Chapter 3: Resultants of Coplanar Force Systems

Example 3.3-1, Page no: 32

In [1]:
import math

#Initilization of variables

F1=150 #lb
F2=200 #lb
F3=80 #lb
F4=180 #lb
theta1=((30*pi)/180) #radians
theta2=((150*pi)/180) #radians
theta3=((240*pi)/180) #radians
theta4=((315*pi)/180) #radians

#Calculations

F1x=F1*cos(theta1) #lb
F1y=F1*sin(theta1) #lb
F2x=F2*cos(theta2) #lb
F2y=F2*sin(theta2) #lb
F3x=F3*cos(theta3) #lb
F3y=F3*sin(theta3) #lb
F4x=F4*cos(theta4) #lb
F4y=F4*sin(theta4) #lb
Fx=F1x+F2x+F3x+F4x #lb
Fy=F1y+F2y+F3y+F4y #lb
R=sqrt(Fx**2+Fy**2) #lb
theta=((arctan(Fy/Fx))*180)/pi #degrees
theta_R=360+theta #degrees

#Result

print'The resultant of the force system is',round(R),"lb"
print'The resultant is at',round(theta_R),"degrees"

The resultant of the force system is 49.0 lb
The resultant is at 334.0 degrees

Example 3.3-2, Page no: 33

In [2]:
import math

#Initilization of variables

F1=50 #N
F2=100 #N
F3=30 #N

#Calculation

#The book has a misprint for squareroot of 1**2
F1x=F1/sqrt(2) #N 
F1y=F1/sqrt(2) #N
F2x=-(F2*3)/sqrt(10) #N
F2y=(-F2)/sqrt(10) #N
F3x=F3/sqrt(5) #N
F3y=(-F3*2)/sqrt(5) #N
Fx=F1x+F2x+F3x #N
Fy=F1y+F2y+F3y #N
R=sqrt(Fx**2+Fy**2) #N
theta=arctan(Fy/Fx) #radians
theta_x=180+(theta*180)/pi #degrees

#Result

print'The resultant is',round(R,1),"N"
print'The resultant makes an angle of',round(theta_x),"degrees"

The resultant is 51.6 N
The resultant makes an angle of 207.0 degrees

Example 3.3-3, Page no: 33

In [3]:
import math

#Initilization of variables

F1=70 #lb
F2=100 #lb
F3=125 #lb
theta1=0 #radians
theta2=((10*pi)/180) #radians
theta3=((30*pi)/180) #radians

#Calculations

Fx=F1-(F2*cos(theta3))-(125*sin(theta2)) #lb
Fy=125*cos(theta2)-(100*sin(theta3)) #lb
R=sqrt(Fx**2+Fy**2) #lb
theta=arctan(Fy/Fx) #radians
theta_x=180+(theta*180)/pi #degrees

#Result

print'The resultant of the force system is',round(R,1),"lb"
print'The resultant with respect to X axis is at',round(theta_x),"degrees"

The resultant of the force system is 82.5 lb
The resultant with respect to X axis is at 118.0 degrees

Example 3.3-4, Page No: 34

In [7]:
import math

#Initilization of variables

F1=-20 #N
F2=30 #N
F3=5 #N
F4=-40 #N
#Distances with respect to point O
x1=6 #m
x2=0 #m
x3=8 #m
x4=13 #m

#Calculations

R=F1+F2+F3+F4 #N
# Applying moment about point O equal to zero
M_O=-(F1*x1)+(F2*x2)+(F3*x3)+(F4*x4) #N-m
#Applying moment about point O equal to R*x
x=M_O/R #m

#Result

print'The resultant of force system is',round(R),"N"
print'The moment about point O is',round(M_O),"N.m"
print'The resultant of moment acts at',round(x,1),"m (to the right of O)"

# The answer for M_O & R is correct but x waries due to some discrepancy in python.

The resultant of force system is -25.0 N
The moment about point O is -360.0 N.m
The resultant of moment acts at 14.0 m (to the right of O)

Example 3.3-5, Page No: 34

In [8]:
import math

#Initilization of variables

F1=-100 #lb
F2=200 #lb
F3=-200 #lb
F4=400 #lb
F5=-300 #lb
#Distance with respect to point O
x1=0 #ft
x2=2 #ft
x3=5 #ft
x4=9 #ft
x5=11 #ft

#Calculation

R=F1+F2+F3+F4+F5 #lb
M_O=(F1*x1)+(F2*x2)+(F3*x3)+(F4*x4)+(F5*x5) #N-m

#Result

print'The resultant of the force system is',round(R),"lb"
print'The moment about point O is',round(M_O),"lb-ft"

The resultant of the force system is 0.0 lb
The moment about point O is -300.0 lb-ft

Example 3.3-6, Page no: 35

In [9]:
import math

#Initilization of variables

F1=20 #lb
F2=20 #lb
F3=-40 #lb
#Distance from point O
x1=3 #ft
x2=3 #ft

#Calculations

R=F1+F2+F3 #lb
M_O=-(F1*x1)+(F2*x2) #lb-ft

#Results

print'The resultant of the force system is',round(R),"lb"
print'The Moment about point O is',round(M_O),"lb-ft"

The resultant of the force system is 0.0 lb
The Moment about point O is 0.0 lb-ft

Example 3.3-7, Page No: 35

In [11]:
import math

#Initilization of variables

F1=500 #N
F2=-400 #N
F3=-200 #N
C=1500 #N-m
#Distance from point O
x1=2 #m
x2=4 #m
x3=6 #m

#Calculations

R=F1+F2+F3 #N
M_O=(F1*x1)+(F2*x2)+(F3*x3)+C #N-m
#Applying Varignons theorem
x=M_O/R #m

#Result

print'The resultant of the force system is',round(R),"N"
print'The moment about point O is',round(M_O),"N.m"
print'The resultant acts at',round(x),"m (from point O)"

The resultant of the force system is -100.0 N
The moment about point O is -300.0 N.m
The resultant acts at 3.0 m (from point O)

Example 3.3-8,Page No: 35

In [12]:
import math

#Initilization of variables

F1=50 #lb
F2=100 #lb
theta1=((45*pi)/180) #radians
#Distance from point O
x1=5 #ft
x2=4 #ft

#Calculation

F_x=F1-(F2*cos(theta1)) #lb
F_y=F1-(F2*sin(theta1)) #lb
R=sqrt(F_x**2+F_y**2) #lb
M_O=F1*x1-(x2*F1) #lb-ft
#Applying Varignons Theorem
x=M_O/R #ft

#Result

print'The resultant of the force system is',round(R,1),"lb"
print'The Moment about point O is',round(M_O),"lb-ft"
print'The Resultant acts at',round(x,2),"ft (from point O)"

The resultant of the force system is 29.3 lb
The Moment about point O is 50.0 lb-ft
The Resultant acts at 1.71 ft (from point O)

Example 3.3-9, Page no: 36

In [1]:
import math

#Initilization of variables

A=80 #N
B=120 #N
C=100 #N
D=50 #N
thetaA=((90*pi)/180) #radians
thetaB=((150*pi)/180) #radians
thetaC=((45*pi)/180) #radians
thetaD=((340*pi)/180) #radians

#Calculations

Ax=A*cos(thetaA) #N
Ay=A*sin(thetaA) #N
Bx=B*cos(thetaB) #N
By=B*sin(thetaB) #N
Cx=C*cos(thetaC) #N
Cy=C*sin(thetaC) #N
Dx=D*cos(thetaD) #N
Dy=D*sin(thetaD) #N
M_Ax=0 #N-m
M_Ay=0 #N-m
M_Bx=-Bx*5 #N-m
M_By=By*8 #N-m
M_Cx=-Cx*1 #N-m
M_Cy=Cy*1 #N-m
M_Dx=-Dx*-1 #N-m
M_Dy=Dy*8 #N-m
Fx=Ax+Bx+Cx+Dx #N
Fy=Ay+By+Cy+Dy #N
R=sqrt(Fx**2+Fy**2) #N
M_O=M_Dx+M_Dy+M_Cx+M_Cy+M_Bx+M_By+M_Ax+M_Ay #N-m
theta=arctan(Fy/Fx) #radians
theta_x=(theta*180)/pi #degrees
#Appliying Varignons theorem
x=M_O/R #m

#Result

print'The resultant of the force system is',round(R),"N"
print'The moment about point O is +',round(M_O),"N.m"
print'The resultant acts at and angle of',round(theta_x),"degrees (with respect to X axis)"
print'The resultant of the force system acts at',round(x,1),"m (from point O)"

The resultant of the force system is 194.0 N
The moment about point O is + 910.0 N.m
The resultant acts at and angle of 86.0 degrees (with respect to X axis)
The resultant of the force system acts at 4.7 m (from point O)

Example 3.3-10, Page No: 37

In [16]:
import math

#Initilization of variables

F1=100 #lb
F2=80 #lb
F3=120 #lb
F4=150 #lb
theta1=((60*pi)/180) #radians
theta2=((45*pi)/180) #radians
theta3=((90*pi)/180) #radians
theta4=((75*pi)/180) #radians
#Distance from point O
x1=-5 #ft
y1=20 #ft
x2=10 #ft
y2=10 #ft
x3=25 #ft
y3=25 #ft
x4=35 #ft
y4=15 #ft

#Calculations

Fx=F1*cos(theta1)+F2*cos(theta2)+F4*cos(theta4) #lb
Fy=-F1*sin(theta1)+F2*sin(theta2)-F3-F4*sin(theta4) #lb
R=sqrt(Fx**2+Fy**2) #lb
theta=arctan(Fy/Fx) #radians
theta_x=(theta*180)/pi #degrees
M_O=-(F1*cos(theta1)*y1)+(-x1)*(F1*sin(theta1))-(x2)*(F2*cos(theta2))+(y2)*(F2*sin(theta2))-(x3*F3)-(y4*F4*cos(theta4))-(x4*F4*sin(theta4)) #lb-ft
#Applying varignons theorem
x=M_O/Fy #ft
y=-M_O/Fx #ft

#Results

print'The resultant of the force system is',round(R),"lb"
print'The resultant acts at',round(theta_x,1),"degrees(with respect to X axis)"
print'The moment about point O is',round(M_O),"lb-ft"
print'The x intercept of resultant is',round(x,1),"ft"
print'The y intercept of resultant is',round(y,1),"ft"
#Answer for angle should be negative which has not been mentioned in the tectbook but a schematic shows the angle in fourth quadrant to clarify the doubt 

The resultant of the force system is 329.0 lb
The resultant acts at -63.8 degrees(with respect to X axis)
The moment about point O is -9220.0 lb-ft
The x intercept of resultant is 31.3 ft
The y intercept of resultant is 63.4 ft

Example 3.3-11, Page No: 38

In [17]:
import math

#Initilization of variables

F1=150 #lb
F2=80 #lb
F3=100 #lb
F4=50 #lb
theta1=((45*pi)/180) #radians
r=3  #units

#Calculations

Fh=F1-F3*cos(theta1) #lb
Fv=F4-F2-F3*sin(theta1) #lb
R=sqrt(Fh**2+Fv**2) #lb
#Applying the Varignons Theorem
a=(F4*r-F1*r+F2*r-F3*r)/R # ft

#Result

print'The resultant of the force system is',round(R),"lb"
print'The resultant acts at',round(a,2),"ft (from point O)"
#Negative sign indicates a negative moment caused by the resultant

The resultant of the force system is 128.0 lb
The resultant acts at -2.81 ft (from point O)

Example 3.3-12, Page no: 39

In [18]:
import math

#Initilization of variables

F1=150 #lb
F2=200 #lb
F3=200 #lb
F4=225 #lb
M=900 #lb-ft
Theta1=(45*pi)/180 #radians
Theta2=(30*pi)/180 #radians
x1=3 #ft
x2=15 #ft
x3=12 #ft
x4=6 #ft

#Calculations

Fx=F1*cos(Theta1)+F2-F4*cos(Theta2) #Applying sum of all forces equal to zero in X direction
Fy=F1*sin(Theta1)-F4*sin(Theta2)+F2 #Applying sum of all forces equal to zero in Y direction
R=sqrt(Fx**2+Fy**2) #lb
theta=(arctan(Fy/Fx))*(180/pi) #degrees
M_o=x1*F2-x2*F1*cos(Theta1)+x3*F1*sin(Theta1)-x4*F2+M+x4*F4*cos(Theta2)-x1*F4*sin(Theta2) #Moment about point O
x=M_o/Fy # in -Varignons Theorem 

#Result

print'The x intercept of resultant position is',round(x,1),"in"
print'The Resultant is',round(R),"lb"
print'The resultant acts at an angle of',round(theta),"degrees"

The x intercept of resultant position is 4.2 in
The Resultant is 223.0 lb
The resultant acts at an angle of 60.0 degrees

Example 3.3-13, Page No: 39

In [2]:
import math
from scipy.integrate import quad
def integrand(x, a, b):
    return 20
a=1
b=1
I=quad(integrand, 0, 6, args=(a,b))

def integrand(x, a, b):
    return 20*x
a=1
b=1
J=quad(integrand, 0, 6, args=(a,b))
d=J[0]/I[0]

# Results
print'The value of R is',round(I[0]),"lb"
print'The value of d is',round(d),"ft"

The value of R is 120.0 lb
The value of d is 3.0 ft

Example 3.3-14, Page no: 39

In [4]:
import math
from scipy.integrate import quad
def integrand(x, a, b):
    return (x/9)*30
a=1
b=1
I=quad(integrand, 0, 9, args=(a,b))

def integrand(y, a, b):
    return y*(y/9)*30
a=1
b=1
J=quad(integrand, 0, 9, args=(a,b))
d=J[0]/I[0]

# Results
print'The value of R is',round(I[0]),"N"
print'The value of d is',round(d),"m"

The value of R is 135.0 N
The value of d is 6.0 m