Joe Tapley
Jul 24, 2014, 4:41:13 AM7/24/14
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Have lost count of the number of times I've seen the statement that, in flight, arrows rotate around their center of mass (center of gravity) as part of some archery discussion.It's one of those archery stories that has become "true" solely on the back of endless repetition despite being fairly obviously wrong.
In general objects can be described as simultaneously having multiple axes of rotation. The number of axes is N+1 where N is the number of force related rotation axes and the +1 relates to the "no force" rotation axis. As an example lets look at the Earth's rotation axes. Let's say that the only force acting on the earth is the Earth-Sun gravity force. In this case (one force) N=1 and 1+1= 2. The Earth has two rotation axes. The Earth rotates about its center of mass (the 24 hour rotation) and the Earth rotates about an axis located somewhere inside the Sun (the one year rotation).
This requirement follows directly from one of Newton's rules i.e. that force equals mass times acceleration. If you apply a force to an object then by definition the center of mass has to accelerate i.e.move. It follows that for any object rotation generated from an applied force the axis of rotation can never be at the center of mass. For a very simple demonstration of this balance an arrow on a finger (i.e. at the COM) and push the shaft upwards fairly hard with the other hand close to the balance point. The arrow will rotate of course under the applied force but the arrow will also lift off the supporting finger (the COM moves upwards in an arc). The basic mechanics of where the axis of rotation is located under an applied force is given on the main site under the heading "Rotation Mechanics" as part of the general discussion about bow stabilization (many people make exactly the same mistake with bow rotation as they do with arrow rotation as in rotating around the center of mass). The converse of this is if an object is rotating without any force applied then the center of mass has zero acceleration and so the rotation axis must be at the center of mass.
So for the Earth the gravitational force results in a rotation axis located somewhere inside the Sun (N = 1). When the Earth was (re) formed the the angular momentum of the relevant bits consolidated into giving the Earth some net spin. As this rotation is not related to an applied force it's the "+1" axis at the center of mass.
Rotation of an arrow in flight is complicated as it is a dynamic non stable process. For an arrow N = 2 (gravity and aerodynamic drag) and so in flight an arrow has nominally three rotation axes. The no force (centre of mass) axis is any spin resulting from the historical effects of drag, gravity and any launch spin. The gravity rotation axis is somewhere inside the earth. The drag rotation axis has a time variable position but generally will probably lie somewhere around the front area of the arrow. A consequence of arrow rotation is lateral arrow movement (drag acceleration) so to the external observer, i.e. the archer, the rotational behaviour of the arrow appears very complex and very variable.
What complicates things is that the drag torque generating rotation and the historical spin are coupled together in a oscillatory fashion. Say you have a fishtailing arrow. Let's look at the two extremes and ignore the effect of gravity. When the arrow is at the end of its (pendulum) swing it has maximum fletching torque and zero rotation. So the arrow has no rotation axis at the center of mass, only the aerodynamic force rotation axis. When the arrow is at maximum rotation rate the fletching torque is zero so the arrow has no aerodynamic rotation axis, only the zero force rotation axis at the center of mass. Averaged out the zero force axis and the aerodynamic drag force axis have around equal importance.
How an arrow rotates in detail during flight is in any practical sense irrelevant beyond the control of the spin at launch (i.e. the tuning aspect). Understanding bow rotation is relevant for bow stabilization as in this case the centre of mass rotation axis is non existant and only the force related rotation axes matter. The incorrect "balance point of the bow being at the grip" idea was purely down to archers/coaches believing the "everything rotates at the center of mass" fiction.
In general objects can be described as simultaneously having multiple axes of rotation. The number of axes is N+1 where N is the number of force related rotation axes and the +1 relates to the "no force" rotation axis. As an example lets look at the Earth's rotation axes. Let's say that the only force acting on the earth is the Earth-Sun gravity force. In this case (one force) N=1 and 1+1= 2. The Earth has two rotation axes. The Earth rotates about its center of mass (the 24 hour rotation) and the Earth rotates about an axis located somewhere inside the Sun (the one year rotation).
This requirement follows directly from one of Newton's rules i.e. that force equals mass times acceleration. If you apply a force to an object then by definition the center of mass has to accelerate i.e.move. It follows that for any object rotation generated from an applied force the axis of rotation can never be at the center of mass. For a very simple demonstration of this balance an arrow on a finger (i.e. at the COM) and push the shaft upwards fairly hard with the other hand close to the balance point. The arrow will rotate of course under the applied force but the arrow will also lift off the supporting finger (the COM moves upwards in an arc). The basic mechanics of where the axis of rotation is located under an applied force is given on the main site under the heading "Rotation Mechanics" as part of the general discussion about bow stabilization (many people make exactly the same mistake with bow rotation as they do with arrow rotation as in rotating around the center of mass). The converse of this is if an object is rotating without any force applied then the center of mass has zero acceleration and so the rotation axis must be at the center of mass.
So for the Earth the gravitational force results in a rotation axis located somewhere inside the Sun (N = 1). When the Earth was (re) formed the the angular momentum of the relevant bits consolidated into giving the Earth some net spin. As this rotation is not related to an applied force it's the "+1" axis at the center of mass.
Rotation of an arrow in flight is complicated as it is a dynamic non stable process. For an arrow N = 2 (gravity and aerodynamic drag) and so in flight an arrow has nominally three rotation axes. The no force (centre of mass) axis is any spin resulting from the historical effects of drag, gravity and any launch spin. The gravity rotation axis is somewhere inside the earth. The drag rotation axis has a time variable position but generally will probably lie somewhere around the front area of the arrow. A consequence of arrow rotation is lateral arrow movement (drag acceleration) so to the external observer, i.e. the archer, the rotational behaviour of the arrow appears very complex and very variable.
What complicates things is that the drag torque generating rotation and the historical spin are coupled together in a oscillatory fashion. Say you have a fishtailing arrow. Let's look at the two extremes and ignore the effect of gravity. When the arrow is at the end of its (pendulum) swing it has maximum fletching torque and zero rotation. So the arrow has no rotation axis at the center of mass, only the aerodynamic force rotation axis. When the arrow is at maximum rotation rate the fletching torque is zero so the arrow has no aerodynamic rotation axis, only the zero force rotation axis at the center of mass. Averaged out the zero force axis and the aerodynamic drag force axis have around equal importance.
How an arrow rotates in detail during flight is in any practical sense irrelevant beyond the control of the spin at launch (i.e. the tuning aspect). Understanding bow rotation is relevant for bow stabilization as in this case the centre of mass rotation axis is non existant and only the force related rotation axes matter. The incorrect "balance point of the bow being at the grip" idea was purely down to archers/coaches believing the "everything rotates at the center of mass" fiction.
A.Ron Carmichael
Feb 27, 2016, 8:18:48 PM2/27/16
to Archery Folk
Hey, Joe - just dropped by to harass you - your premise IS TOTALLY WRONG. (no facts to base that on, just trying to say HI! in my own inimical fashion!) :)
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