import math
#Declaration Of Variables
W=100 #N #Weight of Body
P=F=60 #N #Horizontal Force
#Calculations
#Normal Reaction Force
R=W=100 #N
#Coefficient of Friction
mu=F*R**-1
#Result
print"Coefficient of Friction is",round(mu,2)
import math
#Declaration Of Variables
W=200 #N #Weight of Body
mu=0.3 #Coefficient of Friction
#Calculations
#Normal Reaction
R=W=200 #N
#Horizontal Force
F=mu*R #N
#Result
print"Horizontal Force is",round(F,2),"N"
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
W=50 #N #WEight
F=15 #N #Force required to pull
theta=15 #Degree #Angle made by Force
#Calculations
#Normal Reaction
R=W-F*sin(theta*pi*180**-1) #N
#Coefficient of friction
mu=F*cos(theta*pi*180**-1)*R**-1 #N
#Result
print"Coefficient of Friction is",round(mu,2)
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
W=70 #N #Weight of Body
F=20 #N #force applied
theta=20 #degrees #Angle made by Force
#Calculations
#resolving Forces Normal to plane
R=W+F*sin(20*pi*180**-1) #N
#Resolving Forces along the plane
mu=F*cos(20*pi*180**-1)*R**-1 #N
#Result
print"coefficient of Friction is",round(mu,2)
import math
#Declaration Of Variables
P1=20 #N #pull
P2=25 #N #Required push
theta2=25 #Inclination of push
#Calculations
#Case-1 (When body is pulled)
#Resolving Forces along the plane
#mu*R1=P1*cos(theta*pi*180**-1) #N .........1
#Resolving Force normal to plane
#R1=W-P1*sin(theta*pi*180**-1) #N
#Sub value of mu*R1 in above Equation we get
#mu*(W-8.452)=18.126 ......................2
#Case-2 (When body is pushed)
#Resolving Forces along the plane
#mu*R2=P2*cos(theta2*pi*180**-1) #N .........3
#Resolving Force normal to plane
#R2=W-P2*cos(theta*pi*180**-1) #N
#Sub value of mu*R2 in above Equation we get
#mu*(W-10.565)=22.657 .................4
#dividing equation 2 by 4 and Further simplifying we get
#Weight of body
W=383*4.53**-1
#Sub value of W in Equation 2
mu=18.126*(W-8.452)**-1
#Result
print"Weight of Body is",round(W,2),"N"
print"Coefficient of Friction is",round(mu,2)
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
W=1000 #N #Weight of stone Block
mu=0.6 #Coefficient of Friction
theta=20 #Degrees #Angle with Horizontal
#Calculations
#PArt-1
#resolving Horizontal Forces
#P*cos(theta)=mu*R ...........................1
#Resolving Verticla Forces
#R+P*sin(theta)=W .............................2
#Sub value of P from equation 2,we get
P=mu*W*(cos(theta*pi*180**-1)+mu*sin(theta*pi*180**-1))**-1 #N ........3
#PArt-2
#Let phi=Angle of Friction
#Form Equation 3 ,angle 20 is replaced by angle phi
#Force required to pull the body
#P2=mu*W*(cos(phi)+mu*sin(phi))**-1
#The Force P will be min if Dericative of (cos(phi)+mu*sin(phi)) is equal to zero
phi=np.arctan(mu)*(180*pi**-1) #degrees
#Force required to pull the body
P2=mu*W*(cos(phi*pi*180**-1)+mu*sin(phi*pi*180**-1))**-1 #N
#Result
print"Minimum Pull necessary is",round(P,2),"N"
print"Pull Required if inclination of rope is equal to angle of friction",round(P,2),"N"
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
W=500 #N #weight of Body
P=350 #N #Force applied
alpha=30 #Degrees #Inclination
#Calculations
#Resolving Weights
W1=W*sin(30*pi*180**-1) #N
W2=W*cos(30*pi*180**-1) #N
#Resolving Forces Vertically
#R=W*cos(30)
#Resolving Forces Horizontally
mu=(P-W*sin(alpha*pi*180**-1))*(W*cos(alpha*pi*180**-1))**-1
#Result
print"Coefficient Of Friction",round(mu,2)
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
W=450 #N #Weight of Body
alpha=30 #Degrees #Inclination of plane
mu=0.25 #coefficient of friction
d=10 #m #Distance travelled by body
#Calculations
#Resolving Force normal to plane
R=W*cos(alpha*pi*180**-1)
#Resolving Forces along the plane
P=W*sin(alpha*pi*180**-1)+mu*R #N
#Work done on the body
W=P*d #J
#Result
print"Force required is",round(P,2),"N"
print"Work done is",round(W,2),"J"
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
#Case-1
P1=200 #N #Force applied
theta1=15 #Degrees #Inclination
P2=230 #N #Force applied
theta2=20 #Degrees #Inclination
#Calculations
#For Case-1,
#W1=W*sin(theta1*pi*180**-1) #N
#W2=W*cos(theta1*pi*180**-1) #N
#Resolving Forces Vertically
#R=W2
#Resolving Foces Horizontally
#mu*W2+W1=P1 .......................1
#For case-2
#W3=W*sin(theta2*pi*180**-1) #N
#W4=W*cos(theta2*pi*180**-1) #N
#Resolving Forces Vertically
#R=W3
#Resolving Foces Horizontally
#mu*W3+W4=P2 .......................2
#After sub values inequations 1 & 2 and dividing equations 2 by 1 we get
mu=mu=(P1*sin(20*pi*180**-1)-P2*sin(15*pi*180**-1))*(P2*cos(15*pi*180**-1)-P1*cos(20*pi*180**-1))**-1
#weight of Body
W=P2*(sin(theta2*pi*180**-1)+mu*cos(theta2*pi*180**-1))**-1 #N
#Result
print"Weight of Body is",round(W,2),"N"
print"Coefficient of Friction is",round(mu,2)
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
W=15 #N #Weight of Block
T=5 #N #Tension in string
alpha=45 #Degrees #Inclination
#Calculations
#Frictional Foce on Block
F=-(T*cos(alpha*pi*180**-1)-W*sin(alpha*pi*180**-1)) #N
#Normal Reaction of inclined plane
R=(W*cos(alpha*pi*180**-1)+T*sin(alpha*pi*180**-1)) #N
#Coefficient of friction
mu=F*R**-1
#Result
print"Frictional Force on Block is",round(F,2),"N"
print"Normal Reaction of inclined plane",round(R,2),"N"
print"Coefficient of friction",round(mu,2)
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
mu_s=0.4 #Coefficient of static Friction
mu_k=0.3 #Coefficient of Kinetic friction
M=40 #Kg #MAss of body
g=9.81 #acceleration due to gravity
W=M*g #N
theta=40
alpha=30 #Inclination
P=800 #N
#Calculations
#normal Reaction
R=P*sin(theta*pi*180**-1)+M*g*cos(alpha*pi*180**-1) #N
#Max Frictional Force
F=mu_s*R #N
#Total Force along plane
F=P*cos(theta*pi*180**-1)-M*g*sin(alpha*pi*180**-1) #N
#Magntude of Frictional Force
F2=mu_k*R #N
#Result
print"Magnitude of Friction Force",round(F2,2),"N"
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
W=30 #N #Weight acting vertically downward
P2=6 #N #Force at angle of 30 with inclined plane
theta=30 #Degrees #Inclination of force with the inclined plane
mu=0.3 #Coefficient of friction
#Calculations
#Part-1
#Reaction Force is given by
#R=W*cos(alpha)-P2*sin(theta)
#Now resolving Force we get
alpha=np.arcsin(P2*cos(theta*pi*180**-1)*W**-1)
alpha2=180*pi**-1*alpha #degrees
#PArt-2
#resolving Forces normal to inclined plane
R=W*cos(round(alpha2,2)*pi*180**-1) #N
#Resolving Forces normal to inclined plane
P1=W*sin(round(alpha2,2)*pi*180**-1)+mu*round(R,2)
#Result
print"Force required to move a load 30 N up a rough plane is",round(P1,2),"N"
import math
#Declaration Of Variables
W1=400 #N #Weight of first body
W2=800 #N #Weight of second body
mu1=0.15 #Coefficient of friction of first body
mu2=0.40 #Coefficient of friction of seconnd body
#Calculations
#Forces acting on the first body
#Resolving force along plane
#W1*sin(alpha)=T+mu1*R1 ............................1
#Resolving Forces normal to plane
#W1*cos(alpha)=R1
#Sub value of R1 in equation1,we get
#T=400*sin(alpha)-60*cos(alpha) .......................2
#Forces on second body
#Resolving forces along the plane
#W2*sin(alpha)+T=mu2*R2 .........................3
#Resolving forces normal to plane
#R2=W2*cos(alpha)
#sub value of R2 in equation3
#T=320*cos(alpha)-W2*sin(alpha) ...............4
#Equating values of T,given by equation 2 and 3
#W1*sin(alpha)-60*cos(alpha)=320*cos(alpha)-W2*sin(alpha)
#Further simplifying we get
alpha=arctan(380*1200**-1)*(180*pi**-1) #Degrees
#Sub value of alpha in equation 2
T=W1*sin(round(alpha,2)*pi*180**-1)-60*cos(round(alpha,2)*pi*180**-1)
#Result
print"Inclination of the plane to the horizontal is",round(alpha,2),"Degrees"
print"Tension in the cord is",round(T,2),"N"
import math
#Declaration Of Variables
W_B=1500 #N #Weight of block B
mu_A=0.25 #Coefficint of friction of block A
mu_B=0.35 #Coefficient of Friction of block B
alpha=60 #Degrees
#Calculations
#BLock A
#F_A=mu_A*W_A #Force of friction
#Block B
#HOrizontal Force of friction of block A is transmitted through rod to block B
#Force of friction of block B
#F_B=mu2*R_B
#Resolving Horizontal Forces
#mu1*W_A+F_B*cos(alpha)=R_B*cos(30)
#After further simplifying we get
#mu_A*W_A=0.691*R_B ...........................1
#Resolving Forces vertically
#R_B*sin(30)+F_B*sin(alpha)=W_B
#After further simplifying we get
R_B=W_B*0.803**-1 #N
#Sub value of R_b in equation 1 we get
W_A=0.691*R_B*mu_A**-1
#Result
print"Smallest Weight of Block A is",round(W_A,2),"N"
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
W_A=100 #N Weight of block A
W_B=300 #N #Weight of block B
alpha=45 #Degrees #Inclination of plane
phi=30 #Degrees #Inclination of rigid bar with horizontal
mu=tan(15*pi*180**-1) #Degrees
#Calculations
#LEt R_A and R_B be the reactions at A And B respectively and t be the thrust in rod
#Equilibrium of bLoack A
#Resolving forces along plane
#W_A*sin(alpha)+F_A=T*cos(15)
#After further simplifying we get
#70.7+0.2679*R_A=0.969*T .................1
#Resolving force snormal to plane
#R_A=W_A*cos(alpha)+T*sin(15)=R_A
#Now sub value of R_A in equation 1 we get
#70.7+0.269*(100*0.707+T*0.2588)=0.9659*T
#After further simplifying we get
T=89.64*0.8966**-1 #N
#Equilibrium of block B
#Resolving forces normal to plane
R_B=W_B+T*sin(phi*180**-1*pi)
#Resolving forces along plane
P=T*cos(phi*pi*180**-1)+mu*R_B
#Result
print"HOrizontal Force required to be apllied to block B to just move block A is",round(P,2),"N"
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
mu=0.2 #Coefficient of friction
W1=100 #N #weight of Block1
W2=150 #N #Weight of Block2
theta=60 #Degrees
#Calculations
#Case-1
#Reaction Force
R=W2*cos(theta*pi*180**-1) #N
#Tension in the string
T=W2*sin(theta*pi*180**-1)+mu*R #N
#Case-2
theta2=np.arctan(mu)*(pi**-1*180) #Angle made by Force with horizontal acting on block 1
#Force on block with weight 100 N
P=164.9*((cos(theta2*pi*180**-1)+mu*sin(theta2*pi*180**-1))**-1)
#Result
print"Least Value of Force P to cause motion to impend rightwards is",round(P,2),"N"
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
W1=90 #N #Weight of Block 1
W2=30 #N #Weight of Block2
mu=1*3**-1 #Coefficient of friction
#Calculations
#Considering Equilibrium of weight W2
#Tension in the string
#T=W2*sin(theta)+mu*R1 .....................................1
#Normal reaction to the plane
#R1=W2*cos(theta) .......................................2
#Sub value of R1 in equation 1 we get
#T=W2*sin(theta)+10*cos(theta) ........................3
#Considering Equilibrium of weight W1
#Resolving Forces along the plane
#W1*sin(theta)=10*cos(theta)+mu*R2 ..................4
#Resolving Forces normal to plane
#R2=120*cos(theta) ...........5
#Sub value of R2 in equation 4 we get
theta=np.arctan(0.5555)*(pi**-1*180) #Degrees
#Result
print"Value of angle theta should be",round(theta,2),"degrees"
import math
#Declaration Of Variables
L_AC=10 #m #Length of AC
L_BC=8 #m #Length of BC
W=20 #N #weight
#Calculations
L_AB=(L_AC**2-L_BC**2)**0.5 #m #Length of AB
L_CD=L_BC*2**-1 #m
#Now Resolving Forces
#Vertically
#Reaction Force at C
R_C=W #N
#Horizontally
#R_A=F_C=mu*R_C
#Taking Moment at pt C
#Coefficient of friction
mu=W*L_CD*(R_C*L_AB)**-1
#Frictional Force acting at C
F_C=round(mu,2)*R_C #N
#Result
print"Coefficient of Friction",round(mu,2)
print"Frictional Force acting at pt C",round(F_C,2),"N"
import math
#Declaration Of Variables
L_AB=13 #m #Length of AB
W=25 #N #weight of Ladder
L_AC=5 #m #Distance of lower ladder from wall
mu=0.3 #Coefficient of friction
#Calculations
#Forces on the ladder
#Vertical Forces
R_A=W #N
#Horizontal Forces
R_B=F_A=mu*R_A #N
#MAx amount of frictional Force availale at A
F_A #N
L_AD=L_CD=2.5 #m #Length of AD and CD
L_BC=(L_AB**2-L_AC**2)**0.5 #m
#Moment at pt A
R_B2=R_A*L_AD*L_BC**-1 #N
#Horizontal Forces
F_A2=R_B2 #N
#Result
print"Frictional Force acting on ladder is",round(F_A,2),"N"
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
L_AB=14 #m #Length of AB
W=600 #N #weight of Ladder
L_AD=8 #m #Distance of lower ladder from wall
L_BD=6 #m #LEngth of BD
mu=1*3**-1 #Coefficient of friction
CBA=60 #Degrees
#Calculations
#Resolving forces
#Vertically
R_B=W #N
#Actual Force of friction at pt B
F_B=mu*R_B #N
#Horizontally
R_A=F_B #N
L_BE=L_BD*cos(CBA*pi*180**-1) #m
L_AC=L_AB*sin(CBA*pi*180**-1) #m
R_A2=R_B*L_BE*L_AC**-1
F_B2=R_A2 #N
#Result
print"Force available at B the force required force for equilibrium,the ladder will be stable:F_B is",round(F_B,2),"N"
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
W=850 #N #Weight of ladder
L=L_AB=6 #m #Length of AB
alpha=65 #Degrees #Angle made by ladder with ladder
W1=750 #N #Weight of man
L1=4 #m #Distance of man from top of ladder
L2=L-L1 #m #Distance of man form foot of ladder
L_BE=4 #m #Length of BE
#Calculations
#Forces acting on the ladder
#Resolving Forces Vertically
R_A=W+W1 #N
#Horizontally
#R_B=F_A=mu*R_A #N
L_BC=L_AB*sin(alpha*pi*180**-1) #m #Length of BC
L_AC=L_AB*cos(alpha*pi*180**-1) #m #Length of AC
L_AD=L_AC*2**-1 #m #Length of AD
L_AH=(L_AB-L_BE)*cos(alpha*pi*180**-1) #m
#Coefficient of friction
mu=(W1*L_AD+W*L_AH)*(L_BC*R_A)**-1
#Result
print"Coefficient of friction is",round(mu,2)
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
W=200 #N #Weight of ladder
L=L_AB=4.5 #m
mu=0.4 #Coefficient of friction between ladder and floor
mu2=0.2 #coefficient of frictionbetween LAdder and wall
W1=900 #N #Weight on ladder
L_BE=1.2 #m #distance
#Calculations
#FOrces acting on ladder
#Resolving FOrces Vertically
#R_A+F_B=W1+W #N ................1
#Horizontally
#R_B=mu*R_A .....................2
#Resolving Force R_B in equation 1 we get
R_A=(W1+W)*1.08**-1 #N
#Reaction at B
R_B=mu*R_A #N
#Moment at pt A
#W*L_AD+W1*L_AH=R_B*L_BC+F_B*L_AC
#After further simplifying we get
alpha=np.arctan(1.665)*(180*pi**-1)
#Result
print"Angle made by ladder with Horizontal",round(alpha,2),"Degrees"
print"Reaction at the Foot of ladder",round(R_A,2),"N"
print"Reaction at the Foot top of ladder",round(R_B,2),"N"
#Value of alpha is incorrect in book i.e 59degree 65seconds
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
L=5 #m #Length of ladder
alpha=30 #Degrees #Angle made by ladder with horizontal
W1=250 #N #weight of ladder
W2=800 #N #weight of man
mu=0.2 #Coefficinet of friction
L_AG=5*2**-1 #m
#Calculations
#Let R_A and R_B be the reactions at A and b respectively
#F_B=mu*R_B
#F_A=mu*R_A
#Resolving forces vertically
#R_A+F_B=W1+W2 .............1
#Resolving forces horizontally
#R_B=0.2*R_A.....................2
#After sub values and further simplifying we get
R_A=1050*1.04**-1 #N
#Sub value of R_A in equation 2 we get
R_B=0.2*R_A
#Triangle AGD
L_AD=L_AG*cos(60*pi*180**-1)
#TRiangle AEH
#L_AH=x*cos(60)
L_BC=L*cos(alpha*pi*180**-1)
L_AC=L*cos(60*pi*180**-1)
F_B=mu*R_B
#Taking moment at A
#W2*x*2**-1+W*L_AD=R_B*L_BC+F_B*L_AC
#After sub values and further simplifying we get
x=66.276*40**-1
#Result
print"The slipping will be induced at",round(x,2),"m"
import math
from math import sin, cos, tan, radians, pi
import numpy as np
#Declaration Of Variables
alpha=10 #Degrees #Angle of Wedge
W=1500 #N #weight of Block
mu=0.3 #Coefficient of friction
phi=np.arctan(mu)*(180*pi**-1)
#Calculations
#Applying Lamis theorem to the point O
#W*(sin(2*phi+90+alpha))**-1=R3*(sin(180-(alpha+phi)))**-1=R2*(sin(90-phi))**-1
#After further simplifying we get
Y=180-(alpha+phi)
Z=sin(Y*pi*180**-1)
Y1=2*phi+90+alpha
Z1=sin(Y1*pi*180**-1)
R3=W*(Z)*(Z1)**-1 #N
Y2=90-phi
Z2=sin(Y2*pi*180**-1)
R2=W*Z2*Z1**-1 #N
#Applying Lamis theorem to the point L
#R1*sin(90+alpha+phi)**-1=R2*sin(90+phi)**-1=P*sin(180-2*phi-alpha)**-1
#After further simplifying we get
Y3=180-(2*phi+alpha)
Z3=sin(Y3*pi*180**-1)
P=Z3*R2*Z2**-1 #N
#Result
print"Minimum Horizontal force be applied on wedge to raise the block is",round(P,2),"N"
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
D=0.1 #m #Diameter
R=0.05 #m #Radius
N=150 #r.p.m
mu=0.05 #Coefficient of friction
W=15*10**3 #N #Load
#Calculations
#Power Loast in friction assuming uniform pressure
T=2*3**-1*mu*W*R #N*m
#Power Lost in Friction
P=2*pi*N*T*60**-1 #W
#Power Lost in friction assuming wear
T2=0.5*mu*W*R #W
#Power Lost in friction
P2=2*pi*N*T2*60**-1 #W
#Result
print"Power Loast in friction assuming uniform pressure is",round(P,2),"W"
print"Power Lost in friction assuming wear is",round(P2,2),"W"
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
alpha=60 #Degrees
mu=0.05 #m #coefficient of friction
R=0.15 #m #Radius of shaft
W=20*10**3 #N
N=210 #r.p.m
#Calculations
#Frictional Torque
T=2*3**-1*mu*W*R*(sin(alpha*pi*180**-1))**-1 #N*m
#Power Lost in Friction for uniform pressure
P=2*pi*N*T*60**-1*10**-3 #KW
#frictional torque
T2=1*2**-1*mu*W*R*(sin(alpha*pi*180**-1))**-1 #N*m
#Power Loast in friction for uniform wear
P2=2*pi*N*T2*60**-1*10**-3 #KW
#Result
print"Power Lost in Friction assuming:Uniforming pressure",round(P,2),"KW"
print" :Uniform wear",round(P2,2),"KW"
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
W=25*10**3 #N #load
alpha=60 #degrees
p=350*10**3 #N/m**2 #pressure
N=180 #r.p.m
mu=0.05
#r1*2*r2
#Calculations
#From Equation of uniform pressure
r2=(W*(pi*p*3)**-1)**0.5 #m
r1=2*r2 #m
#Frictional Torque
T=2*3**-1*mu*W*(sin(alpha*pi*180**-1))**-1*(r1**3-r2**3)*(r1**2-r2**2)**-1 #m
#Power absorbed in friction
P=2*pi*N*T*60**-1*10**-3 #KW
#Result
print"Power absorbed in friction",round(P,2),"KW"
import math
#Declaration Of Variables
r1=0.25 #m #External Radius
r2=0.15 #m #Internal Radius
W=50*10**3 #N #Total axial load
mu=0.05 #Coefficient of friction
N=150 #r.p.m
#Calculations
#Torque
T=2*3**-1*mu*W*((r1**3-r2**3)*(r1**2-r2**2)**-1) #N*m
#Power lost in Frction
P=2*pi*N*T*60**-1*10**-3 #KW
#Result
print"Power lost in Frction is",round(P,2),"KN"
import math
from math import sin, cos, tan, radians, pi
#Declaration Of Variables
r1=0.21 #m #External Radius
r2=0.16 #m #Internal Radius
W=60*10**3 #N #Total axial load
mu=0.05 #Coefficient of friction
N=380 #r.p.m
p=350*10**3 #N/m**2 #Intensity of pressure
#Calculations
#Power Loast in Overcoming friction
#Torque
T=2*3**-1*mu*W*((r1**3-r2**3)*(r1**2-r2**2)**-1) #N*m
P=2*pi*N*T*60**-1*10**-3 #KW
#Number of collars required
#Load per collar
W2=p*pi*(r1**2-r2**2)
n=W*W2**-1
#Result
print"Power Loast in Overcoming friction is",round(P,2),"KW"
print"Number of collars required for the thrust",round(n,1)