The MOI of a round bar is:
I1 = (1/12)(m)(3r^2+L^2)
I1 = MOI of the calibration bar
m = mass of bar (slugs) = weight (lbs) / 32.1740
r = 1 /2 of the diameter of bar (feet)
L = length of bar (feet)
The spring's torsional stiffness, k, and pendulum MOI, Ip, are determined using the calibration bar MOI and the average periods of the calibration bar on the pendulum and of the empty pendulum:
k = I1
( t1 / 2Pi )^2 - ( t2 / 2Pi )^2
Ip = k( t1 / 2Pi )^2 - I1
k = torsional stiffness
I1 = MOI of the calibration bar
Pi = 3.1416
t1 = average period of the bar and pendulum (seconds)
t2 = average period of empty pendulum (seconds)
The gun's MOI, Ig, is found by measuring the MOI of the gun on the pendulum, I, and the subtracting the pendulum's MOI, Ip:
Ig = I - Ip = k( t / 2Pi )^2 - Ip
Ig = MOI of gun
I = MOI of the gun and pendulum
Ip = MOI of pendulum
k = torsional stiffness
t = average period of the gun and pendulum (seconds)
Pi = 3.1416
Note: The symbology, X^N, means "X raised to the Nth power." For example:
4^2 = 4 x 4 = 16