# Example 8.4
dy1 = 2 m
a1 = 1 * 4 m^2
ix11 = (1/12) * (1 * 4^3) m^4
ix1 = ix11 + dy1^2 * a1
#
dy2 = 0.5 m
a2 = 2 * 1 m^2
ix22 = (1/12) * (2 * 1^3) m^4
ix2 = ix22 + dy2^2 * a2
#
ix = ix1 + ix2
k = sqrt(ix / (a1 + a2))
#  xy
dx1 = 0.5 m
dy1 = 2 m
a1 = 1 * 4 m^2
ixy1 = dx1 * dy1 * a1
#
dx2 = 2 m
dy2 = 0.5 m
a2 = 2 * 1 m^2
ixy2 = dx2 * dy2 * a2
ixy = ixy1 + ixy2


# Example 8.5
#
dx1 = 60 mm
a1 = 120 * 80 mm^2
iy11 = (1/12) * 80 * 120^3 mm^4
iy1 = iy11 + dx1^2 * a1
#
dx2 = (120 + 4 * 40 / (3 * pi())) mm
a2 = 0.5 * pi() * 40^2 mm^2
iy22 = (pi() / 8 - 8 / (9 * pi())) * 40^4 mm^4
iy2 = iy22 + dx2^2 * a2
#
dx3 = 120 mm
a3 = pi() * 20^2 mm^2
iy33 = 0.25 * pi() * 20^4  mm^4
iy3 = iy33 + dx3^2 * a3
#
iy = iy1 + iy2 - iy3
a = a1 + a2 - a3
k = sqrt(iy / a); mm


# Example 8.6
# square
ix = (1/12) * 1.442 mm * (1.44.2 mm)^3
# I beam
dy1 = 80 mm
a1 = 200 * 40 mm^2
ix11 = (1/12) * 200 * 40^3 mm^4
ix1 = ix11 + dy1^2 * a1
#
dy2 = 0 mm
a2 = 40 * 120 mm^2
ix22 = (1/12) * 40 * 120^3 mm^4
ix2 = ix22 + dy2^2 * a2
#
dy3 = -80 mm
a3 = 200 * 40 mm^2
ix33 = (1/12) * 200 * 40^3 mm^4
ix3 = ix33 + dy3^2 * a3
#
ix = ix1 + ix2 + ix3; mm^4



# Example 8.7
MKS(deg)
ix = (1/12) * 4 * 3^3 m^4
iy = (1/4) * 3 * 4^3 m^4
ixy = (1/8) * 4^2 * 3^2 m^4
q = (atan(2 * ixy / (iy - ix)) / 2)
ixp = (ix + iy) / 2 + (ix - iy) / 2 * cos(2 * q) - ixy * sin(2 * q)
iyp = (ix + iy) / 2 - (ix - iy) / 2 * cos(2 * q) + ixy * sin(2 * q)



# Example 8.8
FPS(deg)
ix = 22 ft^4
iy = 10 ft^4
ixy = 6 ft^4
q = 30 deg

ixp = (ix + iy) / 2 + (ix - iy) / 2 * cos(2 * q) - ixy * sin(2 * q)
iyp = (ix + iy) / 2 - (ix - iy) / 2 * cos(2 * q) + ixy * sin(2 * q)
ixyp = (ix - iy) / 2 * sin(2 * q) + ixy * cos(2 * q)
qp = (atan(2 * ixy / (iy - ix)) / 2)
ixp = (ix + iy) / 2 + (ix - iy) / 2 * cos(2 * qp) - ixy * sin(2 * qp)
iyp = (ix + iy) / 2 - (ix - iy) / 2 * cos(2 * qp) + ixy * sin(2 * qp)



# Example 8.13
x1 = 0.3 m
m1 = 3 kg
x2 = (0.6 + 0.2) m
m2 = 2 kg
x = (x1 * m1 + x2 * m2) / (m1 + m2)
i1 = (1/12) * m1 * (0.6 m)^2 + m1 * (0.2 m)^2
i2 = 0.2 * m2 * (0.2 m)^2 + m2 * (0.3 m)^2
ii = i1 + i2


# Homework Problems:

TBD