Please see attachments about a question of gun barrel dynamics.
Please use i, j and k vector coordinates to solve
About vertical axis (y):
w1 = wy = -d(beta)/dt .j
=> w1 = -(30*pi/180).j rad/sec
=> w1 = -0.52358j rad/s == -0.524j rad/s
(Here, -ve sign is taken because, beta increases clock wise)
About an axis perpendicular to gamma angle plane:
w2 = (d(yamma)/dt)[k.cos(beta) +(-i).sin(beta)]
w2= (10*pi/180)*[k.cos(beta) - i.sin(beta)]
=> w2 = ...
The solution provides step-by-step working to find all parts of the solution on gun barrels.