Chapter 18:Work,Power And Energy

Example 18.1,Page No.673

In [1]:
import math
from math import sin, cos, radians, pi

#Initilization of Variables

W=500 #N #Weight of Body
S=5 #m #Distance
P=250 #N #Force
P2=200 #N #Force 2
theta=30 #Degrees #Angle made by Force with Horizontal

#Calculation

#Part-1

#Work Done
w=P*S #N*m

#Work Done
w2=P2*cos(theta*pi*180**-1)*S #N*m

#Result
print"Work Done when Force 250 N is applied",round(w,2),"N*m"
print"Work Done when Force 200 N is applied",round(w2,2),"N*m"

Work Done when Force 250 N is applied 1250.0 N*m
Work Done when Force 200 N is applied 866.03 N*m

Example 18.2,Page No.673

In [2]:
import math

#Initilization of Variables

W=1500 #N #Weight of Body
S=500 #m #Distance
P=15 #N #Force

#Calculation

#Work Done by Resistance 
w=P*S #N*m

#Result
print"Work Done on body by Resistance is",round(w,2),"N*m"

Work Done on body by Resistance is 7500.0 N*m

Example 18.3,Page No.673

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

#Initilization of Variables

W=800 #N #Weight of Body
theta=30 #Degrees #angle made by Force
S=5 #Distance

#Calculation

#Force apllied on Block
P=W*sin(theta*pi*180**-1) #N

#Work done on body
w=P*S #N*m

#Result
print"Work Done in Pulling up the Body is",round(w,2),"N*m"

Work Done in Pulling up the Body is 2000.0 N*m

Example 18.4,Page No.674

In [4]:
import math
from math import sin, cos, radians, pi

#Initilization of Variables

mu=0.3 #coefficient of Friction
S=5 #m #Distance
W=800 #N #weight of Body
theta=30 #Degrees #Angle made by Weight

#Calculation

#reaction Force 
R=W*cos(theta*pi*180**-1) #N

#Force of friction
F=mu*R

#Forces along plane
P=W*sin(theta*pi*180**-1)+F #N*m

#Work done
w=P*S #N*m

#Result
print"Work done in pullint the Body",round(w,2),"N*m"

Work done in pullint the Body 3039.23 N*m

Example 18.5,Page No.674

In [5]:
import math
from math import sin, cos, radians, pi

#Initilization of Variables

L=50.5 #m #Total Length of chain
R=0.16 #m #Radius of pulley
L_AC=40 #m #Length of chain between A and C
W=505 #N #Weight of chain
w=10 #N #weight of chain per metre length

#Calculation

#Length of BD
L_BD=L-2*pi*R*2**-1-L_AC #m

#Weight of chain 
W_AC=w*L_AC #N

#Weight of chain BD
W_BD=w*L_BD #N

#Force applied at D
P=W_AC-W_BD #N

#Length of chain
l=(L_AC-w)*2**-1

#Work Done
W=P*l*2**-1 #N

#Result
print"Work Done by the man",round(W,2),"N"

Work Done by the man 2250.2 N

Example 18.6,Page No.675

In [6]:
import math

#Initilization of Variables

W=2000 #N #Weight of Body
P_o=750 #N #Initial Force
x_o=0 #Initial Force,distance moved is zero
S=25 #m #Distance Moved

#Calculation

#Force after a distance of 25 m
P=P_o+10*S #N

#Work Done
w=(P_o+P)*2**-1*S #N*m

#Result
print"Work Done by applied Force",round(w,2),"N*m"

Work Done by applied Force 21875.0 N*m

Example 18.7,Page No.676

In [7]:
import math
from scipy.integrate import * 

#Initilization of Variables

L=10 #m #Length of free cable
W=50 #N #weight of cable per m length

#Calculation

#Weight of element 
#X=50*dx

#work done on the elemnt
#dw=500-50*x*dx

def f(x) :
    return 500-50*x

I,err = quad(f,0,10)

#Result
print"Work done by Electric Motor is",round(I,2),"N*m"

Work done by Electric Motor is 2500.0 N*m

Example 18.8,Page No.677

In [8]:
import math

#Initilization of Variables

W=2000 #KN #Weight of train
v=10 #m/s #speed of train
F=20000 #N #Resistance due to friction
p=F #Net Force in Direction of motion

#Calculation

#Power
P=p*v*10**-3 #KW

#Result
print"Power of the Engine",P,"KW"

Power of the Engine 200.0 KW

Example 18.9,Page No.677

In [9]:
import math

#Initilization of Variables

W=2000 #KN #Weight of train
F=20000 #N #Resistance Force
v=10 #m/s #Velocity
a=0.5 #m/s**2 #Acceleration
g=9.81 #m/s**2 #acceleration due to gravity

#Calculation

#Mass of train
m=W*10**3*g**-1 #Kg

#Net Force
F=m*a+F #N

#Power of the engine 
P=F*v*10**-3 #KW

#Result
print"Power of the Engine is",round(P,2),"KW"

Power of the Engine is 1219.37 KW

Example 18.10,Page No.678

In [10]:
import math

#Initilization of Variables

W=1500*1000 #N #Weight of train 
v=10 #m/s #speed
F=7500 #N #Force exerted by engine
#sin(theta)=1*100**-1 

#Calculation

#from equation of Net Force
#p=W*sin(theta*pi*180**-1)+F
#After sub values and further simplifying we get
p=15000+7500 #N

#Power Exerted by Engine
P=p*v*10**-3 #KW


#Result
print"Power Exerted by the Engine is",round(P,2),"KW"

Power Exerted by the Engine is 225.0 KW

Example 18.11,Page No.678

In [13]:
import math

#Initilization of Variables

W=2*10**6 #N
v=5 #m/s velocity
P=35*10**3 #W

#Calculation

#After simplifying Net Force acting on engine in direction of motion, we get
#F=13333.3-F+P  ................1

#Power
P2=P*v**-1

#Sub value in equation 1 we get
F=13333.3+P2 #N

#case-2

#frpm Net force in direction of motion after simplifying we get,value of 
F2=W*150**-1+F #N

#Power developed by engine 
P3=F2*v*10**-3 #KW


#Result
print"Power required to pull the train is",round(P3,2),"KW"

Power required to pull the train is 168.33 KW

Example 18.12,Page No.680

In [1]:
import math
from math import pi

#Initilization of Variables

P=1800 #N #Force
D=0.01 #m #Diameter
R=0.005 #m #Radius
theta=2*pi #Radians

#Calculation

#Torque
T=P*R #N*m

#Work done
W=T*theta #N*m

#Result
print"Work Done is",round(W,2),"N*m"

Work Done is 56.55 N*m

Example 18.13,Page No.681

In [2]:
import math
from math import pi

#Initilization of Variables

F=1800 #N #Force
R=0.005 #m #Radius
T=9 #N*m #Torque
N=200 #r.p.m

#Calculation

#Power of the shaft
P=2*pi*N*T*60**-1 #W

#Result
print"Power of the shaft is",round(P,2),"N*m"

Power of the shaft is 188.5 N*m

Example 18.14,Page No.683

In [16]:
import math

#Initilization of Variables

M=2 #Kg #Mass
v=50 #m/s**2 #Velocity

#Calculation

#Let K.E be E
E=1*2**-1*M*v**2 #N*m

#Result
print"Kinetic Energy is",round(E,2),"N*m"

Kinetic Energy is 2500.0 N*m

Example 18.15,Page No.683

In [17]:
import math

#Initilization of Variables

m=0.081 #Kg
u=300 #m/s #Initial Velocity of bullet

#Calculation

#Part-1

S=0.1 #m #Penetration of bullet
v=0 #Final Velocity of bullet

#Kinetic Energy of bullet
KE=(m*v**2-m*u**2)*2**-1 #N*m

#Force of Resistance
P=-KE*S**-1 #N

#Part-2

#Depth of penetration
S2=0.05 #m

#work Done by force of Resistance
W2=-P*S2 #N*m

#velocity of bullet after 5cm penetration
v1=((W2+(m*u**2*2**-1))*2*m**-1)**0.5

#Result
print"Force of Resistance is",round(P,2),"N"
print"Velocity with which bullet will emerge is",round(v1,2),"m/s"

Force of Resistance is 36450.0 N
Velocity with which bullet will emerge is 212.13 m/s

Example 18.15(A),Page No.684

In [18]:
import math

#Initilization of Variables

m=0.01 #Kg 
u=1000 #m/s #Velocity
t=0.002 #s #time taken b bullet to travel
v=0 #Final Velocity
g=9.81 #acceleration due to gravity

#Calculation

#Kinetic energy of bullet
KE=m*u**2*2**-1 #N*m

#acceleration
a=-(v-u)*t**-1 #m/s**2

#Frictional Force
F=m*a #N

#Distance travelled by bullet
S=F*KE**-1 #m

#Part-2

#Probable speed of the car just before brakes are applied 
V=(30*2*g)**0.5*1000**-1*3600 #m/s

#Result
print"Average Force acted on the bullet is",round(F,2),"N"
print"Distance penetrated by it",round(S,2),"m" 
print"Probable speed of the car just before brakes are applied",round(V,2),"km/hr"

Average Force acted on the bullet is 5000.0 N
Distance penetrated by it 1.0 m
Probable speed of the car just before brakes are applied 87.34 km/hr

Example 18.16,Page No.686

In [19]:
import math

#Initilization of Variables

W=20*10**3 #N #Weight of Truck
u=45*10**3*(3600)**-1 #speed of truck #m/s
v=0 #Final Velocity of truck
g=9.81 #Acceleration due to gravity
m=W*g**-1 #mass of truck
S=20 #m #DIstance

#Calculation

#Kinetic energy of Truck
KE=-m*(v**2-u**2)*2**-1 #N*m

#Average Force of Resisting acting on the truck
P=KE*S**-1 #N

#Result
print"Average Force of Resisting acting on the truck",round(P,2),"N"

Average Force of Resisting acting on the truck 7963.81 N

Example 18.17,Page No.687

In [20]:
import math

#Initilization of Variables

W=9810 #N #Weight of train
m=1000 #Kg  #Mass of car
u=0 #m/s #Intial Velocity
v=12.5 #m/s #Final by car
S=50 #m #Distance
P=100 #N #Resistance

#Calculation

#Change in Kinetic Energy
KE=m*(v**2-u**2)*2**-1 #N*m

#Average driving Force exerted by engine
P2=KE*S**-1+P #N

#Power Developed by Engine
P3=P2*v*10**-3 #KW

#Result
print"Average driving Force exerted by engine",round(P2,2),"N"
print"Power Developed by Engine",round(P3,2),"N"

Average driving Force exerted by engine 1662.5 N
Power Developed by Engine 20.78 N

Example 18.18,Page No.688

In [3]:
import math
from math import cos, pi,sin, radians

#Initilization of Variables

W=196.2 #N #Weight of train
m=20 #Kg #Mass
P=300 #N #force
theta=30 #Degrees #Angle of inclination
mu=0.2 #Coefficient of friction
u=0 #initial Velocity
t=4 #seconds

#Calculation

R=W*cos(theta*pi*180**-1) #N

#Net Force in Direction of motion
F=P-W*sin(theta*pi*180**-1)-mu*R #N

#Acceleration
a=F*m**-1 #m/s**2

#Distance travelled in four seconds
s=u*t+a*t**2*2**-1

#Velocity after 4 seconds
v=u+a*t #m/s

#Kinetic Energy after 4 seconds
KE=m*v**2*2**-1 #N*m

#Work Done on Body
W2=F*s #N*m

#Momentum of the body after four seconds
e=m*v #Kg*m/s

#Impulse applied in four seconds
I=F*t #N*s

#Result
print"Acceleration of Body",round(a,2),"m/s**2"
print"Distance travelled in four seconds",round(s,2),"m"
print"Velocity after 4 seconds",round(v,2),"m/s"
print"Kinetic Energy after 4 seconds",round(KE,2),"N*m"
print"Work Done on Body",round(W2,2),"N*m"
print"Momentum of the body after four seconds",round(e,2),"Kg*m/s"
print"Impulse applied in four seconds",round(I,2),"N*s"

Acceleration of Body 8.4 m/s**2
Distance travelled in four seconds 67.17 m
Velocity after 4 seconds 33.58 m/s
Kinetic Energy after 4 seconds 11278.47 N*m
Work Done on Body 11278.47 N*m
Momentum of the body after four seconds 671.67 Kg*m/s
Impulse applied in four seconds 671.67 N*s

Example 18.19,Page No.689

In [4]:
import math
from math import cos, pi,sin, radians

#Initilization of Variables

W=20 #N #Weight 
theta=20 #Degrees #Angle
u=12 #m/s #Initial Velocity
mu=0.15 #Coefficient of friction
g=9.81 #acceleration due to gravity
m=W*g**-1 #Kg

#Calculation

#PArt-1

v=0 #Final Velocity
R=W*cos(theta*pi*180**-1)
F=mu*R

#Net Force
F2=W*sin(theta*pi*180**-1)+mu*R #N

#Change in Kinetic Energy
KE=m*(v**2-u**2)*2**-1 #N*m

S=KE*F2**-1 #Max Distance

#PArt-2

#Net Force in direction of motion is
F3=W*sin(theta*pi*180**-1)-mu*R #N

#Work Done on the body 
W2=F3*S #N*m

#Velocity of the body
V1=(-W2*2*g*W**-1)**0.5

#Result
print"MAx Distance that the bodt will move up the inclined plane",round(S,2),"m"
print"Velocity of the body",round(V1,2),"m/s"

MAx Distance that the bodt will move up the inclined plane -15.2 m
Velocity of the body 7.74 m/s

Example 18.20,Page No.691

In [23]:
import math

#Initilization of Variables

m1=0.025 #Kg #Mass of bullet
u1=600 #m/s #Initial Veloctiy of Bullet
m2=5 #Kg #Mass of Wooden Block
u2=0 #m/s #Final Velocity of bullet
S=0.9 #m #Distance travelled by block and bullet
g=9.81 #Acceleration due to gravity

#Calculation

#Total mass of bullet
M=m1+m2 #Kg

#common Velocityof bullet and block after impact
V=(m1*u1+m2*u2)*M**-1 #m/s

#Average resistance between block and horizontal surface

#Initial Velocity of Block And Bullet
Vi=V #m/s

#Final Velocity
Vf=0 #m/s

#Change of KE of bullet and Block
KE=M*(Vf**2-Vi**2)*2**-1 #N*m

#Frictional resistance 
P=-KE*S**-1 #N

#Coefficient of Friction 

W=M*g #N
R=W #N 

mu=P*R**-1 

#Result
print"Average Resistance between Block and horizontal surface",round(P,2),"N"
print"Coefficient of friction is",round(mu,2)

Average Resistance between Block and horizontal surface 24.88 N
Coefficient of friction is 0.5

Example 18.21,Page No.692

In [24]:
import math

#Initilization of Variables

m1=0.01 #Kg #mass of bullet
m2=1 #Kg #Mass of Block
S=1 #m #Distance travelled by block and bullet
mu=0.2 #coefficient of friction
g=9.81 #Acceleration due to gravity

#Calculation

#total mass of buulet and wooden block
M=m1+m2 #Kg

#Friction Force
F=mu*M*g #N

#Work Done by force of friction
W=F*S #N

#Velocity of bullet
u1=(W*2*M**-1)**0.5*M*m1**-1 #m/s

#Result
print"Velocity of bullet is",round(u1,2),"m/s"

Velocity of bullet is 200.07 m/s

Example 18.22,Page No.694

In [25]:
import math

#Initilization of Variables

M=1500 #Kg #Mass of Hammer
h=0.6 #m #Height from which hammer drops
m=750 #kg #Mass of pile
S=0.05 #m #Depth of penetration
g=9.81 #Acceleration due to gravity

#Calculation

#Velocity of hammer after falling through height of 0.6 m from rest
v=(2*g*h)**0.5 #m/s

#Total momentum of hammer and pile just before impact
p=M*v #Kg*m/s

#Common Velocity
V=p*(M+m)**-1 #m/s

#Part-2

#K.E of system
KE=(M+m)*round(V,2)**2*2**-1 #N*m

#Loss of P.E of system
PE=(M+m)*g*S #N*m

#Total Energy loss
E=KE+PE #N*m

#Resistance of ground
R=E*S**-1 #N

#Result
print"Common Velocity after impact",round(V,2),"m/s"
print"Average Resistance of the ground",round(R,2),"N"

Common Velocity after impact 2.29 m/s
Average Resistance of the ground 140064.75 N

Example 18.23,Page No.695

In [26]:
import math

#Initilization of Variables

M=750 #Kg #MAss of hammer
h=1.2 #m #Height through which hammer drops
m=200 #kg #mass of pile
R=79*10**3 #N #Average resistance of ground
g=9.81 #Acceleration due to gravity

#Calculation

#Velocity of hammer after falling through height of 0.6 m from rest
v=(2*g*h)**0.5 #m/s

#Total momentum of hammer and pile just before impact
p=M*v #Kg*m/s

#Common Velocity
V=p*(M+m)**-1 #m/s

#Part-2

#K.E of system
KE=(M+m)*round(V,2)**2*2**-1 #N*m

#Depth of penetration into the ground
S=KE*(R-(M+m)*g)**-1

#Result
print"Depth of penetration into the ground",round(S,2),"m"

Depth of penetration into the ground 0.1 m

Example 18.24,Page No.696

In [27]:
import math

#Initilization of Variables

M=400 #Kg #Mass of Hammer
h=3 #m #Height from which hammer drops
m=0 #kg #Mass of pile
S=1 #m #Depth of penetration
g=9.81 #Acceleration due to gravity


#Calculation

#Velocity of hammer after falling through height of 0.6 m from rest
v=(2*g*h)**0.5 #m/s

#Total momentum of hammer and pile just before impact
p=M*v #Kg*m/s

#Common Velocity
V=p*(M+m)**-1 #m/s

#Part-2

#K.E of system
KE=(M+m)*V**2*2**-1 #N*m

#Loss of P.E of system
PE=(M+m)*g*S #N*m

#Total Energy loss
E=KE+PE #N*m

#Resistance of ground
R=E*S**-1 #N


#Result
print"Resistance of ground for penetration is",round(R,2),"N"

Resistance of ground for penetration is 15696.0 N

Example 18.25,Page No.697

In [28]:
import math

#Initilization of Variables

M=10 #Kg #Mass of Body
k=100 #N/cm #stiffnes
h=2 #cm #Height through which mass 10 kg is dropped 
  
#Calculation

#For Position 1
#PE+KE=M*g*(x+h) ............1

#For Position 2
#PE of spring=50*x**2  .............2

#Equating equations 1 and 2 we get
#5x**2-9.81*x-19.62=0

a=5
b=-9.81
c=-19.62

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

#Max Displacement of spring
x1=(-b+X**0.5)*(2*a)**-1
x2=(-b-X**0.5)*(2*a)**-1

#Result
print"Max Displacement of spring",round(x1,2),"m"

Max Displacement of spring 3.19 m

Example 18.26,Page No.698

In [5]:
import math
from math import sin, cos, radians, pi

#Initilization of Variables

m=0.01 #kg #Mass of bullet
M=1 #kg #Mass of body
L=1 #m #Length of string
theta=18.2 #degrees
g=9.81 #acceleration due to gravity
L_OA=1 #m #Length of OA
L_OB=1 #m #Length of OB

#Calculation

#From Geometry of figure
h=L_OA-L_OB*cos(theta*pi*180**-1) #m

#Potential Energy of body and bullet at B
PE=(M+m)*g*h

#from Kinetic Energy of body and bullet after impact
V=(PE*2*(M+m)**-1)**0.5 #m/s #Velocity of body and bullet

#Velocity of bullet
u=(M+m)*V*m**-1

#Result
print"Velocity of Bullet is",round(u,2),"m/s"

Velocity of Bullet is 100.06 m/s

Example 18.27,Page No.700

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

#Initilization of Variables

m=0.03 #kg #Mass of bullet
u=483 #m/s #Velocity of bullet
M=10 #kg #MAss of body
L=0.8 #m #Length of string
g=9.81

#Calculation

#Momentum of bullet and body before impact
p=m*u #Kg*m/s

#from Momentum of bullet and body after impact
V=p*(M+m)**-1 #m/s

#K.E of the bullet and body after impact
KE=(M+m)*V**2*2**-1 #N*m

#Angle through which body swings
theta=np.arccos(-((KE*((M+m)*g)**-1)-L)*L**-1)*(pi**-1*180) #Degrees

#Result
print"Angle through which body swings is",round(theta,2),"Degrees"

Angle through which body swings is 29.88 Degrees

Example 18.28,Page No.701

In [1]:
import math
from math import sin, cos, radians, pi
import numpy as np

#Initilization of Variables

m=0.03 #kg #mass of bullet
u=483 #m/s #velocity of bullet
M=10 #Kg #Mass of body
L=0.8 #m #Length of string
v=96.5 #m/s #Velocity of body
g=9.81 #acceleration due to gravity

#Calculation

#Velocity of Body after impact
V=(m*u-m*v)*M**-1 #m/s

#Height 
h=V**2*(2*g)**-1 #m

#from geometry
theta=np.arccos((L-h)*L**-1)*(pi**-1*180)

#Result
print"Angle through which the body will swing",round(theta,2),"Degrees"

Angle through which the body will swing 23.89 Degrees