created at February 20, 2022

# Bow arrow mass center calculation

If you happen to make your own bolt for your custom crossbow (or bow arrow), one should balance the bolt properly. It's believed that the bolt mass center relative position should be around 0.3 - 0.33. We gonna calculate it by given dimensions and material densities. Let's introduce the following scheme:

L1 - the bullet point length, it's considered to be solid. The tip sharpening influence to the mass center position will be omited for the sake of simplicity. Consider also that there is no insert - the bullet point just penetrates the shaft directly. The penetration rod of the bullet point has length L3 and diameter d1.

The shaft is a tube with length L2 and with outer and inner diameters D and d. If the shaft is solid one can set d = 0. The nock has the outer (visible) length L4, and the inner length and diameter L5 and d5. The outer nock diameter is the same as shaft diameter.

The fletching can be taken into account with a single fletching mass m6, the distance L6 between fletching mass center and shaft end point. If you consider to omit the fletching - set m6 = 0.

In general, for the system of bodies the resulting mass center coordinate can be calculated as:

$X_c = \frac{1}{M}\cdot \sum_{i = 1}^{N}x_i\cdot m_i$

where M - is a total mass of all bodies, xi and mi - the coordinates of mass centers and mases of all bodies. In our case the bodies are cilinders (bullet point, shaft, nock), having the mass center in the middle between the bases. However, the xi coordinates all should be measured from a common point - consider it to be the tip of the bullet point. Then the mass center coordinates of all bolt parts are displayed below:

Please use the attached document (open it in Dysolve) to solve the problem. You gonna have to put your own values for all dimensions and also set the density for the bullet point, shaft, and nock.