Source: Wikipedia via Xiaphias

A bowling ball is made from urethane, plastic, reactive resin or a combination of these materials. Ten-pin bowling balls generally have three holes drilled into them; two finger holes and one thumb hole for gripping. For ten-pin bowling, regulating bodies allow for a maximum weight of 16 lb (7.2 kg), and a maximum diameter of 8.6 inches (21.8 cm) (ref: http://en.wikipedia.org/wiki/Bowling_ball).

The physics of bowling discussed here will be with regards to ten-pin bowling, which is one of the most common sports in the game of bowling.

The figure below shows two ten-pin balls.

Source: Wikipedia via Ommnomnomgulp

The bowling ball consists of a hard outer shell with a weight block in the core (the inside of the bowling ball). The mass and shape of the weight block affects the spin of the bowling ball and how it curves as it rolls down the lane. These play an important role in the physics of bowling and (consequently) a bowler's performance, as will be discussed.

There are two basic types of weight blocks used, symmetric weight blocks and asymmetric weight blocks. To illustrate them, imagine cutting a bowling ball in half with an imaginary cutting plane (as shown below) so as to expose the full cross-section of the weight block inside the bowling ball.

The figure below shows the cross-section view for a symmetric weight block.

The weight block is molded into the bowling ball. The "pin" represents the topmost position of the weight block, as shown. This topmost position is closest to the outside surface of the bowling ball.

A symmetric weight block is called "symmetric" because it is axi-symmetric, meaning it has an axis of symmetry along its centerline. To illustrate, consider the figures below with a coordinate system

Let the plane

The figure below shows the cross-section view for an asymmetric weight block.

Just like the symmetric weight block, the "pin" represents the top position of the asymmetric weight block. However, an asymmetric weight block is

The "PSA indicator pin" is also called the "mass bias" location. This is simply a different naming convention in bowling terminology.

The pins allow one to determine the orientation of the weight block inside the bowling ball. A symmetric weight block only needs a single pin to define its orientation inside the bowling ball, but an asymmetric weight block needs two pins to define its orientation (due to its non-symmetry).

The pins give important information on where to drill the finger and thumb holes of the bowling ball so that the bowler can control the spin and curve of the ball, in order to make the best possible shot. This will be explained in more detail later on, as the physics of bowling is discussed in greater depth.

Bowling balls with symmetric and asymmetric weight blocks result in similar performance, but bowling balls with an asymmetric weight block allow for a bit more "tweaking" to get the ball to react a certain way when in motion. However, the physics of bowling is very similar whether the weight block is symmetric or asymmetric.

In addition to the pins, two other points of interest on a bowling ball are the Positive Axis Point (PAP) and the CG point. These directly relate to the physics of bowling and are shown in the figure below.

The PAP is the

The figure below shows an imaginary line originating from the geometric center

Note that the distance between

The optimal trajectory of a bowling ball is a curved path where it strikes the pins at an angle. Striking the pins at an angle improves the chances that there will be a "strike" in which all the pins are knocked down.

Analyzing the physics of bowling is very useful because it allows one to understand the factors that influence how the bowling ball curves, and how one can make the best possible shot when striking the pins.

If the ball follows a curved path, it will be able to strike the pins at a greater angle than a bowling ball that travels in a straight line. Therefore, controlling the degree of curve is essential to making the best possible shot.

For ten-pin bowling the length of the lane is 18 m (60 ft). The lane is oiled to protect it from wear, especially during the initial sliding (skidding) stage of the bowling ball, before it begins pure rolling.

The bowler must make the shot behind the foul line and must avoid getting the ball into the gutter. The two figures below illustrate the motion of a bowling ball as it travels down the lane. The ball motion shown is typical for a right-handed bowler. For a left-handed bowler the motion is simply "mirrored" so that the ball curves towards the left gutter instead of the right gutter, before hitting the pins.

The ball begins rolling with a rotation speed (angular velocity) of

Typically, during the first part of the motion the bowling ball slides along the lane, since its rotational speed does not match with the linear velocity of the ball. But eventually, lane friction stops the ball from sliding, and pure rolling begins. The ball then continues rolling until it hits the pins.

The ball hits the pins at an angle

Ideally, the front-most pins are hit first by the bowling ball (at an oblique angle), since this will most likely result in a strike.

In order to model the physics of bowling, a (fixed) coordinate axes

The location of

The location of CG relative to the PAP location also influences

For each individual bowler, the PAP is (approximately) in a fixed location relative to the finger and thumb holes. For each individual bowler the position of the PAP relative to the finger and thumb holes, must be determined with a "test" ball. And once this relative position is determined, the finger and thumb holes are drilled in a

In a nutshell, the location of the PAP relative to the pins on the bowling ball determines how much the bowling ball

So in general, more precession leads to more friction and results in more hook, and less precession leads to less friction and results in less hook. This is one of the main criterion that all good players are aware of when calibrating their game for best performance.

The figure below illustrates precession, and how it affects the level of friction between the lane and the bowling ball.

Precession is the change in direction of the bowling balls spin vector

In the case of the bowling ball, precession causes a back and forth rocking (as shown) as it travels down the lane. This changes the contact surface after each full ball revolution, and thus increases the area of the ball in contact with the lane. This becomes evident by the various lines of oil (shown in the figure above) that the ball picks up from contact with the lane. The oil lines appear “flared out”, and the width of the flare (called "track flare") is a measure of the degree of rocking (due to precession). Each line of oil represents one full revolution of the ball as it travels down the lane.

Therefore, precession results in greater friction between the lane and the bowling ball, since the oil is "spread out" on the surface of the bowling ball in a larger area than if the ball did not precess. If the bowling ball did not precess, there would only be one oil line, and this would result in lower friction due to repeated lane contact with the same area of the bowling ball. Hence, precession results in a greater area of the bowling ball making contact with the lane, which results in lower oil accumulation on the ball per contact area, thus resulting in greater friction. This becomes most apparent on the "dry" (non oiled) part of the lane, which is typically the last third (or so) of the lane length. In this part of the lane, friction between lane and ball is most sensitive to the amount of oil accumulated on the ball per contact area (since the lane itself is dry). This means that precession raises the level of friction between lane and ball. Therefore, in the dry part of the lane a precessing ball will hook the most.

The figure below shows another illustration of the typical oil line pattern one might see on a bowling ball that has undergone precession.

As mentioned already, the position of the PAP relative to the pin locations determines how much the bowling ball precesses as it travels down the lane. In particular, we can specify the relative position of the PAP for bowling balls with symmetric and asymmetric weight blocks. This is a key factor in the physics, since it directly affects the motion of the ball as it travels down the lane. It is described next.

If one wishes to have

If one wishes to have precession, then the PAP must be located between these two locations - that is, between the topmost pin and the dotted line. Experienced ball drillers know where to place the PAP to get the desired amount of precession. To have maximum precession (maximum track flare) the PAP must be placed exactly in between the topmost pin and the dotted line, so that the distance from the PAP to the topmost pin is equal to the distance from the PAP to the dotted line.

In physics terms, an axis passing through

For stable rotation (no precession), the PAP must coincide with the principal axes for the minimum

If one wishes to have

If one wishes to have precession, then the PAP must be located between these two locations - that is, between the topmost pin and the PSA pin. Experienced ball drillers know where to place the PAP to get the desired amount of precession. To have maximum precession (maximum track flare) the PAP must be placed exactly in between the topmost pin and the PSA pin, so that the distance from the PAP to the topmost pin is equal to the distance from the PAP to the PSA pin.

In physics terms, an axis passing through

So far in discussing the physics of bowling, we have talked about the influence of PAP location and friction on ball motion. But there is another factor which influences ball motion, although not as much. It is the position of the CG. If the CG is on the left side of the bowling ball as it travels down the lane, the ball tends to hook a bit more (to the left), so it's beneficial. The figure below illustrates this.

The main influence on ball motion is friction between the ball and the lane, whether it's due to friction influenced by ball precession, or natural lane conditions (e.g. oiled vs. non-oiled).

Thus, we wish to look at three different cases of friction, in order to observe its effect on ball motion.

Let's assume that the initial angular velocity of the ball

Let's assume

And let's assume that

The beginning of the bowling balls trajectory is at

The following additional input values are used with regards to the

• The distance between the center of the bowling ball

• The bowling ball has a symmetric weight block with the pin pointing in the positive

• The radius of the bowling ball is 10.85 cm.

• The mass of the bowling ball is 7 kg.

• The minimum principal moment of inertia is 0.031 kg-m

• The maximum principal moment of inertia is 0.033 kg-m

Lastly, the input value for the acceleration due to gravity (on earth's surface) is 9.8 m/s

The above input values will be held constant for the following three friction cases to be considered. In all three cases the static friction will be assumed sufficient enough to maintain pure rolling once pure rolling begins.

The coefficient of kinetic (sliding) friction is equal to 0.12. This value of kinetic friction will be taken as constant over the entire length of the lane, as illustrated below.

The bowling ball slides 9 m before pure rolling begins.

The ball hooks to the left

The impact angle

The coefficient of kinetic friction is equal to 0.08. This value of kinetic friction will be taken as constant over the entire length of the lane, as illustrated below.

The bowling ball slides 13.7 m before pure rolling begins.

The ball hooks to the left

The impact angle

The coefficient of kinetic friction is equal to 0.04 for the first 12 m of the lane, and is equal to 0.2 for the remainder of the lane (the remainder of the lane is "dry", no oil). This is illustrated below.

The bowling ball slides 15 m before pure rolling begins.

The ball hooks to the left

The impact angle

Looking at the three friction cases we can see that the greater the level of friction, the greater the ball hook. The impact angle also increases slightly with friction.

As mentioned before, when the bowling ball precesses, the effective friction between the lane and ball increases. However, in the simulation, the friction coefficient was input as constant values (independent of ball precession), so the dependence of friction on ball precession was not captured in the model (due to the difficulty in doing so). So, to approximate the effect of friction, the friction coefficient was varied over a range of values in order to observe the effect on ball motion, and it is reasonable that this will approximate the influence of friction on ball motion when friction is affected by ball precession.

Hence, the simulation results indirectly show that precession causes the bowling ball to hook more (since precession increases friction).

In general, a rigid body has three distinct principal moments of inertia. It is useful to know the magnitude and direction of these three quantities because they simplify the equations for three-dimensional rigid body dynamics.

The shape of the weight block inside a bowling ball makes it fairly straightforward to determine the principal directions of inertia. If a rigid body has two or three planes of symmetry, the principal directions will be aligned with these planes. By inspection, one can usually find the symmetry planes in the weight blocks, and as a result easily determine the principal directions for them. Since the remainder (outer part) of the bowling ball is itself almost completely symmetric due to its spherical shape, the result is that the (entire) bowling ball has principal directions closely aligned with the principal directions of the weight block.

For example, consider a bowling ball with a symmetric weight block, as shown in the two figures below.

The minimum principal moment of inertia of the bowling ball is about the

The

Now, consider a bowling ball with an asymmetric weight block, as shown in the figure below.

Once more, the minimum principal moment of inertia of the bowling ball is given by

The maximum principal moment of inertia is given by

In physics, the moment of inertia (

In bowling terminology, the radius of gyration is called the "RG value", and the term "RG differential" refers to the difference between the maximum and minimum moment of inertia of a bowling ball (i.e.

In bowling lingo: The greater the "RG differential", the greater the potential for a bowling ball to create "track flare", and the greater the hook potential. Translated into physics lingo this means: The greater the difference

However, bowling regulations limit the amount of "RG differential" in a bowling ball.

Of course, whether or not this hook potential is fully utilized depends on where the PAP is placed relative to the pin locations on the bowling ball.

I created a numerical model in Excel which captures the physics of bowling and simulates the motion of a bowling ball as it travels down the lane. A time step of 0.0001 seconds was used to ensure sufficient numerical accuracy. It was a challenging and time consuming task to develop this model, but the result is that I am able to accurately capture the essential physics behind bowling, without any gross simplifications and shortcuts which sacrifice accuracy.

To download the Excel spreadsheet right-click on this link.

I'm making the spreadsheet available, and the instructions to use it, for free. The download file is in a compressed file format. You need to uncompress this file before you can do anything. This file can be uncompressed with several different types of compression software (such as WinZip, which may already be installed on your computer). If you don't have compression software on your computer, or you are unable to uncompress the file with the software you do have, you can download and install the 7-zip compression software on to your computer. It's available for free at: http://www.7-zip.org.

Lastly, to use the bowling simulator you need to have Microsoft Excel installed on your computer. The program is compatible with all versions of Excel.

The development of the equations to fully analyze the physics behind bowling is quite an onerous task both to derive and fully present on a website. Therefore, the full mathematical development will not be presented here. Instead, the basic equations will be introduced in order to give the reader a basic understanding of the core physics and mathematics required.

The following assumptions are made in the model:

• The lane is perfectly flat.

• Friction is proportional to normal force, and the coefficient of friction between ball and lane is constant.

The figure below illustrates the full set up for analyzing the physics, with (fixed) global coordinate system

Where:

To define the orientation of

Thus,

Where:

By definition, the orientation of

Since

and since

Similar to before, the orientation of

Where:

Similar to before, the angular velocity and angular acceleration are vectors expressed relative to the (fixed) global

Where:

Thus, the angular velocity and angular acceleration of the bowling ball are always with respect to (fixed) ground.

It's necessary to resolve the angular velocity and angular acceleration into their components along the

Therefore,

where the subscripts

Let

Forward integration is used to solve the physics equations. In this model, various

For instance, we can solve for w(

Similarly, we can find the new orientation of

where

Similarly, for the vector

Note that the above expressions for

where

The center of mass

Where:

The external forces acting on the bowling ball are gravity and the contact forces at point

Therefore,

Where:

If we take the sum of the moments about the center of mass

We can thus apply the Euler equations of motion for a rigid body:

where

and

Note that the sum of moments in the Euler equations are expressed with respect to the local

Lastly, we must consider two separate cases of ball motion when analyzing the physics. The first case involves kinetic (sliding) friction, where the bowling ball skids along the lane. The second case involves static friction, where the bowling ball has stopped skidding and is in a state of pure rolling.

For this case, there is relative motion between the bowling ball and the lane at point

Now,

Where:

Therefore,

The magnitude of kinetic friction is given as

where

Kinetic friction acts opposite the direction of (relative) motion, therefore

where

The components along the

For this case, there is no relative motion between the bowling ball and the lane at point

Using the same expression as before,

Since

Similarly,

Therefore,

For pure rolling of the bowling ball the following condition must be satisfied:

where

This completes the analysis of the physics of bowling.

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