In [1]:

```
import math
#Declaration Of Variables
F1=100 #N #Force acting on body
#Calculations
#As the Force F1 & F2 are acting on the same body and at same point but in opposite directions
#These two forces will be equal
F2=F1
#Result
print"Magnitude of Force F2 is",round(F2,2),"N"
```

In [2]:

```
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
#Force
F3=400 #N
theta1=30 #Degree #Angle made by forces F2 & F3
#Calculations
#By Lami's Theorem
#F1*sin(120)**-1=F2*sin(120)**-1=F3*sin(120)**-1
F2=F3*sin(120*pi*180**-1)**-1*sin(120*pi*180**-1)
F1=F2*sin(120*pi*180**-1)**-1*sin(120*pi*180**-1)
#Result
print"Magnitude of Forces:F1",round(F1,2),"N"
print" :F2",round(F2,2),"N"
```

In [3]:

```
import math
#Declaration Of Variables
#FOrces
F1=250 #N
F3=1000 #N
L_AB=1 #m #Length of AB
#Calculations
#Sum of forces in y direction
F2=F1+F3 #N
#Moment at pt A
#-F2*L_AB+F3*(L_AB+x)=0
#After further simplifying we get
x=F2*L_AB*F3**-1-L_AB
#Result
print"Magnitude of Force F2 is",round(F2,2),"N"
print"Distance of F2 From F3 is",round(x,2),"N"
```

In [4]:

```
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
#Forces
F1=18 #N
F2=22.5 #N
F3=15 #N
F4=30 #N
#Angles
theta2=45 #Degree
theta3=90 #Degree
theta4=30 #Degree
#Calculations
#Sum of Forces in x-direction
#F1+F2*cos(45)-F4*cos(30)-F5*cos(theta5)=0
#After further simplifying we get
#F5*cos(theta5)=F1+F2*cos(45)-F4*cos(30)....................1
#Sum of Forces in y-direction
#F3+F2*sin(45)-F4*sin(30)-F5*sin(theta5)=0
#After further simplifying we get
#F5*sin(theta5)=F3+F2*sin(45)-F4*sin(30)....................2
#Dividing equation 2 and 1 we get
X=F3+F2*sin(45*pi*180**-1)-F4*sin(30*pi*180**-1)
Y=F1+F2*cos(45*pi*180**-1)-F4*cos(30*pi*180**-1)
theta=np.arctan((X)*(Y)**-1)*(pi**-1*180)
F5=(F1+F2*cos(45*pi*180**-1)-F4*cos(30*pi*180**-1))*(cos(theta*pi*180**-1))**-1
#Result
print"Magnitude of force F5 is",round(F5,2),"N"
print"Direction of F5 is",round(theta,2),"Degrees"
```

In [5]:

```
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
#Forces
F_C=1500 #N
theta_C=60 #degrees
F_B=1805 #N
theta_B=33.67 #Degrees
F_A=2240 #N
theta_A=63.43 #Degrees
#Distances of forces from D
L_DC=2 #m
L_DB=3 #m
L_DE=4 #m
L_DO=3 #m
#Calculations
#NEt forces along Y-axis
R_y=-F_C*cos(theta_C*pi*180**-1)-F_B*cos(theta_B*pi*180**-1)+F_A*cos(theta_A*pi*180**-1)
#Net Forces along x-axis
R_x=F_C*sin(theta_C*pi*180**-1)-F_B*sin(theta_B*pi*180**-1)-F_A*sin(theta_A*pi*180**-1)
#Resultant Forces
R=(R_x**2+R_y**2)**0.5 #N
#Angle made by resultant
theta=np.arctan(R_y*R_x**-1)*(pi**-1*180) #Degrees
#Net Moment about point O
M_O=-F_C*cos(theta_C*pi*180**-1)*L_DO-F_B*cos(theta_B*pi*180**-1)*L_DO-F_C*sin(theta_C*pi*180**-1)*L_DC+F_B*sin(theta_B*pi*180**-1)*L_DB+F_A*sin(theta_A*pi*180**-1)*L_DE
#Moment of R about O
#X-intercept
x=M_O*-R_x**-1
#Y-intercept
y=M_O*-R_y**-1
#Result
print"Resultant is",round(R,2),"N"
print"X intercept is",round(x,2),"m"
print"Y intercept is",round(y,2),"m"
```

In [6]:

```
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
theta=60 #Degrees #Angle made by chain with ceiling
W=5 #N #Weight of lamp
theta2=120 #Degree #Angle made by chain with cord
theta3=150 #Degree #Angle made by chain with wire holding lamp
theta4=90 #Degree #Angle made by cord with wire holding lamp
#Calculations
#LEt T1=tension in cord
#T2=tension in chain
#By lami's theorem
#T1*sin(theta3)**-1=T2*sin(theta4)**-1=W*sin(theta2)**-1
T1=W*sin(theta2*pi*180**-1)**-1*sin(theta3*pi*180**-1) #N
T2=W*sin(theta2*pi*180**-1)**-1*sin(theta4*pi*180**-1) #N
#Result
print"Tension in chain is",round(T2,2),"N"
print"Tension in cord is",round(T1,2),"N"
```

In [7]:

```
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
#Forces
F_P=1000 #N
F_Q=1500 #N
F_R=1000 #N
F_S=500 #N
theta=90 #Degree #Angle made by F_P with PS
theta2=60 #Degree #Angle made by F_Q with QS
theta3=45 #Degree #Angle made by F_R with RS
theta4=30 #Degree #Angle made by F_S with PS
#Calculations
#Resultant of forces along x-axis
R_x=-F_Q*cos(theta2*pi*180**-1)-F_R*cos(theta3*pi*180**-1)-F_S*cos(theta4*pi*180**-1)
#Resultant of forces along y-axis
R_y=-F_P*sin(theta*pi*180**-1)-F_Q*sin(theta2*pi*180**-1)-F_R*sin(theta3*pi*180**-1)-F_S*sin(theta4*pi*180**-1)
#Resultant
R=(R_x**2+R_y**2)**0.5 #N
#Direction of resultant
theta=np.arctan(R_y*R_x**-1)*(180*pi**-1) #Degree
#Result
print"Magnitude of Resultant is",round(R,2),"N"
print"Direction of Resultant is",round(theta,2),"degree"
```

In [8]:

```
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
W=100 #N #Weight of roller
L_BC=10 #cm #Radius of roller
L_AB=20 #cm #Length of tie rod
#Calculations
theta=np.arcsin(L_BC*L_AB**-1)*(pi**-1*180) #Degrees
F=W*cos(theta*pi*180**-1)**-1 #N #Force in tie rod
R_C=F*sin(theta*pi*180**-1)
#Result
print"Force in tie rod is",round(F,2),"N"
print"Reaction at C is",round(R_C,2),"N"
```

In [9]:

```
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
W=120 #N #Weight of ball
theta=30 #Degrees #angle made by groove
theta2=60 #Degrees #Angle made by groove
#Calculations
#LEt R_A and R_B be the reactions at A and B respectively
#by LAmi's theorem
#R_B*sin(120)**-1=R_A*sin(150)**-1=W*sin(90)**-1
R_C=W*sin(90*pi*180**-1)**-1*sin(120*pi*180**-1) #N
R_A=W*sin(90*pi*180**-1)**-1*sin(150*pi*180**-1) #N
#Result
print"Reaction at A is",round(R_A,2),"N"
print"Reaction at c is",round(R_C,2),"N"
```

In [10]:

```
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
W=100 #N #weight of roller
#Calculations
#LEt R_A and R_B be the reactions at A and B respectively
theta=45 #Degrees #Angle made by R_B with horizontal
theta2=30 #Degrees #Angle made by R_A with horizontal
#by LAmi's theorem
#R_B*sin(120)**-1=R_A*sin(135)**-1=W*sin(105)**-1
R_B=W*sin(105*pi*180**-1)**-1*sin(120*pi*180**-1) #N
R_A=W*sin(105*pi*180**-1)**-1*sin(135*pi*180**-1) #N
#Result
print"Reaction at A is",round(R_A,2),"N"
print"Reaction at B is",round(R_B,2),"N"
```

In [11]:

```
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
W=100 #N #Weight of roller
F=200 #N #Horizontal Force
L_AB=10 #cm #LEngth of bar AB
L_BC=5 #cm #radius of roller
#Calculations
theta=np.arcsin(L_BC*L_AB**-1)*(pi**-1*180)
#LEt R_C be the reaction at c
#sum of Forces along x-axis
F_AB=F*cos(theta*pi*180**-1)**-1 #N
#sum of forces along y-axis
R_C=W+F_AB*sin(theta*pi*180**-1)
#Result
print"Force in bar AB is",round(F_AB,2),"N"
print"Reaction at C is",round(R_C,2),"N"
```

In [12]:

```
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
W=1000 #N #Weight of rollers
theta=30 #Degree #Angle made by groove
#Calculations
#LEt R_A,R_B,R_C,R_D be the reactions at A,B,C,D respectively
#Roller-2
R_D=W*sin(90*pi*180**-1)*sin(150*pi*180**-1) #N
R_A=W*sin(90*pi*180**-1)*sin(120*pi*180**-1) #N
#ROller-1
R_B=(W+R_D*sin(theta*pi*180**-1))*sin(60*pi*180**-1)**-1 #N
R_C=R_B*cos(60*pi*180**-1)+R_D*cos(theta*pi*180**-1)
#Result
print"Reactions at A:R_A",round(R_A,2),"N"
print" B:R_B",round(R_B,2),"N"
print" C:R_C",round(R_C,2),"N"
```

In [13]:

```
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
W=1000 #N #Weight of each sphere
L_AF=L_BF=L_FD=L_DE=L_CE=25 #cm
L=90 #Width of channel
L_FG=40 #cm
L_EF=L_FD+L_DE #cm
#Calculations
theta=np.arcsin(L_FG*L_EF**-1)*(180*pi**-1) #Degrees
#LEt R_A,R_B,R_C,R_D be the reactions at A,B,C,D respectively
#Roller-2
R_D=W*(cos(theta*pi*180**-1))**-1 #N
R_C=R_D*sin(theta*pi*180**-1) #N
#Roller-1
R_A=R_D*sin(theta*pi*180**-1) #N
R_B=R_D*cos(theta*pi*180**-1)+W #N
#Result
print"Reactions at A:R_A",round(R_A,2),"N"
print" B:R_B",round(R_B,2),"N"
print" C:R_C",round(R_C,2),"N"
```

In [14]:

```
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
W=1000 #N #Weight of 2 circular cyclinders
L_AF=L_FC=L_CG=L_GB=15 #cm
W2=2000 #N #Weight of 3rd cyclinder
r2=15 #cm
L_AB=40 #cm
L_AH=L_HB=20 #cm
#Calculations
theta=np.arcsin(L_AH*(L_AF+L_FC)**-1)*(pi**-1*180) #Degrees
#Let R_G,R_F,R_D,R_E be the reactions at G,F,D,E
R_F=W2*(2*cos(theta*pi*180**-1))**-1 #N
R_G=R_F #N
#Roller-1
R_D=W+R_F*cos(theta*pi*180**-1) #N
S=R_F*sin(theta*pi*180**-1) #N
#Roller-2
R_E=W+R_G*cos(41.81*pi*180**-1) #N
#Result
print"Reaction at E is ",round(R_E,2),"N"
print"Reactions at D is",round(R_D,2),"N"
print"Force S in the string is ",round(S,2),"N"
```

In [15]:

```
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
L_OB=L_OC=L_OA=40 #cm #Radius of roller
h=20 #cm #Height of block
L_OD=L_OA-h #cm #Length
W=3000 #N #Weight of roller
#Calculations
#LEt R_B be the reaction at B
L_BD=((L_OB**2-L_OD**2)**0.5) #cm
theta=np.arctan(L_BD*(L_OC+L_OD)**-1)*(180*pi**-1) #degree
#Sum of all vertical Forces
R_B=W*(cos(theta*pi*180**-1))**-1 #N
#Sum of all horizontal Forces
P=R_B*sin(theta*pi*180**-1) #N
#Result
print"Reaction at B is",round(R_B,2),"N"
print"Horizontal Reaction at C is",round(P,2),"N"
```

In [16]:

```
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
#Declaration Of Variables
L_OB=L_OC=L_OA=40 #cm #Radius of roller
h=20 #cm #Height of block
L_OD=L_OA-h #cm #Length
W=3000 #N #Weight of roller
#Calculations
theta=np.arccos(L_OD*L_OB**-1)*(pi**-1*180) #N
R_B=W*(cos(theta*pi*180**-1))**-1 #N
P=R_B*sin(theta*pi*180**-1) #N
#LEt P_min be the least Force applied
#let alpha be the angle made by least force
#P_min=W*L_BD*L_BC**-1
L_BD=((L_OB**2-L_OD**2)**0.5) #cm
#But L_BC=L_BO*sin(alpha)
#Force P will be min when sin(Alpha) is max.
#thus sin(alpha)=90 or sin(alpha)=0. therefore sub value in above equation,we get min Force
#LEt P_min be the Least Force to be applied
P_min=W*L_BD*(L_OB*1)**-1 #N
#Direction of least force is right angle to L_BO
#Result
print"Minimum least force is",round(P_min,2),"N"
print"Magnitude of force applied horizontally at centre of roller",round(P,2),"N"
```

In [17]:

```
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
L_BC=25 #cm
L_AB=40 #cm
#Calculations
#Let alpha be the angle made by force F so that body will be in equilibrium
#Theta is angle made by R_A with horizontal,so Force F has to make same angle with horizontal
alpha=np.arctan(L_AB*L_BC**-1)*(pi**-1*180) #degrees
theta=alpha
#Result
print"Angle theta is",round(theta,2),"Degrees"
```

In [18]:

```
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
L_AB=1.6 #m
L_BD=1.2 #m
L_BC=0.8 #m
L_CD=0.4 #m
F_C=200 #N #Force t C
theta=60 #Degrees
#Calculations
#Sum of all forces in x-direction
R_Bx=F_C #N
L_BD2=L_BD*sin(theta*pi*180**-1) #m
L_DD=L_BD*cos(theta*pi*180**-1) #m
L_BC2=sin(theta*pi*180**-1)*L_BC #m
R_D=F_C*L_BC2*L_DD**-1 #N
R_By=R_D
#Resultant reaction at B
R_B=(R_Bx**2+R_By**2)**0.5 #N
#Sum of moments at A
M_A=R_By*L_AB
#Result
print"Couple to be apllied to hold the system is",round(M_A,2),"N"
print"Magnitude of pin reaction at B",round(R_B,2),"N"
```

In [15]:

```
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
W=2000 #N #Weight of chain
L_AB=2 #m
F_B=320 #N
#Calculations
#By LAmi's theorem
#F_A*sin(90)=W*sin(180-theta)**-1=F_B*sin(90+theta)**-1
#But sin(180-theta)=sin(theta),sin(90+theta)=cos(theta)
#Tan(theta)=W*F_B**-1
theta=np.arctan(W*F_B**-1)*(180*pi**-1) #Degrees
F_A=W*(sin(theta*pi*180**-1))**-1 #N
#Let x be the lateral distance
x=cos(theta*pi*180**-1)*2
#Result
print"Force in the chain is",round(F_A,2),"N"
print"Horizontal displacement is",round(x,2),"m"
```