# Chapter 1:Fundamental Of Engineering Mechanics¶

## Example 1.1,Page No.8¶

In [2]:
import math
from math import sin, cos, tan, radians, pi

#Declaration of Variables

P=10 #N #Force1
Q=8  #N #Force2
alpha=60 #Degrees #Angle Between Two Forces

#Calculations

#Magnitude of Resultant Force
R=(P**2+Q**2+2*P*Q*cos(alpha*pi*180**-1))**0.5 #N

#Result
print"Magnitude of Resultant Force",round(R,2),"N"

Magnitude of Resultant Force 15.62 N


## Example 1.2,Page No.8¶

In [3]:
import math

#Declaration of Variables

alpha=60 #Degrees #Angle between Forces
R=20*(3)**0.5

#Let P & Q be the Two forces
#As Two Forces are equal i.e P=Q

#Magnitude of Resultant Force
#R=(P**2+Q**2+2*P*Q*cos(alpha*pi*180**-1))**0.5 #N
#After Sub values and Furhter simplifying above equations we get
#R=2*P*cos(alpha*2**-1*pi*180**-1)

#Further on Simplifying we get
P=R*((3)**0.5)**-1 #N

#Result
print"Magnitude of Force is",round(P,2),"N"

Magnitude of Force is 20.0 N


## Example 1.3,Page No.9¶

In [4]:
import math
import numpy as np

#Declaration of Variables

#Case-1
R1=14 #N #Resultant1
alpha1=60 #Degrees #Angle between two forces

#Case-2
R2=(136)**0.5
alpha2=90 #Degrees #Angle between two Forces

#Let P And Q be the two forces
#R=(P**2+Q**2+2*P*Q**cos(alpha))

#Now For case-1,we get Resultant as
#P**2+Q**2+P*Q=196       ............................(1)

#For case-2,we get Resultant as
#P**2+Q**2=136       ...................................(2)

#Subtracting Equation 2 from equation 1 we get
#P*Q=60      .........................................(3)

#Multiplying abovw equation by 2 we get
#2*P*Q=120       .......................................(4)

#Adding equation 4 to equation 2 we get
#P**2+Q**2+2*P*Q=256
#After Further simplifying we get
#P=16-Q           ..........................(5)

#Sub value of P in equation 3 we get
#Q**2-16*Q+60=0
a=1
b=-16
c=60

X=b**2-4*a*c

Q1=(-b+X**0.5)*(2*a)**-1
Q2=(-b-X**0.5)*(2*a)**-1

#Now sub value of Q in equation 5 we get
P1=16-Q1
P2=16-Q2

#Result
print"Magnitude of two Forces is:P2",round(P2,2),"N"
print"                          :Q1",round(Q2,2),"N"

Magnitude of two Forces is:P2 10.0 N
:Q1 6.0 N


## Example 1.4,Page No.10¶

In [2]:
import math
from math import sin, cos, tan, radians, pi
import numpy as np

P=50 #N #Force acting at pt O
Q=100 #N #force acting at pt O
alpha=30 #DEgree #Angle Between Two Forces

#Calculations

#MAgnitude of Resultant
R=(P**2+Q**2+2*P*Q*cos(alpha*pi*180**-1))**0.5 #N

#Angle Made by resultant with the direction of P
X=(Q*sin(alpha*180**-1*pi)*(P+(Q*cos(alpha*pi*180**-1)))**-1)
theta=np.arctan(X)*(180*pi**-1) #Degrees

#Angle made by resultant with x-axis is
Y=theta+alpha*2**-1 #Degrees

#Result
print"Resultant in the Magnitude is",round(R,2),"N"
print"Resultant in the Direction is",round(Y,2),"Degrees"

Resultant in the Magnitude is 145.47 N
Resultant in the Direction is 35.1 Degrees


## Example 1.5,Page No.10¶

In [6]:
import math

#Declaration of Variables

R=1500 #N #REsultant of two Forces
alpha=90 #Degrees #Angle between two Forces
theta=36 #Degrees #Angle Made by Resultant  with one Force

#Calculations

#Now From Equation of Direction of Resultant,we get
#tan(theta)=(Q*sin(alpha))*(P+Q*sin(alpha))**-1
#After Further sub values and simplifying we get
#Q=0.726*P    .......................................(1)

#Now From Equation of Resultant
#R=(P**2+Q**2+2*P*Q*cos(alpha))
#After sub values and further simplifying we get
#R=1.527*P**2
#Therefore,we get value of  P After simplifying above equation
P=(R**2*(1.527)**-1)**0.5

#Sub value of P in equation 1 we get
Q=0.726*P

#Result
print"Magnitude Of Forces:P",round(P,2),"N"
print"                   :Q",round(Q,2),"N"

Magnitude Of Forces:P 1213.87 N
:Q 881.27 N


## Example 1.6,Page No.11¶

In [7]:
import math
from math import sin, cos, tan, radians, pi
import numpy as np

#Declaration of Variables
#P+Q=120
R=180 #N #Resultant of two Forces
theta=90 #Degrees #Angle between force and Resultant

#Calculations

#Now From Equation of Direction of Resultant,we get
#Tan(theta)=Q*sin(alpha)*(P+Q*sin(alpha))**-1
#After Further ssub values a in above equation and further simplifying we get
#P=-Q*cos(alpha)   ..........(1)

#Now From equation of Resultant we get
#R=(P**2++Q**2+2*P*Q*cos(alpha))**0.5
#After sub values and further simplifying
#Q-P=120  ......................................(1)
#P+Q=270   ......................................(2)

#After Adding above equations i.e equations 1 and 2 we get
Q=390*2**-1 #N

P=270-Q #N

#Value of angle alpha
alpha=np.arccos(-P*Q**-1)*(180*pi**-1) #Degrees

#Result
print"Magnitude of Each Force:P",round(P,2),"N"
print"                       :Q",round(Q,2),"N"
print"Angle between Two Forces",round(alpha,2),"Degrees"

Magnitude of Each Force:P 75.0 N
:Q 195.0 N
Angle between Two Forces 112.62 Degrees


## Example 1.7,Page No.13¶

In [8]:
import math
from math import sin, cos, tan, radians, pi

#Declaration of Variables

W=1000 #N #Weight
#Angles
CAB=30 #Degrees
CBA=CBD=60 #Degrees
ACB=90 #Degrees

#Calculations
#Angle

#In Right-Angle Triangle,Angle BDC
#Angle
BCD=90-CBD #Degrees
ACE=180-ACB-90-60 #DEgrees
BCE=180-ACE-ACB #DEGrees

#Applying LAmi's Theorem at Point C
#T1*(sin150)**-1=T2*(sin(120)**-1=1000*sin(90)**-1

#After Further simp;ifying we get
T1=W*sin(150*pi*180**-1) #N
T2=W*sin(120*pi*180**-1) #N

#Result
print"Tension in Chain is:T1",round(T1,2),"N"
print"                   :T2",round(T2,2),"N"

Tension in Chain is:T1 500.0 N
:T2 866.03 N


## Example 1.8,Page No.18¶

In [9]:
import math
from math import sin, cos, tan, radians, pi
import numpy as np

#Declaration of Variables
W=900 #N #Weight at C
#Length
AC=4 #m
BC=3 #m
AB=5 #m

#Calculations

#In Triangle ABC
X=AC**2+BC**2
Y=AB**2

#Therefore,
#X=Y

#Therefore,
#Triangle ABC is Right Angle Triangle,In Which Angle ACB=90 Degrees
alpha=np.arcsin(BC*AB**-1)*(180*pi**-1)
Beta=90-alpha

theta1=90-alpha

#In Right Angle Triangle BDC,
theta2=90-Beta

#Now,Angles
ACE=180-theta1
BCE=180-theta2

#Now applying ami's Theorem
#T1*(sin(BCE))**-1=T2*(sin(ACE))**-1=W*(sin(90))**-1

#Tensions in chains
T1=W*sin(BCE*180**-1*pi) #N
T2=W*sin(ACE*180**-1*pi) #N

#Result
print"Tension in Chains are:T1",round(T1,2),"N"
print"                     :T2",round(T2,2),"N"

#Answer in hte book For T2 is incorrect

Tension in Chains are:T1 540.0 N
:T2 720.0 N


## Example 1.9,Page No.18¶

In [10]:
import math
from math import sin, cos, tan, radians, pi

#Declaration of Variables

W=15 #N #Weight at Pt C
OAC=FAC=60 #Degrees
CBD=BCF=45 #Degrees
FCA=90-FAC #Degrees

#Calculations

#Using Lami's theorem,
#W*(sin(BCA))**-1=T1*(sin(ACE))**-1=T2*(sin(ACE))**-1

#Angles
BCA=BCF+FCA #Degrees
ACE=180-FCA #Degrees
BCE=180-BCF #Degrees

#Force's in the string AC
T1=W*sin(ACE*180**-1*pi)*(sin(BCA*180**-1*pi))**-1 #N
T2=W*sin(BCE*180**-1*pi)*(sin(BCA*180**-1*pi))**-1 #N

#Result
print"Force's in the string:T1",round(T1,2),"N"
print"                     :T2",round(T2,2),"N"

Force's in the string:T1 7.76 N
:T2 10.98 N


## Example 1.10,Page No.16¶

In [11]:
import math
from math import sin, cos, tan, radians, pi
import numpy as np

#Declaration of Variables

#Forces
P=50 #N
Q=100 #N
alpha=30 #Angle Between Two Forces
theta=15 #Degrees #Angle Made By Force P with x-axis
theta2=alpha+theta #Degrees

#Calculations

#Sum Of COmponents of forces along X-Axis is
H=P*cos(theta*pi*180**-1)+Q*cos(theta2*pi*180**-1) #N

#Sum Of COmponents of forces along Y-Axis is
V=P*sin(theta*pi*180**-1)+Q*sin(theta2*pi*180**-1) #N

#MAgnitude Of Resultant Force is
R=(H**2+V**2)**0.5 #N

#Let Direction Of Resultant Force be beta
#Direction Of Resultant Force is
beta=np.arctan(V*H**-1)*(180*pi**-1) #Degrees

#Result
print"Magnitude of Resultant Force is",round(R,2),"N"
print"Direction of Resultant Force is",round(beta,2),"Degrees"

Magnitude of Resultant Force is 145.47 N
Direction of Resultant Force is 35.1 Degrees


## Example 1.11,Page No.17¶

In [3]:
import math
from math import sin, cos, tan, radians, pi
import numpy as np

#Declaration of Variables

#Let 3Forces Be
R1=40 #KN
R2=15 #KN
R3=20 #KN

#Angles Made by respective forces with X-Axis
theta1=60  #Degrees
theta2=120 #Degrees
theta3=240 #Degrees

#Calculations

#Now sum of components of all forces along X-Axis
H=R1*cos(theta1*pi*180**-1)+R2*cos(theta2*pi*180**-1)+R3*cos(theta3*pi*180**-1)

#Now sum of components of all forces along Y-Axis
V=R1*sin(theta1*pi*180**-1)+R2*sin(theta2*pi*180**-1)+R3*sin(theta3*pi*180**-1)

#MAgnitude of Resultant Force is
R=(H**2+V**2)**0.5 #N

#Direction of Resultant Force is
theta=np.arctan(V*H**-1)*(pi**-1*180)

#Result
print"Magnitude of Resultant Force is",round(R,2),"KN"
print"Direction of Resultant Force is",round(theta,2),"Degrees"

#Declaration of Variables

#Let 3Forces Be
R1=40 #KN
R2=15 #KN
R3=20 #KN

#Angles Made by respective forces with X-Axis
theta1=60  #Degrees
theta2=120 #Degrees
theta3=240 #Degrees

#Calculations

#Now sum of components of all forces along X-Axis
H=R1*cos(theta1*pi*180**-1)+R2*cos(theta2*pi*180**-1)+R3*cos(theta3*pi*180**-1)

#Now sum of components of all forces along Y-Axis
V=R1*sin(theta1*pi*180**-1)+R2*sin(theta2*pi*180**-1)+R3*sin(theta3*pi*180**-1)

#MAgnitude of Resultant Force is
R=(H**2+V**2)**0.5 #N

#Direction of Resultant Force is
theta=np.arctan(V*H**-1)*(pi**-1*180)

#Result
print"Magnitude of Resultant Force is",round(R,2),"KN"
print"Direction of Resultant Force is",round(theta,2),"Degrees"

Magnitude of Resultant Force is 30.41 KN
Direction of Resultant Force is 85.28 Degrees
Magnitude of Resultant Force is 30.41 KN
Direction of Resultant Force is 85.28 Degrees


## Example 1.12,Page No.18¶

In [13]:
import math
from math import sin, cos, tan, radians, pi
import numpy as np

#Declaration of Variables

#Let 4 Forces Be
R1=10 #KN
R2=15 #KN
R3=20 #KN
R4=40 #KN

#Angles Made by respective forces with X-Axis
theta1=30  #Degrees
theta2=60  #Degrees
theta3=90  #Degree
theta4=120 #Degrees

#Calculations

#Now sum of components of all forces along X-Axis
H=R1*cos(theta1*pi*180**-1)+R2*cos(theta2*pi*180**-1)+R3*cos(theta3*pi*180**-1)+R4*cos(theta4*pi*180**-1)

#Now sum of components of all forces along Y-Axis
V=R1*sin(theta1*pi*180**-1)+R2*sin(theta2*pi*180**-1)+R3*sin(theta3*pi*180**-1)+R4*sin(theta4*pi*180**-1)

#MAgnitude of Resultant Force is
R=(H**2+V**2)**0.5 #N

#Direction of Resultant Force is
theta4=np.arctan(V*H**-1)*(pi**-1*180)
theta=180+theta4 #Degrees

#Result
print"Magnitude of Resultant Force is",round(R,2),"KN"
print"Direction of Resultant Force is",round(theta,2),"Degrees"

Magnitude of Resultant Force is 72.73 KN
Direction of Resultant Force is 93.03 Degrees


## Example 1.13,Page No.19¶

In [14]:
import math

#Declaration of Variables

L=10 #m #Length of beam

#Distances
L_AC=4 #m
L_CB=6 #m

#Calculations

#Let R_A & R_B be the forces acting at A & B
#Taking Moment at A
R_B=(W*L_CB)*L**-1 #N
R_A=W-R_B #N

#Result
print"Beam Reactions are:R_A",round(R_A,2),"N"
print"                  :R_B",round(R_B,2),"N"

Beam Reactions are:R_A 80.0 N
:R_B 120.0 N


## Example 1.14,Page No.20¶

In [15]:
import math
from math import sin, cos, tan, radians, pi
import numpy as np

#Declaration of Variables

#Let 4 Forces be
F1=10 #N
F2=20 #N
F3=30 #N
F4=40 #N

#Calculations

#Net Forces in Horizontal direction is
H=F1-F3 #N

#Net Forces in Vertical direction is
V=F2-F4 #N

#Resultant Force is given by
R=(H**2+V**2)**0.5 #N

#Direction of resultant Forces
theta=np.arctan(V*H**-1)*(pi**-1*180) #Degrees

#Since H & V are negative theta lies between 180 & 270
theta2=180+theta #Degrees

#Result
print"Magnitude of Force is",round(R,2),"N"
print"Direction of Force is",round(theta2,2),"Degrees"

Magnitude of Force is 28.28 N
Direction of Force is 225.0 Degrees