# Example 4.1
Ma = (100 N) * (2 m); N m
Mb = (50 N) * (0.75 m); N m
Mc = (40 lb) * (4 ft + 2 ft * cos(30 deg)); lbf ft
Md = (60 lb) * (1 ft * sin(45 deg)); lb ft
Me = (7 kN) * (4 m - 1 m); kN m

# Example 4.2
MKS (N m)
f = 800 N
Ma = f * 2.5 m
Mb = f * 1.5 m
Mc = f * 0 m
Md = f * 0.5 m

# Example 4.3
MKS (N m)
Mr = (-50 N) * (2 m) + (60 N) * (0 m) + (20 N) * (3 m * sin(30 deg))

# Example 4.4
MKS (N m)
rb = [1, 3, 2] m
rc = [3, 4, 0] m
rf = rb - rc
uf = rf / mag(rf)
f = (60 N) * uf
ma = cross(rb, f)
ma_prime = cross(rc, f)
mag(ma)

# Example 4.5
FPS (lbf ft, deg)
ra = [0, 5, 0] ft
rb = [4, 5, -2] ft
f1 = [-60, 40, 20] lb
f2 = [0, 50, 0] lb
f3 = [80, 40, -30] lb
Mro = cross(ra,f1) + cross(ra, f2) + cross(rb, f3)
mag(Mro)
u = Mro / mag(Mro)
alp = acos(u[1])
bet = acos(u[2])
gam = acos(u[3])

# Example 4.6
MKS(N, N m, deg)
q = 45 deg
d = 100 mm * cos(q)
f = 200 N
Ma = f * d
Ma_prime = f * sin(q) * 0.2 m - f * cos(q) * 0.1 m

# Example 4.7
MKS(N, N m, deg)
q = 30 deg
f = 400 N
Mo = f * sin(q) * 0.2 m - f * cos(q) * 0.4 m
r = [0.4, -0.2] m
ff = f * [sin(q), - cos(q)]
cross(r, ff)

# Example 4.8
MKS(N,N m, deg)
f = [-40, 20, 10] N
ra = [-3, 4, 6] m
ux = [1, 0, 0]
ua = [-3/5, 4/5, 0]
Mx = dot(ux, cross(ra, f))
Ma = dot(ua, cross(ra, f))
Mx_prime = (10 N) * (4 m) - (20 N) * (6 m)
My = (10  N) * (3 m) - (40 N) * (6 m)
Mz = (40 N) * (4 m) - (20 N) * (3 m)

# Example 4.9
MKS(N,N m, deg)
f = [-600, 200, -300] N
rb = [0.4, 0.2, 0] m
ub = rb / mag(rb)
rd = [0, 0.2] m
Mab = dot(ub, cross(rd, f))
Mab * ub

# Example 4.10
MKS(N,N m, deg)
M = 24 N m
d = 0.2 m
f = M / d

# Example 4.11
FPS(lb ft, deg)
M = (120 lb) * (0 ft) - (90 lb) * (2 ft) + (90 lb) * (5 ft) + (120 lb) * (1 ft)

# Example 4.12
IPS(lb in, deg)
ra = [0, 8] in
fa = [0, 0, -25] lb
q = 30 deg
rb = [6 * cos(q), 8, -6 * sin(q)] in
fb = [0, 0, 25] lb
M = cross(ra, fa) + cross(rb, fb)
MM = (25 lb) * (5.2 in)

# Example 4.13
MKS(N,N m, deg)
f = 150 N
d = 0.4 m
M1 = f * d
rdc = [0.3, 0, 0]
uc = [0, 4/5, -3/5] m
fc = 125 N * uc
M2 = cross(rdc, fc)
Mr = [M1, 0, 0] + M2

# Example 4.14
MKS(N,N m, deg)
q = 45 deg
Frx = -100 N - 400 N * cos(q)
Fry = -600 N - 400 N * sin(q)
mag([Frx, Fry])
PolarAngle([Frx, Fry])
Mra = (100 N) * (0 m) - 600 N * 0.4 m - 400 N * sin(q) * 0.8 m \
   - 400 N * cos(q) * 0.3 m

# Example 4.15
MKS(N,N m, deg)
f1 = [0, 0, -800] N
rcb = [-0.15, 0.1, 0] m
ucb = rcb / mag(rcb)
f2 = 300 N * ucb
M = [0, -500 * (4/5), 500 * (3/5)] N m
Fr = f1 + f2
rc = [0, 0, 1] m
rb = rc - rcb
Mr0 = M + cross(rc, f1) + cross(rb, f2)

# Example 4.16
MKS(N,N m, deg)
f = 500 N
q = 60 deg
frx = f * cos(q) + 100 N
fry = -f * sin(q) + 200 N
mag([frx, fry])
PolarAngle([frx, fry])
BEGIN
  d = 1 m
  eq = (233.0 N * d + 350 N * 0 m) - (500N * sin(q) * 4 m \
         + 500 N * cos(q) * (0 m) \
         - 100 N * 0.5 m - 200 N * 2.5 m)
END
solve(eq,d)

# Example 4.17
FPS(lb ft, deg)

fx = -250 lb * (3/5) - 175 lb
fy = -250 lb * (4/5) - 60 lb
f = [fx, fy]
mag(f)
PolarAngle(f)
BEGIN
  y = 1 ft
  eq = (325 lb * y + 260 lb * 0 ft) - \
       (175 lb * 5 ft - 60 lb * 3 ft + 250 lb * (3/5) * 11 ft - 250 lb * (4/5) * 8 ft)
END
solve(eq,y)
BEGIN
   x = 1 ft
   eq = (325 lb * 11 ft - 260 lb * x) - \
      (175 lb * 5 ft - 60 lb * 3 ft + 250 lb * (3/5) * (11 ft) - 250 lb * (4/5) * 8 ft)
END
solve(eq,x)

# Example 4.18
MKS(N,N m, deg)
fr = -600 N + 100 N - 400 N - 500 N
y = (600 N * 0 m + 100 N * 5 m - 400 N * 10 m + 500 N * 0 m) / fr
x = (600 N * 8 m - 100 N * 6 m + 400 N * 0 m + 500 N * 0 m) / fr

# Example 4.19
FPS(lb ft, deg)
fa = [0, 0, -300] lb
fb = [0, 0, -200] lb
fc = [0, 0, -150] lb
fr = fa + fb + fc
ra = [8, 0, 0] ft
rb = [0 , -8, 0] ft
rc = 8 ft * [-sin(45 deg), cos(45 deg)]
BEGIN
   x = 1 ft
   y = 1 ft
   r = [x, y]
   eq = cross(r, fr) - cross(ra, fa) - cross(rb, fb) - cross(rc, fc)
   eq1 = eq[1]
   eq2 = eq[2]
END
solve(eq1,eq2, x, y)

# Example 4.20
MKS(N,N m, deg)
a = 2 m
b = 240 N/m
AA = a * b / 3
xbar = (3/4) * a

# Example 4.21
MKS(kN,kN m, deg)
fr = (1/2) * (9 m) * 1440 N/m
xbar = 9 m - (1/3) * (9 m)
fr = (1/2) * (7200 N / m^2) * (9 m) * 0.2 m

# Example 4.22
FPS()
p = 50 lb/ft
x = 9 ft
f1 = (1/2) * x * p
f2 = x * p
x1bar = (1/3) * x
x2bar = (1/2) *x
fr = 225 lb + 450 lb
xbar = (x1bar * f1 + x2bar * f2) / fr

# Homework Problems:

TBD