The Bow.

In a nutshell, a bow is a device that converts slow and steady human force over a distance (Work) into stored Mechanical Potential Energy (in the form of tension in the Bowstave, Limbs, or Prod). This energy is converted into Kinetic Energy upon release of the Bowstring, and a great deal of that kinetic energy is transfered to the arrow.

Q: If it's just human work, why can't someone throw the arrow that fast and far?

A: For a number of reasons. One-- When you throw something, you're also throwing your arm's Mass along with it. Two-- Humans are only capable of exerting a certain amount of Force. With a bow, a human can exert that force over a longer period of time, storing it up in the force of Elastic Potential Energy.

DRAW CURVE

An archery draws back on his bow, applying a force to the bowstring which in turn bends the bow as it adds elastic potential energy. Thus, a bow is basically a spring which stores enegry to be put into the arrow.

The Force (Draw or Draw Weight) upon a bow is represented by the draw curve:

 

The Draw Weight(F) is directly proportional to the Draw Length(x), so like a spring, it follows Hooke's Law.

However, it is only an apparent relationship. The Draw Curve is actually a slight curve (because of the shape of the bowstave), but combined with other factors, it nets a straight line relationship.

The Energy stored in the bow is equal to Fx/2 (because there are two limbs).

For a Recurve Bow, the straight line graph above works. However, for a Compound Bow, which utilizes levers and such, the weight actually decreases with the draw length, allowing a bow with the same amount of energy to require less force.

The graph at left represents the weight of a compound bow over its draw length.

The Draw Force Curve determines:

1. The Weight on the Archers Fingers at Full Draw.

2. How much energy is stored in the bow at full draw.

3. The shape of the Draw Curve determines the Stacking quality of the bow.

The Draw Length is the distance of the offset from the initial position of the string:

There is a relatively simple way to determine the Draw Weight over any given distance, using simple trigonometry.

At right, we see the Draw Length of the bow. If you notice, at a different Draw Length, the angle of the string with the bowstave and the angle of the arrow with the string change. At 0 draw length, there is a 90 angle between the arrow and the bowstring.

Depending on the make of the bow, the angle between the string and the force direction of the bowstave at 0 draw differs.

The angle between the arrow and the string is angle B. The angle between the spring force on the limb of the bow and the bowstring is angle A. At any given time, the Force that the archer must hold the arrow with is equal to the 2x Force x Cos (A) x Cos (B). The two is a result of the presence of, obviously, two limbs, assuming identicle. (Yes, assuming identical, to make this easier.)

F, A, and B change steadily as the arrow is drawn back.

 

At left are the two factors which lead to the net draw force curve. On the left, 2Cos(A)Cos(B). On the Right, in the Dark Green, the various force (F) of the Limb as a spring. This is a result of the bow's shape, being thicker at the center and thinner towards the limbs. This causes there to be less spring force per unit of distance that the bow is pulled from the Bracing Height.

 

The Bracing Height is the distance from where the string at an undrawn bow to the bowstave. In the image at right, the Bracing Height is the red distance "b".
Increasing the Bracing Height "weakens," reducing velocity. Decreasing the Bracing Height strengths the arrow, that is, more kinetic energy is put into it.

This happens because the Draw Curve is determined by the amount of spring force in the limb, which decreases the more the limb is bent. With a higher Bracing Height, the limb starts off with more bend, and therefore, less spring force.

Also, a Bracing Height 2" higher results in 2" less Draw Length for the bow, given a Draw Curve like the one at right, comparing identical bows with different Bracing Heights and the Draw Weight, or Force, on both at maximum Draw.

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