 Source: http://en.wikipedia.org/wiki/Bowling. Author: http://en.wikipedia.org/wiki/User:XiaphiasPhysics of Bowling – The Bowling BallA bowling ball is do from urethane, plastic, reactive resin or a combination of these materials. Ten-pin bowling balls normally have 3 holes drilled into them; 2 finger holes and one ignorance hole for gripping. Because that ten-pin bowling, regulation bodies enable for a maximum load of 16 lb (7.2 kg), and a best diameter that 8.6 inch (21.8 cm) (ref: http://en.wikipedia.org/wiki/Bowling_ball).The physics the bowling disputed here will be v regards come ten-pin bowling, which is just one of the most usual sports in the video game of bowling.The figure listed below shows two ten-pin balls.

You are watching: The mass of a bowling ball Source: http://en.wikipedia.org/wiki/Bowling_ball. Author: http://en.wikipedia.org/wiki/User:OmmnomnomgulpBowling round Interior
The bowling ball is composed of a difficult outer shell with a load block in the main point (the within of the bowling ball). The mass and also shape of the load block affect the spin of the bowling ball and also how that curves as it rolls under the lane. This play vital role in the physics of bowling and also (consequently) a bowler's performance, as will certainly be discussed.There room two an easy types of load blocks used, symmetric weight blocks and asymmetric load blocks. To show them, imagine cutting a bowling ball in fifty percent with an imagine cutting plane (as shown below) so as to expose the full cross-section of the load block within the bowling ball. The figure listed below shows the cross-section see for a symmetric weight block. The weight block is molded right into the bowling ball. The "pin" to represent the topmost position of the load block, as shown. This topmost position is closest to the external surface the the bowling ball.A symmetric load block is dubbed "symmetric" because it is axi-symmetric, an interpretation it has an axis of symmetry follow me its centerline. To illustrate, think about the figures listed below with a coordinate device xyz together shown. Note that point G to represent the facility of massive of the bowling ball.Let the airplane x-z represent the imaginary cutting plane. For any angle θ the result cross-section check out of the symmetric weight block would be the same. In other words, a symmetric weight block is axi-symmetric v respect to angle θ.The figure below shows the cross-section see for one asymmetric load block. Just favor the symmetric weight block, the "pin" represents the peak position of the asymmetric load block. However, one asymmetric weight block is not symmetric v respect to edge θ. And also due to its non-symmetry, a "PSA indicator pin" is put on the next of the weight block, in ~ the suggest (or area) closest to the external surface the the bowling ball. These 2 pins are located 90° from each other.The "PSA indicator pin" is also called the "mass bias" location. This is merely a various naming convention in bowling terminology.The pins enable one to recognize the orientation the the weight block within the bowling ball. A symmetric load block only requirements a solitary pin to define its orientation inside the bowling ball, however an asymmetric load block requirements two pins to define its orientation (due come its non-symmetry).The pins give important information on whereby to drill the finger and also thumb feet of the bowling ball so the the bowler can control the spin and also curve that the ball, in bespeak to make the best feasible shot. This will be defined in an ext detail later on, together the physics the bowling is disputed in greater depth.Bowling balls v symmetric and also asymmetric load blocks an outcome in similar performance, yet bowling balls with an asymmetric weight block enable for a bit more "tweaking" to obtain the ball to reaction a certain means when in motion. However, the physics of bowling is very similar whether the load block is symmetric or asymmetric. In enhancement to the pins, two other points of attention on a bowling ball are the hopeful Axis suggest (PAP) and also the CG point. These straight relate come the physics that bowling and also are shown in the number below. The PAP is the early stage axis that rotation of the bowling round as quickly as it starts traveling down the lane. The orientation the this axis depends entirely on the bowler's relax technique. The is distinctive to every bowler.wo is the early stage rotation speed (angular velocity) of the bowling ball. This likewise depends top top the bowler's technique.The figure below shows an imaginary line originating from the geometric facility C that the bowling ball, and also passing through the center of mass G that the bowling ball. The intersection that this line v the surface ar of the bowling round is called CG. This allude is useful because it gives information top top the ar of the center of massive G the the bowling ball. Note the the distance in between C and G is considerably exaggerated for clarity. In reality, the distance in between the two is very little and is typically less than 1 mm (ref: What provides Bowling Balls Hook?, Cliff Frohlich, American association of Physics Teachers, 2004).Bowling ball Motion
The optimal trajectory of a bowling round is a curved course where that strikes the pins at an angle. Highlight the pins in ~ an angle improves the possibilities that there will certainly be a "strike" in which every the pins space knocked down.Analyzing the physics that bowling is an extremely useful because it allows one to recognize the components that influence just how the bowling sphere curves, and how one can make the best feasible shot as soon as striking the pins.If the ball follows a bent path, the will have the ability to strike the pins at a greater angle than a bowling sphere that travel in a directly line. Therefore, regulating the level of curve is essential to make the best possible shot.For ten-pin bowling the length of the lane is 18 m (60 ft). The lane is oil to defend it native wear, especially during the initial slide (skidding) phase of the bowling ball, before it starts pure rolling.The bowler must make the shot behind the foul line and also must avoid obtaining the ball into the gutter. The 2 figures listed below illustrate the movement of a bowling round as it travels down the lane. The sphere motion shown is usual for a right-handed bowler. Because that a left-handed bowler the motion is simply "mirrored" so the the ball curves towards the left gutter rather of the best gutter, prior to hitting the pins.  The ball starts rolling with a rotation rate (angular velocity) that wo and also a direct velocity Vo.Typically, during the very first part the the motion the bowling round slides along the lane, because its rotational speed does not enhance with the direct velocity of the ball. But eventually, lane friction stop the ball from sliding, and pure roll begins. The sphere then continues rolling until it hits the pins.The ball hits the pins at an edge θ. The higher this angle, the an ext oblique the influence and the better the possibility that every the pins will certainly be knocked down. The deflection δ is dubbed "hook". It is the sideways deflection of the round from the initial (straight) trajectory shown by the dashed blue line.Ideally, the front-most pins space hit first by the bowling ball (at an slope angle), because this will most likely result in a strike. In bespeak to design the physics the bowling, a (fixed) coordinate axes XYZ is favored with the complying with orientation: The X-axis is aligned through the appropriate gutter (parallel to the lane), the Y-axis is aligned with the foul line, and also the Z-axis is upright (perpendicular to the lane). This chosen coordinate mechanism becomes relevant in the next section, whereby the physics is analyzed in depth.The location of wo (PAP) family member to the pen location(s) ~ above the bowling ball, is very important in affecting the lot of hook δ, and influence angle θ. Therefore, the finger and also thumb holes should be positioned in one optimal way relative to the pen location(s), so as to optimize δ and also θ, in order to acquire the best possible shot.The place of CG family member to the PAP location likewise influences δ and θ, but to a much lesser degree, so that is not accounted for virtually as lot when optimizing ball performance.For each individual bowler, the PAP is (approximately) in a fixed ar relative come the finger and thumb holes. Because that each individual bowler the place of the PAP relative to the finger and thumb holes, need to be determined with a "test" ball. And also once this relative place is determined, the finger and also thumb holes room drilled in a new ball, so that the resulting PAP is in one optimal position relative to the pin location(s).In a nutshell, the location of the PAP family member to the pins ~ above the bowling round determines how much the bowling ball precesses as it travels under the lane. Consequently, the level that precession is directly proportional come the level of friction between the lane and also the bowling ball. The level that friction, in turn, has actually a huge influence ~ above δ and θ.So in general, more precession leads to more friction and results in an ext hook, and less precession leads to much less friction and results in much less hook. This is among the key criterion the all great players are conscious of as soon as calibrating their game for finest performance.The figure listed below illustrates precession, and also how it affects the level the friction in between the lane and also the bowling ball. Precession is the readjust in direction of the bowling balls turn vector wspin, over time. For example, a spinning height (pivoted around a base) has two rotate components. The an initial component is the main spin, i beg your pardon is the spinning of the top around its central axis. The 2nd component is the secondary spin, which reasons the orientation of the optimal to readjust so the its central axis goes approximately in a circle. This an additional spin component is called precession.In the case of the bowling ball, precession reasons a ago and soon rocking (as shown) as it travels down the lane. This alters the contact surface after each full ball revolution, and also thus increases the area that the sphere in contact with the lane. This becomes noticeable by the miscellaneous lines that oil (shown in the figure above) the the round picks up from call with the lane. The oil lines show up “flared out”, and the width of the flare (called "track flare") is a measure of the degree of rocking (due come precession). Each line the oil represents one full change of the round as the travels under the lane.Therefore, precession outcomes in higher friction between the lane and the bowling ball, because the oil is "spread out" on the surface ar of the bowling round in a bigger area 보다 if the sphere did no precess. If the bowling ball did no precess, there would only be one oil line, and this would an outcome in reduced friction because of repeated lane contact with the very same area that the bowling ball. Hence, precession outcomes in a higher area the the bowling round making call with the lane, which results in lower oil buildup on the ball per contact area, therefore resulting in better friction. This becomes most evident on the "dry" (non oiled) part of the lane, i m sorry is frequently the last third (or so) of the roadway length. In this component of the lane, friction between lane and ball is many sensitive come the lot of oil gathered on the sphere per call area (since the roadway itself is dry). This means that precession raises the level of friction between lane and also ball. Therefore, in the dry component of the roadway a precessing sphere will hook the most.The figure below shows an additional illustration of the usual oil line pattern one could see top top a bowling ball that has undergone precession. As pointed out already, the place of the PAP loved one to the pin areas determines how much the bowling ball precesses together it travels under the lane. In particular, we can specify the relative position of the PAP for bowling balls v symmetric and also asymmetric load blocks. This is a key factor in the physics, since it straight affects the activity of the ball as it travels down the lane. It is described next.Bowling Balls with Symmetric load Blocks
If one desire to have actually no precession, the PAP should be located at the topmost pin place or anywhere on the circumference about the center of the bowling ball, stood for by the dotted line, displayed in the number below. If one wishes to have precession, climate the PAP should be located between these two places - the is, between the topmost pin and the dotted line. Skilled ball drillers understand where to ar the PAP to gain the desired amount of precession. To have actually maximum precession (maximum track flare) the PAP must be placed exactly in in between the topmost pin and also the dotted line, so the the street from the PAP come the topmost pin is equal to the street from the PAP come the dotted line.In physics terms, an axis passing through G and the topmost pin place represents the minimum principal moment of inertia, and also an axis passing through G and any suggest on the dotted line represents the preferably principal moment of inertia - this is as result of the the opposite of the weight block.Side note: Precession is a kind of rough rotation, and it occurs once the PAP does no coincide with the major axes because that the minimum or maximum moment of inertia.For stable rotation (no precession), the PAP must coincide through the primary axes for the minimum or maximum moment of inertia.Bowling Balls v Asymmetric weight Blocks
If one wishes to have no precession, the PAP need to be located at the topmost pin place or in ~ the PSA indicator pin ar on the side, as shown in the figure below. If one desire to have precession, climate the PAP should be located in between these two areas - the is, in between the topmost pin and the PSA pin. Experienced ball drillers know where to ar the PAP to get the preferred amount that precession. To have maximum precession (maximum track flare) the PAP should be placed exactly in in between the topmost pin and the PSA pin, so that the street from the PAP to the topmost pin is equal to the street from the PAP come the PSA pin.In physics terms, one axis passing through G and also the topmost pin place represents the minimum principal moment of inertia, and an axis passing through G and also the PSA pin represents the preferably principal minute of inertia - this is as result of the asymmetry the the load block. Keep in mind that PSA means "preferred turn axis" because the axis with the maximum principal moment of inertia is normally the most stable axis the rotation because that a strict body.So much in discussing the physics the bowling, we have talked around the influence of PAP location and also friction on round motion. Yet there is one more factor which influences ball motion, although not as much. It is the position of the CG. If the CG is top top the left side of the bowling sphere as it travels under the lane, the sphere tends to hook a bit more (to the left), so it's beneficial. The figure below illustrates this. Simulation For various Friction Cases
The main influence on ball motion is friction between the ball and the lane, even if it is it's because of friction affected by sphere precession, or natural lane conditions (e.g. Oil vs. Non-oiled).Thus, us wish to look at three different cases of friction, in stimulate to observe its impact on ball motion.Let's assume that the early angular velocity the the sphere wo = 30 rad/s, and the initial linear velocity of the sphere Vo = 8 m/s. This is a reasonable estimate for the common bowler that averages 200 (ref: What provides Bowling Balls Hook?, Cliff Frohlich, American association of Physics Teachers, 2004).Let's assume wo is pointing 15° to the left the the Y-axis, so the it has materials wX = -30×sin 15° = -7.76 rad/s, and also wY = 30×cos 15° = 28.98 rad/s.And let's assume that Vo is pointing in the hopeful X-direction.The beginning of the bowling balls trajectory is at X = 0.The following additional input worths are provided with regards to the initial orientation, and also properties the the bowling ball:• The distance between the center of the bowling sphere C and also the facility of fixed G is 1 mm (0.001 m). The point CG is pointing in the optimistic Y-direction.• The bowling ball has a symmetric weight block v the pen pointing in the optimistic Y-direction.• The radius of the bowling ball is 10.85 cm.• The massive of the bowling round is 7 kg.• The minimum principal minute of inertia is 0.031 kg·m2. This is the moment of inertia about an axis passing with the center of mass of the bowling round G and also the pin.• The preferably principal moment of inertia is 0.033 kg·m2. This is the minute of inertia about any axis which passes v the facility of fixed of the bowling sphere G and which is perpendicular to the axis equivalent to the minimum principal moment of inertia.Lastly, the input worth for the acceleration due to gravity (on earth's surface) is 9.8 m/s2.The above input worths will be held continuous for the complying with three friction situations to it is in considered. In all three situations the revolution friction will be suspect sufficient sufficient to maintain pure rolling once pure rolling begins. Friction situation 1The coefficient that kinetic (sliding) friction is equal to 0.12. This worth of kinetic friction will certainly be take away as continuous over the entire length of the lane, as portrayed below. Simulation ResultsThe bowling ball slides 9 m before pure rojo begins.The ball hooks come the left δ = 68 cm.The affect angle θ is 3.6° Friction case 2
The coefficient that kinetic friction is same to 0.08. This worth of kinetic friction will be take away as consistent over the whole length that the lane, as depicted below. Simulation ResultsThe bowling sphere slides 13.7 m before pure rolling begins.The ball hooks to the left δ = 55 cm.The impact angle θ is 3.3° Friction case 3
The coefficient that kinetic friction is same to 0.04 because that the first 12 m the the lane, and is equal to 0.2 because that the remainder of the roadway (the remainder that the roadway is "dry", no oil). This is portrayed below. Simulation ResultsThe bowling ball slides 15 m prior to pure roll begins.The round hooks come the left δ = 37 cm.The influence angle θ is 3.3°Looking at the three friction cases we can see the the higher the level of friction, the greater the ball hook. The impact angle also increases slightly with friction.As discussed before, when the bowling ball precesses, the efficient friction in between the lane and ball increases. However, in the simulation, the friction coefficient was input as consistent values (independent of ball precession), for this reason the dependency of friction on round precession was not caught in the design (due to the difficulty in act 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 sphere motion, and also it is reasonable that this will approximate the affect of friction top top ball activity when friction is impacted by sphere precession.Hence, the simulation results indirectly show that precession causes the bowling ball to hook an ext (since precession increases friction).A Closer Look in ~ The principal Moments of Inertia

See more: How Many Railroad Ties In A Mile, What Is The Distance Between Railroad Ties   Where:C is the geometric facility of the bowling ballR is the radius that the bowling ballG is the center of mass of the bowling ballr is the vector from allude C to allude G. This vector is an extremely short, usually much less than 1 mm in lengthP is the contact suggest between the bowling ball and also the surface ar of the lanemp is the vector from point G to suggest Pxyz is the regional coordinate axes fixed to the bowling ball, so the it moves v the bowling ball. The orientation the xyz is such that it is aligned v the principal moments the inertia that the bowling ball.w is the angular velocity the the bowling ballα is the angular acceleration that the bowling ballVc is the linear velocity of the geometric facility of the round C. This is likewise considered to be the straight velocity that the bowling ballac is the straight acceleration that the geometric facility of the sphere C. This is likewise considered to it is in the straight acceleration the the bowling ballFs is the friction force acting top top the bowling ball, because of contact in between the ball and lane. This pressure is parallel come the lane surfaceN is the normal pressure pushing increase on the ball, perpendicular come the roadway surfaceg is the acceleration as result of gravity, acting downwards. This worth is same to 9.8 m/s2 top top earth defining the orientation the xyz in ~ any suggest in time
To specify the orientation the xyz v respect to the (fixed) global XYZ axes, us must first set increase the basic equations because that the vectors x, y, and also z in terms of direction cosines.Thus, Where:x, y, and z are unit vectorsl1, m1, n1 space the direction cosines corresponding to the unit vector xl2, m2, n2 room the direction cosines corresponding to the unit vector yl3, m3, n3 space the direction cosines matching to the unit vector zI is the unit vector pointing along the hopeful direction of global XJ is the unit vector pointing follow me the confident direction of an international YK is the unit vector pointing along the positive direction of an international ZBy definition, the orientation the I, J, K continues to be constant, because XYZ is fixed in space.Since x, y, and z room unit vectors, and because x, y, and also z room perpendicular come each various other (by definition), defining the orientation of vector r in ~ any allude in time
Similar to before, the orientation that r is through respect come the (fixed) an international XYZ axes: Where:rX is the component of r along the an international X-directionrY is the ingredient of r along the worldwide Y-directionrZ is the component of r along the global Z-direction specifying the orientation that angular velocity w and angular acceleration α in ~ any suggest in time
Similar to before, the angular velocity and angular acceleration are vectors expressed relative to the (fixed) global XYZ axes. Lock are offered by Where:t is timeαX is the component of α follow me the an international X-directionαY is the component of α along the worldwide Y-directionαZ is the ingredient of α along the global Z-directionwX is the component of w along the an international X-directionwY is the ingredient of w follow me the global Y-directionwZ is the component of w along the an international Z-directionThus, the angular velocity and angular acceleration that the bowling sphere are always with respect to (fixed) ground. Identify the materials of the angular velocity w and also angular acceleration α follow me the xyz axes
It's vital to solve the angular velocity and angular acceleration into their contents along the xyz axis. This deserve to be done utilizing the vector period product (a⋅b): Therefore, where the subscripts x, y, and z (for the angular velocity and also angular acceleration) denote their contents along the x, y, and z directions, respectively. Utilizing forward integration to deal with for the vectors w, r, and the orientation the xyz together a role of time
Let Δt be a small time step.Forward integration is used to fix the physics equations. In this model, various unknown quantities at a new time t+Δt are calculated based upon previous well-known quantities in ~ time t.For instance, we have the right to solve for w(t+Δt) using the adhering to equation because that angular acceleration: Similarly, us can discover the brand-new orientation the xyz at time t+Δt based upon its previous orientation at time t: where a×b to represent the vector cross product.Similarly, because that the vector r: Note the the over expressions for x, y, z, and r incrementally boost their vector lengths (magnitudes) because that each time step. Therefore, we need to normalize the vectors x, y, z, and also r after every time step to keep their size (magnitude). This normalizing operation deserve to be expressed mathematically as follows: where ro is the length of the r vector calculated from its initial specification (at time t = 0). The acceleration of the facility of massive G
The facility of mass G is counter from the geometric center of the bowling round C. Thus we have to calculate the acceleration the G together follows: Where:aG is the acceleration that the center of mass GaCX is the worldwide X-component the the acceleration of suggest CaCY is the global Y-component the the acceleration of allude C application of Newton's second Law
The external forces exhilaration on the bowling ball are gravity and also the contact forces at suggest P. Therefore, we can express the amount of the pressures on the bowling round as: Therefore, Where:m is the massive of the bowling ballFsX is the global X-component that the pressure acting on the bowling round at allude PFsY is the worldwide Y-component that the pressure acting ~ above the bowling sphere at allude PaGX is the an international X-component the the acceleration of point GaGY is the global Y-component the the acceleration of allude GaGZ is the worldwide Z-component that the acceleration of allude G amount of the moments (torque) exhilaration on the bowling ball
If we take the sum of the moments about the center of mass G, the only pressures to account for are those acting at the contact point P, through vector eight mp. (The pressure of gravity exerts no moment around point G due to the fact that its heat of activity passes through point G).We can thus use the Euler equations of motion for a strict body: where Ix, Iy, and also Iz space the major moments of inertia, and and Note that the amount of moments in the Euler equations are expressed v respect to the neighborhood x, y, and z directions. It is essential to execute this as soon as using these equations.Lastly, us must consider two separate situations of ball activity when examining the physics. The very first case requires kinetic (sliding) friction, where the bowling ball skids follow me the lane. The second case involves static friction, whereby the bowling ball has stopped skidding and also is in a state the pure rolling. Case 1 – Kinetic Friction
For this case, there is loved one motion between the bowling ball and also the lane at allude P. We have the right to express this relative activity as velocity VP, where: Now, Where:VCX is the global X-component the the velocity of point CVCY is the global Y-component the the velocity of suggest CTherefore, The size of kinetic friction is given as where μk is the coefficient the kinetic friction.Kinetic friction acts opposite the direction the (relative) motion, therefore where The contents along the X and Y directions are: instance 2 – static Friction
For this case, there is no family member motion in between the bowling ball and also the roadway at suggest P. Therefore, VP = 0.Using the exact same expression as before, Since VP = 0, Similarly, Therefore, For pure rojo of the bowling ball the following condition must it is in satisfied: where μs is the coefficient of revolution friction.This completes the evaluation of the physics of bowling.Return come The Physics of Sports
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