Doctor Hooves and the Physics of Bowling ‘Hooks’ · 7:58pm Jun 22nd, 2015
You know how the game works. You bowl your ball towards ten pins set up in a triangle, with the aim of knocking down as many as possible. Simple. But the fundamental simplicity does not mean there is little to it, and like all the best games, fanatics have spent years analyzing every detail to determine the best strategy. This is an excellent one to study with the language of physics.
You release your ball with a linear momentum, and it will slide along the oiled surface of the alley. But the friction between the ball’s surface and the floor will slow it down, and create a torque, causing it to build up angular momentum, until it stops sliding and rolls towards the pins.
Bowlers have long recognized that if the ball hits the front pin at an angle, instead of head-on, it is more likely to be a strike, knocking down the full set, and minimizing the risk of the seven/ten (harshest-of-the-harsh) split. How to get a curved path? With a skillful flick of your hoof, you set it moving with a slight side-spin. This does not change the path as it slides, but once rolling, the extra component of angular velocity will bend it to one side. Most sharply at the end of the alley where the surface is not oiled, and friction is higher. Thus the best players can achieve a ‘hook’ where the ball swerves to the center at the last moment and hits the pins at an oblique angle.
How to calculate the trajectory? We need to consider the force F and torque (twisting force) τ=r×F on the ball at all times and remember the coefficient of friction μ varies along the alley, and the moment of inertia of the ball [1] I (ratio of the applied torque to resulting angular acceleration α). And this is before we consider the offset between the center of gravity and the geometric center.
[Here is the screenshot. The diagram on the bottom right looks like it comes from Physics of Bowling. It seems the artists put in a few deliberate mistakes to make it more fun. There also seem to be twelve pins in the background image. Twenty percent cooler.]
[1] The moment of inertia is to rotational motion, what mass is to linear motion. A massive object needs more push to get it moving; an object with a large moment of inertia needs more twist to get it spinning. This depends on how the mass is distributed about the axis around which you spin it, and can be different when spun around different axes. Bowling balls are not uniform, but are made by casting resin around a central weight block (then drilling holes). A full description requires a nine component moment of inertia tensor.
This is not a complete calculation but a brain dump. Good way to approach an unfamiliar problem. Write down every formula which seems relevant, and try to see how it fits together. Or if it looks too intimidating, give up, as the Doctor did. There are simply too many variables!
Can we finish the calculation? Indeed this challenge has been taken up by quite a few physics-and-bowling fans over the years.
In 1976 DC Hopkins and JD Patterson of the South Dakota School of Mines and Technology published a paper on Bowling frames: Paths of a bowling ball (American Journal of Physics vol 45 no 3), in which they derive the equations of motion under certain assumptions, and solve these to calculate the curved path of the ball. But this pioneering work had limitations, failing to consider that the ball is non-uniform and the friction varies along the length of the alley. In a 1988 Scientific American article Why sidespin helps the bowler and how to keep scoring strikes Jearl Walker examines this further, and investigates how when the ball approaches at an angle, it improves the chances of a strike.
More recently, Cliff Frohlich from the University of Texas, published a paper in 2004: What makes bowling balls hook? (American Journal of Physics vol 72 no 9), taking into consideration the differences in the principal moments of inertia of the ball and the offset of the center of gravity.
This is not just about the bowler. The properties of the ball also have a big effect. Top bowlers will spend several hundred dollars for a premium ball, and market forces have worked to ensure a supply of balls with the optimum distribution of mass for every bowling strategy. As Professor Frohlich discusses, a center of mass offset provides an extra torque as the ball rolls. He also speculates that where the properties of the ball cause the rotation to 'precess'—the axis of rotation rotates around and around—this means the oil is spread more thinly over the surface, increasing the friction, and hence the hook.
So given all the variables: ball mass, dimensions, center of gravity, all components of the moment of inertia tensor, and coefficient of friction at all points down the alley, you can take these equations and compute the optimum initial linear and angular velocity to send the ball along the perfect path.
Variables? What are you talking about, man? Just throw the ball straight!
But... but you said it's better to curve it! I am confuse.
Dat math tho'.
Since I am not a particularly skilled bowler, I have my own strategy.
1: Pick the heaviest ball I can find.
2: Throw it as hard as I can at the middle pin.
When that works right, the resultant explosion of pins generally takes them all down, and sometimes causes a pin monkey to have to go down the lane and collect a few strays from the gutter (or the next lane over, if I really had a good throw).
Dang I love this blog.
Torque makes my head spin
Unless the physics is really going over my head, I think you meant chances.