from numpy import *
dtr = pi/180
g = 9.81     # m/s2
gfps = 32.2  # ft/s2

TA,TB = 40e3,20e3  # N
l,kG = 8,15        # m
m = 200e3          # kg
t = 5              # s
w = l/(m*kG**2)*(TA-TB)*t
print('angular velocity is {:.3f} rad/s'.format(w))

angular velocity is 0.018 rad/s

A = 5    # N.m
b = 0.5  # 1/s
m = 4    # kg
t1 = 2   # s
r,rA = .25,0.06  # m
J = A/b*(exp(b*t1)-1)
wa = J/(m*r**2)
print('(a) angular velocity is {:.3f} rad/s'.format(wa))
wb = J/(m*(0.5*rA**2+r**2))
print('(b) angular velocity is {:.3f} rad/s'.format(wb))
print('(c) angular velocity is {:.3f} rad/s'.format(wa))

(a) angular velocity is 68.731 rad/s
(b) angular velocity is 66.807 rad/s
(c) angular velocity is 68.731 rad/s

T,W = 50,100    # lb
r,k = 0.5,0.75  # ft
mu = 0.1  
t1 = 3          # s
m = W/gfps 
w = (2*T-mu*W)*r*t1 / (m*(k**2+r**2))
vG = w*r
print('velocity of G is {:.3f} ft/s'.format(vG))

velocity of G is 26.751 ft/s

mu = 0.1
m = 100        # kg
r,R = 0.2,0.4  # m
Q = 30*dtr     # rad
t1 = 4         # s
k = 0.25       # m
w1 = g*t1*( (R+r)*mu*cos(Q) - r*sin(Q) )/( k**2+r**2)
print('angular velocity is {:.3f} rad/s'.format(w1))

angular velocity is -18.391 rad/s

vi = 3   # m/s
w0 = 8   # rad/s
mu = 0.6
r = 0.5  # m
Q = 30*dtr # rad
t = (-w0*r-vi)/(g*(sin(Q)-2*mu*cos(Q)))
print('hoops stops slipping in {:.3f} s'.format(t))

hoops stops slipping in 1.323 s

m = 9  # kg
l = 1  # m
A = 15 # N.m/s2
t1 = 3 # s
w = 2*A*t1**3 / ( 10*m*l**2)
print('angular velocity is {:.3f} rad/s'.format(w) )

angular velocity is 9.000 rad/s

m,mb = 5,10e-3  # kg
v = 800         # m/s
l,r = 2,0.4     # m
L = l+r
M = m+mb
IA = 0.5*m*r**2 + M*L**2
w = mb*L*v / IA
print('angular velocity is {:.3f} rad/s'.format(w))
Q = arccos(1 - 0.5*IA*w**2/(M*g*L))
print('max angle is {:.3f} deg'.format(Q/dtr))

angular velocity is 0.656 rad/s
max angle is 18.810 deg

md,mr = 4,2      # kg
l,r = 0.75,0.15  # m
wi = 5           # rad/s
Id = 0.5*md*r**2
Hi = 2*Id*wi
Ic = 2*(Id+md*l**2) + 1/12*mr*(2*l)**2
wf = Hi/Ic
print('final angular velocity is {:.4f} rad/s'.format(wf))

final angular velocity is 0.0906 rad/s

l = 1.5   # ft
w = sqrt( gfps/(6*l) )
wf = w/2
hc = 8*wf**2*l**2/gfps
print('C will rise {:.3f} ft above the sawhorses'.format(hc))

C will rise 0.500 ft above the sawhorses

mr,md = 2,5          # kg
l,r,h = 0.5,0.2,0.3  # m
M = mr+md
vG = sqrt(2*g*h)
rG = ( mr*l/2 + md*(l+r) )/M
Hi = -rG*M*vG
Is = 1/3*mr*l**2 + 1/2*md*r**2 + md*(l+r)**2
w = -Hi/Is
wf = sqrt( 2*M*g*rG/Is + w**2 )
print('angular velocity is {:.3f} rad/s'.format(wf))

angular velocity is 6.454 rad/s

W = 10  # lb
l = 1   # ft
v = sqrt( gfps*l*(1/sqrt(2)-1/2)*16/3 )
print('minimum speed is {:.3f} ft/s'.format(v))

minimum speed is 5.964 ft/s

Ch 19 problems¶