimetres wide, but is crucial to the motion of the sphere. If the viscosity of air was zero, the air would flow around the ball as shown in figure III, below. Figure III - Flow of an ideal fluid having no viscosity. Now consider a small section of fluid, known as a fluid particle, within the fluid as a whole. As the fluid particle at point A in figure III approaches the ball, and begins to move around it to the equator (A'), it will accelerate. In order for this to happen, a force must be applied to the fluid particle. This ''pressure'' force comes from the pressure difference between A and A'. According to Bernoulli's Principle, the pressure at A is greater than at A', and thus the kinetic energy of the particle must increase for equation 8 to hold. The reverse happens between A' and A", with the kinetic energy being given back. However, this is not quite the case, as air does have a viscosity. As the fluid particle moves past A', it still slows down. The free stream flow sweeps the fluid particle along, but on the downstream side of the sphere, however, the particle does not reach A''. A region known as the wake region develops, where the air experiences a swirling, chaotic flow. The point at which the wake region begins is known as the boundary layer separation point, and the phenomenon itself is known as boundary layer separation. Figure IV - Flow of a real fluid (e.g. air) flowing around a sphere. Due to the chaotic flow, the pressure in the wake region is significantly lower than the pressure on the front of the sphere. This is because the kinetic energy lost by the fluid particles is bound up in the turbulent eddies in the wake, instead of being added back to the pressure. The existence of the wake creates a drag force. The drag force slows the ball down, whilst the shape of the wake region causes curvature in flight. 1.3 Drag We can estimate the drag forces on a ball (6).