Chapter 9:Lifting Machines

Example 9.1,Page No.295

In [1]:
import math

#Initilization of Variables

P=100 #N #effort applied
W=900 #N  #Lad applied
y=100 #cm #Distance moved by effort
x=10 #cm #Distance moved by load

#Calculations

#Mechanical Advantage
MA=W*P**-1 

#Velocity ratio 
VR=y*x**-1 

#Efficiency
rho=MA*VR**-1*100

#Result
print"Mechanical Advantage is",round(MA,2)
print"velocity ratio is",round(VR,2)
print"Efficiency is",round(rho,2)

Mechanical Advantage is 9.0
velocity ratio is 10.0
Efficiency is 90.0

Example 9.2,Page No.296

In [2]:
import math

#Initilization of Variables

P=500 #N #Effort applied
y=5 #m #Distance moved by effort
x=0.5 #cm #Distance moved by load
rho=0.8 #Efficiency

#Calculations

#Load Lifted by machine
W=P*y*rho*(x)**-1 #N

#Mechanical Advantage
MA=W*P**-1 

#Velocity ratio
VR=y*x**-1 

#Result
print"Load Lifted by the machine is",round(W,2),"N"
print"Mechanical Advantage is",round(MA,2)
print"Velocity ratio is",round(VR,2)

Load Lifted by the machine is 4000.0 N
Mechanical Advantage is 8.0
Velocity ratio is 10.0

Example 9.3,Page No.297

In [3]:
import math

#Initilization of Variables

P=20 #N #Actual Effort
W=900 #N #Load Lifted
y=2.40 #m #Distance moved by effort
x=0.04 #m #Distance moved by load

#Calculations

#Mechanical Advantage
MA=W*P**-1 

#Velocity ratio
VR=y*x**-1 

#Efficiency
rho=MA*VR**-1

#Ideal Effort required
P1=rho*P 

#Result
print"Mechanical Advantage is",round(MA,2)
print"Velocity ratio is",round(VR,2)
print"Efficiency is",round(rho,2)
print"Ideal Effort is",round(P1,2)

Mechanical Advantage is 45.0
Velocity ratio is 60.0
Efficiency is 0.75
Ideal Effort is 15.0

Example 9.4,Page No.297

In [4]:
import math

#Initilization of Variables

rho=0.7 #eifficiency
P=10 #N #Effort
W=500 #N #Load

#Calculations

#Mechanical Advantage
MA=W*P**-1 

#Velocity ratio
VR=MA*rho**-1

#Result
print"Mechanical Advantage is",round(MA,2)
print"Velocity ratio is",round(VR,2)

Mechanical Advantage is 50.0
Velocity ratio is 71.43

Example 9.5,Page No.299

In [5]:
import math

#Initilization of Variables

P1=15 #N #Effort
W1=770 # #Load
rho=0.60 #Efficiency

P2=25 #N 
W2=1320

P=15 #N 
W=500 #N

#Calculations

#First Case

#Mechanical Advantage
MA=W1*P1**-1 

#Velocity ratio
VR=MA*rho**-1 

#Second case

#Mechanical Advantage
MA2=W2*P2**-1 

#Efficiency
rho2=MA2*VR**-1*100

#Third case

#from LAw of machine
#P=m*W+C  ................1

#Equation 2
#P2=W2*m+C  ...............2

#Subtracting equation 2 from 1 we get
m=10*550**-1

#Constacnt value 
C=P2-W2*m

#Sub value C in equation 1  we get
P3=m*W+C #N

#MAx Mechanical advantage
MA_max=1*m**-1

#MAx Efficiency 
rho_max=1*(m*VR)**-1*100

#Result
print"Mechanical Advantage is",round(MA,2)
print"Velocity Ratio is",round(VR,2)
print"Efficiency is",round(rho2,2),"%"
print"Effort Required to raise the Load 500 N",round(P3,2),"N"
print"MAx Mechanical Advantage is",round(MA_max,2)
print"MAx Efficiency is",round(rho_max,2)

Mechanical Advantage is 51.33
Velocity Ratio is 85.56
Efficiency is 61.71 %
Effort Required to raise the Load 500 N 10.09 N
MAx Mechanical Advantage is 55.0
MAx Efficiency is 64.29

Example 9.6,Page No.301

In [6]:
import math

#Initilization of Variables

#Effort
P1=15.5 #N
P2=19.5 #N

#Loads
W1=100 #N
W2=90 #N

m=0.2

#Calculations

#Law of machine equation 
#P=m*W+C 

#Equations
#P1=m*W1+C  ................1
#P2=m*W2+C  ....................2

#sub value of m in equation 1 weget
C=P1-m*W1

#Effort required to lift a Load
P1=m*W1+C

#MEchanical advantage
MA=1*m**-1

#Result
print"Effort required to Lift a Load of 100 N",round(P1,2),"N"
print"MAx MEchanical Advantage is",round(MA,2)

Effort required to Lift a Load of 100 N 15.5 N
MAx MEchanical Advantage is 5.0

Example 9.7,Page No.303

In [7]:
import math

#Initilization of Variables

rho=0.8 #Efficiency
P=15 #N #Effort
W=130 #N #Load

#Calculations

#Velocity ratio
VR=W*(P*rho)**-1 

#Frictional force in terms of machine in tems of effort
Fp=P-W*VR**-1 #N

#Frictional Force of the machine in terms of Load
Fw=P*VR-W #N

#Result
print"Velocity ratio is",round(VR,2)
print"Frictional force in terms of machine in tems of effort",round(Fp,2),"N"
print"Frictional Force of the machine in terms of Load is",round(Fw,2),"N"

Velocity ratio is 10.83
Frictional force in terms of machine in tems of effort 3.0 N
Frictional Force of the machine in terms of Load is 32.5 N

Example 9.8,Page No.303

In [8]:
import math

#Initilization of Variables

VR=15 #Velocity ratio
rho=0.6 #Efficiency
W=100 #N #Load Lifted

#Calculations

#Power
P=W*(VR*rho)**-1 #N

#Frictional Force
Fp=P-(W*VR**-1) #N

#Let C=Fp
C=Fp

#From law of machine,we get
#P=m*W+C
#After sub values and furter simplifying we get
m=(P-C)*W**-1

#After sub values in above equaion we get law of  machine as
#P2=m*W2+c #N

#when W2=140 #N
W2=140 #N
P2=m*W2+C #N

#When W3=0
W3=0 #N
P3=m*W3+C #N

#Result
print"Effort required to run the machine at Load:W2=140 is",round(P2,2),"N"
print"Effort required to run the machine at Load:W3=0 is",round(P3,2),"N"

Effort required to run the machine at Load:W2=140 is 13.78 N
Effort required to run the machine at Load:W3=0 is 4.44 N

Example 9.9,Page No.304

In [9]:
import math

#Initilization of Variables

P=12 #N #Effort
VR=18 #Velocity ratio
rho=0.6 #efficiency

#Calculations

#Load lifted
W=rho*P*VR #N

#LEt C=Fp
Fp=P-(W*VR**-1) #N
C=Fp

#From law of machine we get
m=(P-C)*W**-1 #N

#Sub value of m in equation we get
#P2=1*18**-1*W2+C

#Sub W2=90
W2=90 
P2=m*W2+C


#Result
print"Effort required to run the machine is",round(P2,2),"N"

Effort required to run the machine is 9.8 N

Example 9.10,Page No.305

In [10]:
import math

#Initilization of Variables

VR=10 #Velocity ratio
P=100 #N #Effort applied
Fp=20 #N #effort lost in friction

#Calculations

#Load Lifted
W=(P-Fp)*VR #N

#Efficiency
rho=W*P**-1*VR**-1*100 

#Result
print"Load Lifted is",round(W,2),"N"
print"Efficiency is",round(rho,2),"%"

Load Lifted is 800.0 N
Efficiency is 80.0 %

Example 9.11,Page No.305

In [11]:
import math

#Initilization of Variables
P=40 #N #Effort applied
W=600 #N #Load Lifted
VR=20 #Velocity ratio

#Calculations

#MAchine Fiction in terms of effort
Fp=P-W*VR**-1 #N

#M/c Friction in terms of Load
Fw=P*VR-W #N

#efficiency
rho=W*P**-1*VR**-1*100

#Result
print"MAchine Fiction in terms of effort is",round(Fp,2),"N"
print"M/c Friction in terms of Load is",round(Fw,2),"N"
print"Ffficiency of the machine is",round(rho,2),"%"

MAchine Fiction in terms of effort is 10.0 N
M/c Friction in terms of Load is 200.0 N
Ffficiency of the machine is 75.0 %

Example 9.12,Page No.306

In [12]:
import math

#Initilization of Variables
P=15 #N #Effort applied
W=200 #N #Load Lifted
VR=40 #Velocity ratio

#Calculations

#Efficiency
rho=W*P**-1*VR**-1 #%

#Friction Load of m/c 
Fw=P*VR-W #N

#Result
print"Friction Load of m/c is",round(Fw,2),"N"

Friction Load of m/c is 400.0 N

Example 9.13,Page No.307

In [13]:
import math

#Initilization of Variables

W=48 #N #Weight
P=16 #N #Force
D=400 #mm #Diameter of wheel
d=100 #mm #Diameter of axle

#Calculations

#Mechanical Advantage
MA=W*P**-1 

#Velocity ratio
VR=D*d**-1 

#Efficiency of the machine
rho=MA*VR**-1*100 #%

#Result
print"Mechanical Advantage is",round(MA,2)
print"Velocity ratio is",round(VR,2)
print"Efficiency of the machine is",round(rho,2),"%"

Mechanical Advantage is 3.0
Velocity ratio is 4.0
Efficiency of the machine is 75.0 %

Example 9.14,Page No.309

In [14]:
import math

#Initilization of Variables

D=25 #cm #Diameter of wheel
d1=10 #cm #LArge dia. of axle
d2=9 #cm #Small Dia. of axle
P=30 #N #Effort applied
W=900 #N #Load Lifted

#Calculations

#Velocity ratio
VR=2*D*(d1-d2)**-1 

#Mechanical advantage
MA=W*P**-1 

#Efficiency
rho=MA*VR**-1*100

#Result
print"Velocity ratio is",round(VR,2)
print"Mechanical advantage is",round(MA,2)
print"Efficiency is",round(rho,2),"%"

Velocity ratio is 50.0
Mechanical advantage is 30.0
Efficiency is 60.0 %

Example 9.15,Page No.310

In [15]:
import math

#Initilization of Variables

T=60 #No. of teeth on worm wheel
L=12.5 #cm #Radius of effort wheel
r=6.25 #cm #Radius of Load drum
P=20 #N #Effort
W=600 #N #Load

#Calculations

#Velocity ratio
VR=L*T*r**-1 

#Efficiency
rho=W*P**-1*VR**-1*100

#Result
print"Velocity ratio for single threaded worm is",round(VR,2)
print"Efficiency of the worm is",round(rho,2),"%"

Velocity ratio for single threaded worm is 120.0
Efficiency of the worm is 25.0 %

Example 9.16,Page No.312

In [16]:
import math

#Initilization of Variables

T1=10 #No.of teeth on pinion
T2=100 #No.of teeth on spur wheel
D=30 #cm #Dia. of Load axle
L=30 #cm #Length of lever
P=20 #N #Effort applied
W=360 #N #Load Lifted

#Calculations

#Velocity ratio
VR=2*L*T2*(D*T1)**-1

#Efficiency
rho=W*P**-1*VR**-1*100

#Result
print"Velocity ratio is",round(VR,2)
print"Efficinecy is",round(rho,2),"%"

Velocity ratio is 20.0
Efficinecy is 90.0 %

Example 9.17,Page No.314

In [17]:
import math

#Initilization of Variables

P=40 #N #Effort
rho=0.5 #Efficincy
D=20 #cm #Dia. of load axle
L=80 #cm #Length of Lever
T1=10 #No. of teeth om pinion of effort axle
T2=100 #No. of teeth on spur wheel of intermediate axle
T3=20 #No. of teeth om pinion of Load axle
T4=200 #No. of teeth on spur wheel of load axle

#Calculations

#Velocity ratio
VR=2*L*D**-1*T2*T1**-1*T4*T3**-1 

#Mechanical Advatnage
MA=rho*VR

#Load Which can be Lifted
W=MA*P*10**-3 #N

#Result
print"Velocity ratio is",round(VR,2)
print"Load Which can be Lifted is",round(W,2),"KN"

Velocity ratio is 800.0
Load Which can be Lifted is 16.0 KN

Example 9.17(A),Page No.315

In [18]:
import math

#Initilization of Variables

rho=0.4 #Efficincy
D=20 #cm #Dia. of load axle
L=40 #cm #Length of Lever
T1=15 #No. of teeth om pinion of effort axle
T2=45 #No. of teeth on spur wheel of intermediate axle
T3=20 #No. of teeth om pinion of Load axle
T4=40 #No. of teeth on spur wheel of load axle
W=250 #N #Load Lifted

#Calculations

#Velocity ratio
VR=VR=2*L*D**-1*T2*T1**-1*T4*T3**-1 

#Effort applied
P=W*(rho*VR)**-1 #N

#Result
print"Effort applied at the end is",round(P,2),"N"

Effort applied at the end is 26.04 N

Example 9.18,Page No.318

In [19]:
import math

#Initilization of Variables

n=4 #No. of movable pulleys
W=1440 #N #Load
P=100 #N #effort

#Calculations

#Mechanical Advantage
MA=W*P**-1 

#Velocity ratio
VR=2**4

#Efficiency
rho=MA*VR**-1*100 #%

#Ideal Effort
P2=W*VR**-1

#Effort wsted in friction
P3=P-P2 #N

#Load WAsted in friction
W2=VR*P

W3=W2-W

#Result
print"Efficiency of the machine is,",round(rho,2),"%"
print"Effort Wasted in friction is",round(P3,2),"N"
print"Load Wasted in friction is",round(W3,2),"N"

Efficiency of the machine is, 90.0 %
Effort Wasted in friction is 10.0 N
Load Wasted in friction is 160.0 N

Example 9.19,Page No.320

In [20]:
import math

#Initilization of Variables

W=2000 #N #Weight
P=600 #Effort
n=5 #Total No. of pulleys
VR=n 

#Calculations

#Mechanical advantage
MA=W*P**-1

#efficiency
rho=MA*VR**-1*100

#Result
print"Efficiency of the system is",round(rho,2),"%"

Efficiency of the system is 66.67 %

Example 9.20,Page No.321

In [21]:
import math

#Initilization of Variables

n=4 #No. of pulleys
P=160 #N #Effort
rho=0.75 #efficiency
VR=15 #Velocity ratio

#Calculations

#weight Lifted
W=rho*P*VR 

#Result
print"Weight Lifted is",round(W,2),"N"

Weight Lifted is 1800.0 N

Example 9.21,Page No.323

In [22]:
import math

#Initilization of Variables

rho=0.5 #Efficency
D=25 #cm #Diameter of Large pulley
d=20 #cm #Dia. of smaller pulley
P=20 #N #Effort applied

#Calculations

#Velocity ratio
VR=2*D*(D-d)**-1

#Load Lifted
W=rho*d*VR #N

#Result
print"Load Lifted by the machine is",round(W,2),"N"

Load Lifted by the machine is 100.0 N

Example 9.22,Page No.327

In [23]:
import math

#Initilization of Variables

W=1500 #Load
L=0.7 #Length of handle
d=0.06 #m #Mean diaof screw jack
p=0.009 #pitch of the screw jack
mu=0.095 #co-efficient of friction
pi=3.14

#Calculations

#Effort required
X=(W*d*(2*L)**-1)*(p+mu*pi*d)*(pi*d-p*mu)**-1 #N

#Effort required at the end of the handle for lowering the load
P2=W*d*(2*L)**-1*(mu*pi*d-p)*(pi*d+mu*p)**-1 #N

#Result
print"Effort required at the end of the handle for Lifting Load 1500 N",round(X,2),"N"
print"Effort required at the end of the handle for lowering the load is",round(P2,2),"N"

Effort required at the end of the handle for Lifting Load 1500 N 9.22 N
Effort required at the end of the handle for lowering the load is 3.02 N

Example 9.23,Page No.328

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

#Initilization of Variables

W=3000 #N #Load Lifted
n=2 #No. of square thread
D1=6 #cm #Outer diameter
mu=0.09 #Coefficient offriction
L=0.6 #m #Length

#Calculations

#pitch of screw
p=1.2*n**-1*10**-2 #m

#Thickness of thread
t=0.5*p #

#Diameter at base of screw
D2=D1-2*t

#Mean Diameter
d=(D1+D2)*2**-1*10**-2 #m

#Force 
P=W*d*(2*L)**-1*(p+mu*pi*d)*(pi*d-p*mu)**-1 #N

#Result
print"Force required at the end of handle is",round(P,2),"N"

#Answer in textbook is incorrect

Force required at the end of handle is 18.32 N

Example 9.23(A),Page No.328

In [25]:
import math
from math import sin, cos, tan, radians, pi

#Initilization of Variables

W=5000 #N #Load
n=2
t=0.003 #m #Thickness
D1=0.06 #m #outer diameter
D2=0.054 #m #Inner diameter
d=0.057 #MEan diameter
mu=0.08 #Coefficient of friction
L=0.6 #m #Length
p=0.006 #m #pitch

#Calculations

#Let tan(alpha)=X
X=p*(pi*d)**-1

#Let tan(phi)=Y
Y=mu

#Force reuired at the end of handle 
P=d*(2*L)**-1*W*(X+Y)*(1-X*Y)**-1

#Result 
print"Force reuired at the end is",round(P,2),"N"

Force reuired at the end is 27.03 N

Example 9.24,Page No.329

In [26]:
import math
from math import sin, cos, tan, radians, pi

#Initilization of Variables

W=2000 #N #Load Lifted
p=1 #mm #pitch
rho=0.5 #efficiency
L=50 #cm #Length of handle

#Calculations

#Velocity ratio
VR=2*pi*L*p**-1

#Effort applied
P=W*(rho*VR)**-1

#Result
print"Effort applied at the end of handle",round(P,2),"N"

Effort applied at the end of handle 12.74 N

Example 9.25,Page No.332

In [27]:
import math
from math import sin, cos, tan, radians, pi

#Initilization of Variables

rho=0.55 #efficiency
W=1500 #N #Load Lifted
L=0.5 #m #Length of handle
p=0.01 #m #Pitch of the screw

#Calculations

#Velocity ratio
VR=2*pi*L*p**-1 

#Effort applied
P=W*(VR*rho)**-1 #N

#Result
print"Effort applied is",round(P,2),"N"

Effort applied is 8.69 N

Example 9.26,Page No.333

In [28]:
import math
from math import sin, cos, tan, radians, pi

#Initilization of Variables

d=0.075 #m #Mean diameter
p=0.015 #m #Pitch of threads
mu=0.05 #Coefficient of friction
W=600 #N
L=0.36 #m #LEngth

#Calculations

#Tangential Force
P=W*d*(2*L)**-1*(p+mu*pi*d)*(pi*d-p*mu)**-1 #N

#Let Tan(alpha)=X
#tan(phi)=Y
#tan(alpha+phi)=Z
X=p*(pi*d)**-1
Y=mu
Z=(X+Y)*(1-X*Y)**-1

#efficiency
rho=X*Z**-1 

#Effort
P2=W*((X-Y)*(1+X*Y)**-1) #N

#Torque required
T=P2*d*2**-1 #N*m

#Result
print"Tangential Force to be qpplied is",round(P,2),"N"
print"Torque necesscary to lower the load is",round(T,2),"Nm"

Tangential Force to be qpplied is 7.13 N
Torque necesscary to lower the load is 0.31 Nm

Example 9.27,Page No.334

In [29]:
import math
from math import sin, cos, tan, radians, pi

#Initilization of Variables

d=0.06 #m #Mean diameter
p=0.008 #m P#itch
mu=0.09
W=3 #Load Lifted
x=0.12 #m
n=15 #No. of turns

#Calculations

#Let Tan(alpha)=X
#tan(phi)=Y
X=p*(pi*d)**-1
Y=mu
P=W*((X+Y)*(1-X*Y)**-1) #N

#Torque required
T=P*d*2**-1 #N*m

#Total Angular displacement
omega=n*2*pi

#Workk done
W2=omega*T #KNm

#efficiency
rho=W*x*W2**-1*100 #%

#Efficiency can also be determined as
rho2=X*(X+Y)**-1*(1-X*Y)*100

#Result
print"Torque required is",round(T,2),"Nm"
print"Work done in lifting the load is",round(W2,3),"KN"
print"Efficiency of the jack is",round(rho2,1),"%"

Torque required is 0.01 Nm
Work done in lifting the load is 1.127 KN
Efficiency of the jack is 31.9 %

Example 9.28,Page No.336

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

#Initilization of Variables

p1=1 #cm #Pitch of Larger screw
p2=0.7 #cm #Pitch of smaller screw
l=36 #cm #Length of handle
rho=0.28 #efficiency
W=5000 #N #Weight

#Calculations

#Velocity ratio
VR=2*pi*l*(p1-p2)**-1

#Effort applied
P=W*(rho*VR)**-1 #N

#Result
print"Effort required to Lift the Load",round(P,2),"N" 

Effort required to Lift the Load 23.68 N