Chapter 15:Kinetics of Rigid Bodies And Laws of Motion

Example 15.1,Page No.536

In [2]:
import math

#Initilization of Variables

M=150 #kg #Mass of the Body
a=3 #m/s**2 #Acceleration 


#Calculation

#Force 
F=M*a #N

#Result
print"FOrce is",round(F,2),"N"

FOrce is 450.0 N

Example 15.2,Page No.537

In [3]:
import math

#Initilization of Variables

F=100 #N #Force
m=4 #kg #mass
t=10 #seconds #time
u=5 #m/s #Initial Velocity

#Calculation

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

#Distance
s=u*t+a*t**2*2**-1 #m

#Result
print"Acceleration is",round(a,2),"m/s**2"
print"Distance moved is",round(s,2),"m"

Acceleration is 25.0 m/s**2
Distance moved is 1300.0 m

Example 15.3,Page No.537

In [4]:
import math

#Initilization of Variables

W=980 #N #Weight of body on the earth
g=9.80 #m/s**2 #Acceleration due to gravity
g2=1.6 #m/s**2 #Acceleration due to gravity on moon
g3=270 #m/s**2 #Acceleration due to gravity on sun

#Calculation

#Mass
m=W*g**-1 #Kg

#Weight of body on moon
W1=m*g2 #N

#Weight of body on sun
W2=m*g3 #N

#Result
print"Weight of body on moon",round(W1,2),"N"
print"Weight of body on sun",round(W2,2),"N"

Weight of body on moon 160.0 N
Weight of body on sun 27000.0 N

Example 15.4,Page No.537

In [5]:
import math

#Initilization of Variables

F=200 #N #Force 
m=300 #kg #mass
t=90 #sec #time
u=20 #m/s #Initial velocity

#Calculation

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

#Final Velocity in Direction of motion 
v=u+a*t #m/s

#Final Velocity in opposite direction of motion
v2=u-a*t #m/s

#Result
print"Final Velocity in Direction of motion",round(v,2),"m/s"
print"Final Velocity in opposite direction of motion",round(v2,2),"m/s"

Final Velocity in Direction of motion 80.0 m/s
Final Velocity in opposite direction of motion -40.0 m/s

Example 15.5,Page No.538

In [7]:
import math

#Initilization of Variables

m=15 #kg #Mass 
h=19.6 #m #Height of body from ground
F=4900 #N #Force of resistance
g=9.80 #m/s**2 #Acceleration due to gravity

#Calculation

#Final Velocity of body
v=(2*g*h)**0.5 #m/s

#Weight of body
W=m*g #N

#Net Force acting in the upward direction
F2=F-W #N

#Acceleration
a=F2*m**-1 #m/s**2 

#Part-2

v2=0 #Final Velocity after penetration into the ground
u=v #Initial Velocity on the ground

#Distance penetrated into the ground
s=-(v2**2-u**2)*(2*a)**-1 #m 

#Result
print"Distance penetrated into the ground",round(s,3),"m"

Distance penetrated into the ground 0.606 m

Example 15.6,Page No.539

In [8]:
import math

#Initilization of Variables

W=637 #N #Weight of man
h=19.6 #m #Height of Tower
u=0 #m/s #Initial Velocity of man when he reaches the water surface
g=9.8 #m/s**2 #acceleration due to gravity
s=2 #m #Distance travelled

#Calculation

#Final Velocity of man when he reaches the water surface
v=(2*g*h)**0.5 #m/s

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

#Mass of man
m=W*g**-1 #Kg 

#Average resistance of water
F=m*a+W #N

#Result
print"Average Resistance of water",round(F,2),"N"

Average Resistance of water 6879.6 N

Example 15.7,Page No.540

In [9]:
import math

#Initilization of Variables

m=0.081 #kg
v=300 #m/s #velocity
s=0.1 #m #Depth
s2=0.05 #m #Distance travelled

#Calculation

#Acceleration 
a=v**2*(2*s)**-1 #m/s**2 

#Force offered by wood to the bullet
F=m*a #N

#Velocity
v=-(u**2-(2*a*s2)) #m/s
v2=v**0.5 #m/s

#Result
print"Force of resistance",round(v2,2),"m/s"

Force of resistance 212.13 m/s

Example 15.8,Page No.541

In [1]:
import math

#Initilization of Variables

v=0 #Final Velocity
s=60 #m #Distance travelled
mu=0.4 #coefficient of friction
g=9.80


#Calculation

#acceleration
a=mu*g #m/s**2

#speed of car
u=(2*a*s)**0.5*1000**-1*3600 #m/s

#Result
print"Speed of car is",round(u,2),"Km/hr"

Speed of car is 78.08 Km/hr

Example 15.9,Page No.542

In [11]:
import math

#Initilization of Variables

F1=2000 #N #Tractive of force exerted by railway car
W=50 #KN #Weight of car
g=9.81 #m/s**2 #acceleration due to gravity 

#Calculation

#mass of car
m=W*1000*g**-1 #N

#Frictional resistance
F2=5*W

#Net Force in Direction of motion
F=F1-F2 #N

#Acceleration
a=F*m**-1

#Result
print"acceleration when the car is moving",round(a,2),"m/s**2"

acceleration when the car is moving 0.34 m/s**2

Example 15.10,Page No.542

In [12]:
import math

#Initilization of Variables

W=1960*1000 #N #Weight of train
g=9.80 #m/s**2 #Acceleration due to gravity

#Calculation

#mass of train 
m=W*g**-1 #kg

#final Velocity
v=100*3**-1 #m/s
t=5*60 #sec

#Acceleration
a=v*t**-1 #m/s**2 

#Average pull required
F2=m*a+19600


#Result
print"Average pull required",round(F2,2),"N"

Average pull required 41822.22 N

Example 15.11,Page No.544

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

#Initilization of Variables

W=200 #N #Weight of Body
g=9.81 #m/s**2 #aceleration due to gravity
theta=45 #Degrees #Angle of plane
u=0 #m/s #Initial Velocity
v=2 #m/s #Final velocity
mu=0.1 #coefficient of friction

#Calculation

#Acceleration of body
a=g*(sin(theta*pi*180**-1)-mu*cos(theta*pi*180**-1)) #m/s**2

#Distance
s=(v**2-u**2)*(2*a)**-1 #m

#Result
print"Distance along inclined plane is",round(s,2),"m"

Distance along inclined plane is 0.32 m

Example 15.12,Page No.544

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

#Initilization of Variables

u=0 #m/s #Initial Velocity
theta=20 #degree #Angle of inclination
mu1=0.08 #Coefficient of friction between the plane and lower body
mu2=0.08 #Coefficient of friction between the plane and upper body
d=10 #m #distance beween two body
g=9.81 #m/s**2 #Acceleration due to gravity

#Calculation

#Acceleration of lower body down the plane
a1=g*(sin(theta*pi*180**-1)-mu1*cos(theta*pi*180**-1)) #m/s**2

#Acceleration of upper body 
a2=g*(sin(theta*pi*180**-1)-mu2*cos(theta*pi*180**-1)) #m/s**2

#Distance travelled by lower body
#s1=u*t+a1*t**2*2**-1
#After sub values and further simplifying we get
#s1=1.3805*t**2    ...................1

#Distance travelled by upper body
#s1=u*t+a1*t**2*2**-1
#After sub values and further simplifying we get
#s1=1.447*t**2    .......................2

#Further simplfying we get
t=(10*0.1385**-1)**0.5 #s

#sub value of t in equation 1 and 2
s1=1.3805*round(t,2)**2 #m
s2=1.447*round(t,2)**2 #m

#Result
print"distance through which each body travels before they meet:s1",round(s1,2),"m"
print"                                                         :s2",round(s2,2),"m"

#Answer of s1 is incorrect in book

distance through which each body travels before they meet:s1 99.74 m
                                                         :s2 104.55 m

Example 15.13,Page No.546

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

#Initilization of Variables

W=6000 #N #Weight of truck
u=10 #m/s #speed of truck
#sin(theta)=1*40**-1

#Calculation

#Road Resistance
F1=W*1*40**-1 #N

#Frictional Force
F2=F1*(W*10**-3)**-1

#PArt-2

#Speed of truck
u2=2*u #m/s

#Force exerted by engine up theplane
P=W*1*40**-1+F1 #N

#Power Exerted by engine
P2=P*u2*1000**-1 #KW

#Result
print"Frictional Force of truck is",round(F2,2),"N"
print"Power Exerted by engine is",round(P2,2),"KW"

Frictional Force of truck is 25.0 N
Power Exerted by engine is 6.0 KW

Example 15.14,Page No.547

In [16]:
import math

#Initilization of Variables

W=200*10**3 #N #Weight of train
#sin(theta)=1*150**-1 #Slope of track
u=5 #m/s #speed of train
p=3.5 #KW #Power developed by engine

#Calculation

#Case-1

#power developed by engine
P=p*1000*u**-1 #N

#Net Force 
F=W*1*150**-1+P #N

#Case-2

#Force exerted by engine while moving up
P2=W*1*150**-1+F #N

#Power developed by engine
P3=P2*u*1000**-1 #KW

#Result
print"Power Developed by Engine to pull up the train is",round(P3,2),"KW"

#Answer of Power developed by engine is incorrect i.e P so answer of Power Developed by Engine to pull up the train is also incorrect i.e P3

Power Developed by Engine to pull up the train is 16.83 KW

Example 15.15,Page No.549

In [17]:
import math

#Initilization of Variables

L_BC=100 #m #Distance
V=20*3**-1 #m/s #Velocity
W=20000 #N #Weight
g=9.81 #m/s**2 #acceleration due to gravity
m=W*g**-1 #Mass of car
#sin(theta)=5*100**-1

#Calculation

#Frictional resistance due to track
F=8*20 #N

#Final Velocity of car at D
v=0 

#Component of weight of train
W2=W*5*100**-1 #N

#Total Retarding Force against motion
F2=F+W2 #N

#acceleration
a=F2*g*W**-1 #m/s**2

#Distance
s=V**2*(2*round(a,3))**-1 #m

#PArt-2

#Dstance travelled by car From B to E

#Distance BD
s_BD=s+L_BC #m

F3=840 #N #Net Force down the grade

#Acceleration 
a2=F3*g*W**-1 #m/s**2

#Velocity
v2=(2*a2*s_BD)**0.5 #m/s

#PArt 3

#Motion From B to E

#acceleration
a3=F*g*W**-1 #m/s**2

#Initial velocity at B
u_B=10.70 #m/s

#Distance
s2=u_B**2*(2*round(a3,3))**-1 #m

#Result
print"Distance travelled by car before stopping is",round(s,2),"m"
print"Distance travelled by car beyond Bon level track before stopping at E",round(s2,2),"m"

Distance travelled by car before stopping is 39.05 m
Distance travelled by car beyond Bon level track before stopping at E 733.91 m

Example 15.16,Page No.552

In [18]:
import math

#Initilization of Variables

W=100 #N #Weight carried by lift
a=2.45 #m/s**2 #Acceleration
g=9.80 #m/s**2 #Acceleration due to gravity

#Calculation

#Tension in the cables supporting the lift

#Lift moving upwards
T=W*(1+a*g**-1) #N

#Lift moving downwards
T2=W*(1-a*g**-1) #N

#Result
print"Tension in the cables supporting the lift:when moving upwards",round(T,2),"N"
print"Tension in the cables supporting the lift:when moving downward",round(T2,2),"N"

Tension in the cables supporting the lift:when moving upwards 125.0 N
Tension in the cables supporting the lift:when moving downward 75.0 N

Example 15.16(A),Page No.553

In [19]:
import math

#Initilization of Variables

a=1 #m/s**2 #upward acceleration
W=600 #N #weight of man
g=9.81 #m/s**2 #Acceleration due to gravity

#Calculation

#NEt Force in upward direction
F=W*g**-1 #N

#But net Force in upward direction
T=F+W #N

#Result
print"net Force in upward direction",round(T,2),"N"

net Force in upward direction 661.16 N

Example 15.17,Page No.553

In [20]:
import math

#Initilization of Variables

a=1.225 #m/s**2 #upward acceleration
W=500 #N #Weight of man
g=9.8 #m/s**2 #Acceleration due to gravity

#Calculation

#Tension in the cables supporting the lift

#Lift moving upwards
T=W*(1+a*g**-1) #N

#lift moving downwards
T2=W*(1-a*g**-1) #N

#Lift moving upwards with unknown acceleration
T3=600 #N #Pressure exerted by man
a=(T3-W)*g*W**-1 #m/s**2

#Result
print"Acceleration upwards is",round(a,2),"m/s**2"

Acceleration upwards is 1.96 m/s**2

Example 15.18,Page No.554

In [21]:
import math

#Initilization of Variables

W=2500 #N #Weight of an elevator
u=0 #m/s #Initial Velocity
s=35 #m #Distance travelled
t=10 #sec #time
g=9.81 #m/s**2 #Acceleration due to gravity

#Calculation

#Tension in the cables

#When acceleration is zero i.e a=0
a=0 #m/s**2
T=W*(1-a*g**-1) #N

#When acceleration is zero i.e a=0
a2=9.81 #m/s**2
T2=W*(1-a2*g**-1) #N

#Using equation of distance
a3=(s-u*t)*2*(t**2)**-1 #m/s**2

#Tension in the cable at time t=10 sec
T3=W*(1-a3*g**-1) #N

#Result
print"Limits of table Tension is:when a=0",round(T,2),"N"
print"                          :when a=9.81 m/s**2",round(T2,2),"N"
print"Cable Tension at time t=10 sec",round(T3,2),"N"

Limits of table Tension is:when a=0 2500.0 N
                          :when a=9.81 m/s**2 0.0 N
Cable Tension at time t=10 sec 2321.61 N

Example 15.19,Page No.555

In [22]:
import math

#Initilization of Variables

g=9.80 #m/s**2 #Acceleration due to gravity
W=5000 #N #Weight of 10 men on the cage
u=0 #m/s #Initial Velocity of cage
v=12 #m/s #Final Velocity
s=20 #m #Distance travelled

#Calculation

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

#Tension in cable while moving downwards
T=W*(1-a*g**-1) #N

#Tension produced by one men
T2=T*10**-1 #N

#Result
print"Pressure Exerted by each man on the cage",round(T2,2),"N"

Pressure Exerted by each man on the cage 316.33 N

Example 15.20,Page No.556

In [4]:
import math

#Initilization of Variables

W=5000 #N #Weight of elevator
a=3 #m/s**2 #Acceleration
W2=700 #N #weight of perator 
g=9.80

#Calculation

#Reaction offered by floor on operator
R=W*g**-1*a+W2 #N

#Total Weight 
W3=W+W2 #N

#Total Tension in the cable
T=W3*g**-1*a+W3 #N

#Result
print"Total tension in the cable",round(T,2),"N"

Total tension in the cable 7444.9 N

Example 15.21,Page No.558

In [24]:
import math

#Initilization of Variables

W1=50 #N #Heavier Weight
W2=30 #N #lighter Weight
g=9.80 #m/s**2 #Acceleration due to gravity

#Calculation

#Acceleration of the system
a=g*(W1-W2)*(W1+W2)**-1 #m/s**2 

#Tension in the string
T=2*W1*W2*(W1+W2)**-1 #N

#Result
print"Acceleration of the system",round(a,2),"m/s**2"
print"Tension in the string",round(T,2),"N"

Acceleration of the system 2.45 m/s**2
Tension in the string 37.5 N

Example 15.22,Page No.558

In [25]:
import math

#Initilization of Variables

g=9.80 #m/s**2 #Acceleration due to gravity
W1=60 #N #bigger weight
a=3 #m/s**2 #Acceleration of the system

#Calculation

#smaller Weight
W2=-(a*W1*g**-1-W1)*(a*g**-1+1)**-1 #N

#Tension in the string
T=2*W1*W2*(W1+W2)**-1 #N

#Result
print"Smaller Weight is",round(W2,2),"N"
print"Tension in the string",round(T,2),"N"

Smaller Weight is 31.88 N
Tension in the string 41.63 N

Example 15.23,Page No.559

In [26]:
import math

#Initilization of Variables

W1=700 #N #Bigger Load
W2=500 #N #Smaller Load

#Calculation

#Weight of block A when acceleration is g/3
W1_1=((3*W2)+W2)*2**-1 #N

#Weight added 
W=W1_1-W1 #N

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

Weight added is 300.0 N

Example 15.24,Page No.560

In [27]:
import math

#Initilization of Variables

W_A=150 #N #Weight of block A
W_B=50 #N #Weight of block B
g=9.81 #m/s**2 #Acceleration due to gravity

#Calculation

#Acceleration
a=(W_A-W_B*2)*((W_B*2+(W_A*2**-1))*g**-1)**-1 #m/s**2

#Acceleration of block B
a_B=a #m/s**2

#Acceleration of block A
a_A=a*2**-1 #m/s**2 

#Tensions in the string
T=W_B+W_B*g**-1*a #N

#Result
print"Tensions in the string is",round(T,2),"N"
print"Acceleration of block B is",round(a_B,2),"m/s**2"
print"Acceleration of block A is",round(a_A,2),"m/s**2"

Tensions in the string is 64.29 N
Acceleration of block B is 2.8 m/s**2
Acceleration of block A is 1.4 m/s**2

Example 15.25,Page No.561

In [28]:
import math

#Initilization of Variables

W1=15 #N #weight over pulley A
W2=10 #N #total weight over pulley B
w1=6 #N #weight over pulley B
w2=4 #N #weight over pulley B
g=9.80 #m/s**2 #acceleration due to gravity

#Calculation

#Consider motion of weight 15 N
#(W1-T1)=W1*g**-1*a               ..........................(1)

#Consider motion of weight 4 N
#(T2-w2)=w2*g**-1*(a1+a)                 ...................(2)

#Consider motion of weight 6 N
#(w1-w2)=w1*g**-1*(a1-a)            .........................(3)

#Consider motion of pulley B
#T1=2*T2                       ...............................(4)

#Adding equations 2 and 3 we get
#g=5*a1-a    .......................................(5)

#Multiplying equation (2) by 2
#2*T2-8=8*g**-1*(a1+a)

#But sub value 2*T2=T1 in equation above
#T1-8=8*g**-1*(a1+a)   .......................................(6)

#Adding equation 1 and 6
#7*g=23*a+8*a1 .......................................(7)

#Multiplying equation (5) by 23
#23*g=-23*a+5*23*a1   .......................................(8)

#Adding equation 7 and 8 we get
a1=30*g*123**-1 #m/s**2

#sub value of equation 5 we get
a=5*a1-g #m/s**2

#Acceleration of weight 15 N
a_15=a #m/s**2

#Acceleration of weight 6 N
a_6=a1-a #m/s**2

#Acceleration of weight 4 N
a_4=a1+a #m/s**2 

#Result
print"Acceleration of weight 15 N",round(a_15,2),"m/s**2"
print"Acceleration of weight 6 N",round(a_6,2),"m/s**2"
print"Acceleration of weight 4 N",round(a_4,2),"m/s**2"

Acceleration of weight 15 N 2.15 m/s**2
Acceleration of weight 6 N 0.24 m/s**2
Acceleration of weight 4 N 4.54 m/s**2

Example 15.26,Page No.565

In [29]:
import math

#Initilization of Variables

#Weight of bodies
W1=10 #N  #Weight placed on Horizontal surface 
W2=20 #N #Weight hanging free in air
g=9.81 #m/s**2 #Acceleration due to gravity

#Calculation

#Acceleration of the system
a=g*W1*(W1+W2)**-1 #m/s**2

#tension in the string
T=W1*W2*(W1+W2)**-1 #N

#Result
print"Acceleration of the system",round(a,2),"m/s**2"
print"Tension in the string",round(T,2),"N"

Acceleration of the system 3.27 m/s**2
Tension in the string 6.67 N

Example 15.27,Page No.565

In [1]:
import math

#Initilization of Variables

W1=10 #N #Weight on horizontal surface
mu=0.3 #coefficient of friction
W2=20 #N #Weight hanging free in air
g=9.80

#Calculation

#Acceleration of the system
a=g*(W1-mu*W2)*(W1+W2)**-1 #m/s**2

#tension in the string
T=W1*W2*(1+mu)*(W1+W2)**-1 #N


#Result
print"Acceleration of the system",round(a,2),"m/s**2"
print"Tension in the string",round(T,2),"N"

Acceleration of the system 1.31 m/s**2
Tension in the string 8.67 N

Example 15.28,Page No.566

In [2]:
import math

#Initilization of Variables

W1=10 #N #Weight of block A
W2=20 #N #weight of block B
mu=0.25 #coefficient of friction
s=2 #m #Distance moved by block
u=0 #Initial velocity of block B
g=9.80

#Calculation

#Acceleration
a=g*(W1-mu*W2)*(W1+W2)**-1 #m/s**2

#velocity of block B
v=u**2+2*a*s #m/s

#Result
print"velocity of block B",round(v,2),"m/s"

velocity of block B 6.53 m/s

Example 15.29,Page No.566

In [3]:
import math

#Initilization of Variables

W2=10 #Weight placed on rough horizontal surface
W1_1=1.5 #N #Weight hanging free in air
W1=0.5 #N #additional weight added 
T=1.5 #N #Tension in the string
R=10 #N #Normal Reaction
g=9.80

#Calculation

#Total Weight hanging in air
W=W1_1+W1

#Max Frictional Force 
F=T=1.5 #N

#Coefficient of friction 
mu=F*R**-1

#Acceleration of two weights
a=g*(W-mu*W2)*(W+W2)**-1 #m/s**2 

#Tension in the string
T1=W*W2*(1+mu)*(W+W2)**-1 #N

#Result
print"Acceleration of two weights is",round(a,3),"m/s**2"
print"Tension in the string",round(T1,3),"N"

Acceleration of two weights is 0.408 m/s**2
Tension in the string 1.917 N

Example 15.30,Page No.568

In [4]:
import math

#Initilization of Variables

W2=1500 #N #weight of body A
W1=1000 #N #Weight of body B
g=9.80

#Coefficient of friction
mu=mu1=mu2=0.2

#T1=1.3691*T2

#Calculation

#Acceleration
a=(W1-1.3691*mu*W2)*((W1*g**-1)+1.3691*W2*g**-1)**-1 #m/s**2

#Tension in the string to which weight 1500 N is attached
T2=W2*g**-1*round(a,2)+mu*W2 #N

#Tension in the string to which weight 1000 N is attached
T1=1.3691*round(T2,3)

#Result
print"Acceleration of the systems is",round(a,2),"m/s**2"
print"Tension in the string:T1",round(T1,2),"N"
print"                     :T2",round(T2,2),"N"

#Answer for T2 is incorrect in the book

Acceleration of the systems is 1.89 m/s**2
Tension in the string:T1 806.79 N
                     :T2 589.29 N

Example 15.31,Page No.573

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

#Initilization of Variables

W1=15 #N #Weight of hanging free in air
W2=40 #N #weight placed on inclined plane
theta=15 #degree #Inclination
g=9.80 #m/s**2 #Acceleration due to gravity

#Calculation

#Acceleration
a=g*(W1-W2*sin(theta*pi*180**-1))*(W1+W2)**-1 #m/s**2
     
#Tension in string
T=W1*W2*(1+sin(theta*pi*180**-1))*(W1+W2)**-1 #N

#Result
print"Acceleration is",round(a,2),"m/s**2"
print"Tension in the string is",round(T,2),"N"

Acceleration is 0.83 m/s**2
Tension in the string is 13.73 N

Example 15.32,Page No.573

In [6]:
import math
from math import sin, cos, tan, radians, pi

#Initilization of Variables

W1=25 #N #Weight of hanging free in air
W2=40 #N #weight placed on inclined plane
theta=15 #degree #Inclination
g=9.80 #m/s**2 #Acceleration due to gravity
mu=0.2 #coefficient of friction

#Calculation

#Acceleration
a=g*(W1-W2*sin(theta*pi*180**-1)-mu*W2*cos(theta*pi*180**-1))*(W1+W2)**-1 #m/s**2

#Tension
T=W1*W2*(1+sin(theta*pi*180**-1)+mu*cos(theta*pi*180**-1))*(W1+W2)**-1

#Distance
u=0 #m/s
t=3 #sec
s=u*t+a*t**2*2**-1 #m

#Result
print"Acceleration is",round(a,2),"m/s**2"
print"Tension is",round(T,2),"N"
print"Distance moved by 25 N is",round(s,2),"m"

Acceleration is 1.04 m/s**2
Tension is 22.34 N
Distance moved by 25 N is 4.69 m

Example 15.33,Page No.578

In [36]:
import math
from math import sin, cos, tan, radians, pi

#Initilization of Variables

#For Circular Lamina
D=60 #cm
m=0.001 #kg/cm**2 #mass per unit area

#For circular cyclinder
D2=80 #cm
h=15 #cm #height
m2=0.002 #kg/cm**3

#For solid sphere
D3=40 #cm
m3=0.0015 #kg/cm**3

#Calculation

#For Circular Lamina

#Radius
R=D*2**-1 #cm

#Total Mass
M=m*pi*R**2 #kg

#Moment of Inertia of circular section
I_zz=M*R**2*2**-1 #Kg/cm**2

#Radius of Gyration For circular section
k=R*((2)**0.5)**-1 #cm

#For circular cyclinder

#Radius
R2=D2*2**-1 #cm

#Total Mass
M2=m2*pi*R2**2*h #kg

#Moment of Inertia of circular section
I_zz2=M2*R2**2*2**-1 #Kg/cm**2

#Radius of Gyration For circular section
k2=R2*((2)**0.5)**-1 #cm

#For solid sphere

#Radius
R3=D3*2**-1 #cm

#Total Mass
M3=m3*4*pi*R3**3*3**-1 #kg

#Moment of Inertia of circular section
I_zz3=2*5**-1*M3*R3**2 #Kg/cm**2

#Radius of Gyration For circular section
k3=R3*0.6324 #cm

#Result
print"M.I of Circular Lamina is",round(I_zz,2),"Kg/cm**2"
print"Radius of gyration of Circular Lamina is",round(k,2),"cm"

print"M.I of circular cyclinder is",round(I_zz2,2),"Kg/cm**2"
print"Radius of gyration of circular cyclinder is",round(k2,2),"cm"


print"M.I of solid sphere is",round(I_zz3,2),"Kg/cm**2"
print"Radius of gyration of solid sphere is",round(k3,2),"cm"

M.I of Circular Lamina is 1272.35 Kg/cm**2
Radius of gyration of Circular Lamina is 21.21 cm
M.I of circular cyclinder is 120637.16 Kg/cm**2
Radius of gyration of circular cyclinder is 28.28 cm
M.I of solid sphere is 8042.48 Kg/cm**2
Radius of gyration of solid sphere is 12.65 cm

Example 15.34,Page No.579

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

#Initilization of Variables

D=90 #cm #Diameter of grindstone
t=10 #cm #Thickness
m=0.0026 #kg/cm**3 #Mass per unit volume

#Calculation

#Radius
R=D*2**-1 #cm

#Total Mass
M=m*pi*R**2*t #kg

#M.I of of grindstone
I_zz=M*R**2*2**-1 #Kg/cm**2

#Radius of gyration
k=R*((2)**0.5)**-1 #cm

#Result
print"M.I of Grindstone is",round(I_zz,2),"Kg/cm**2"
print"Radius of gyration is",round(k,2),"cm"

M.I of Grindstone is 167472.41 Kg/cm**2
Radius of gyration is 31.82 cm

Example 15.35,Page No.581

In [38]:
import math
from math import sin, cos, tan, radians, pi

#Initilization of Variables

W=1000 #N #Weight of flying wheel
k=0.5 #m
T=1200 #N*m #Torque
g=9.80 #m/s**2

#Calculation

#MAss of flywheel
M=W*g**-1 #Kg

#Moment of Inertia
I=M*k**2 #Kg/m**2

#Angular Acceleration
alpha=T*I**-1 #radians/s**2

#Result
print"Angular Acceleration of flywheel is",round(alpha,2),"radians/s**2"

Angular Acceleration of flywheel is 47.04 radians/s**2

Example 15.35(A),Page No.582

In [39]:
import math

#Initilization of Variables

I=12 #Kg*m**2 #M.I of circular disc
t=3 #s #Time
T=800 #N*m #Torque
w_o=0 #m/s #Angular velocity initially

#Calculation

alpha=T*I**-1 #radians/s**2

#Angular velocity after 3 seconds
w=w_o+alpha*t #rad/s

#Result
print"angular Velocity after 3 seconds",round(w,2),"rad/s"

angular Velocity after 3 seconds 200.0 rad/s

Example 15.36,Page No.582

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

#Initilization of Variables

M=5000 #kg #Mass of flywheel
k=1 #m #radius of gyration 
N_o=400 #r.p.m #Initial Velocity
N=280 #r.p.m #Final speed
t=120 #seconds #Time

#Calculation

#Initial Angular velocity
w_o=2*pi*N_o*60**-1 #rad/s

#Final Angular velocity
w=2*pi*N*60**-1 #rad/s 

#M.I
I=M*k**2 #kg/m**2

#Angular acceleration
alpha=(w-w_o)*t**-1 #rad/s**2

#Torque
T=-M*alpha #N*m

#Final K.E
E2=round(w,2)**2*I*2**-1 #N*m

#Initial K.E
E1=41.88**2*I*2**-1 #N*m

#Change in K.E
E=E2-E1 #N*m

#Initial Momentum
p1=I*round(w,2) #N*m/s

#Final Momentum
p2=I*41.88 #N*m/s

#Change in angular Momentum
p=p2-p1 #N*m/s

#Result
print p1
print"Retading Torque acting is",round(T,2),"N*m"
print"Change in K.E is",round(E,2),"N*m"
print"Change in Angular Momentum is",round(p,2),"N*m/s"

146600.0
Retading Torque acting is 523.6 N*m
Change in K.E is -2235680.0 N*m
Change in Angular Momentum is 62800.0 N*m/s

Example 15.37,Page No.583

In [41]:
import math

#Initilization of Variables

V=0.2 #m/s #Linear Velocity
W=0.1 #N #Weight of cyclinder
R=0.1 #m #Radius
g=9.81 #m/s**2 #Acceleration due to gravity

#Calculation

#Mass 
M=W*g**-1 #Mass

#M.I
I=M*R**2*2**-1 

#Angular Velocity
w=V*R**-1 #rad/s

#Total K.E
E=(I*w**2+M*V**2)*2**-1 #N*m

#Result
print"Total Kinetic Energy is",round(E,6),"N*m"

Total Kinetic Energy is 0.000306 N*m

Example 15.38,Page No.584

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

#Initilization of Variables

k=0.5 #m #Radius of Gyration
W=6000 #N #Weight of flywheel
N_o=0 #Initial r.p.m
N=200 #Final r.p.m  
t=120 #seconds
g=9.80 #m/s**2

#Calculation

#MAss
M=W*g**-1 #Kg

#Initial Angular Velocity
w_o=2*pi*N_o*60**-1 

#Final Angular Velocity
w=2*pi*N*60**-1 #rad/s

#M.I
I=M*k**2 

#Angular acceleration
alpha=(w-w_o)*t**-1 #rad/s**2

#Torque Exerted
T=I*alpha #N*m

#Result
print"Average Torque exerted is",round(T,2),"N*m"

#Answer for M.I is incorrect so value of Torque in book is incorrect

Average Torque exerted is 26.71 N*m

Example 15.38(A),Page No.585

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

#Initilization of Variables

#weights
W_A=100 #N
W_B=180 #N

#Radii
r_A=0.1 #m 
r_B=0.15 #m

#RAdius of gyration
k_A=0.08 #m 
k_B=0.13 #m

theta=30 #degree #Inclination of plane
g=9.81 #m/s**2 #Acceleration due to gravity

#Calculation

#Wheel A

#Angular acceleration 
alpha_A=(W_A*sin(theta*pi*180**-1)*r_A)*(W_A*g**-1*(k_A**2+r_A**2))**-1 #rad/s

#Linear acc. of wheel 
alpha_A_a=alpha_A*r_A #m/s**2

#Wheel B

#Angular acceleration
alpha_B=W_B*sin(theta*pi*180**-1)*r_B*((W_B*g**-1*(k_B**2+r_B**2)))**-1

#Linear acc. of wheel 
alpha_B_b=alpha_B*r_B #m/s**2

#Acceleration of A with respect to B
a_A_B=-(round(alpha_B_b,2)-round(alpha_A_a,2))

#Result
print"Acceleration of A with respect to B",a_A_B,"m/s**2"

Acceleration of A with respect to B 0.19 m/s**2

Example 15.39,Page No.589

In [44]:
import math

#Initilization of Variables

W=5 #N #Weight suspended by arope
W_o=50 #N #Weight of pulley
R=0.3 #m #Radius of pulley
g=9.81 #m/s**2

#Calculation

#Acceleration
a=g*W*(W+W_o*2**-1)**-1 #m/s**2

#Tension in the string
T=W*W_o*(2*W+W_o)**-1 #N

#Result
print"Acceleration is",round(a,2),"m/s**2"
print"Tension in the string is",round(T,2),"N"

Acceleration is 1.64 m/s**2
Tension in the string is 4.17 N

Example 15.40,Page No.590

In [11]:
import math

#Initilization of Variables

W1=100 #N #bigger Weight
W2=40 #N #Smaller Weight
W_o=50 #N #Weight of pulley
g=9.80

#Calculation

#Acceleration 
a=g*(W1-W2)*(W1+W2+W_o*2**-1)**-1 #m/s**2

#TEnsion T1
T1=W1*(2*W2+W_o*2**-1)*(W1+W2+W_o*2**-1)**-1 #N

#Tension T2
T2=W2*(2*W1+W_o*2**-1)*(W1+W2+W_o*2**-1)**-1 #N

#Result
print"Acceleration of block is",round(a,2),"m/s**2"
print"Tension T1",round(T1,2),"N"
print"Tension T2",round(T2,2),"N"

Acceleration of block is 3.56 m/s**2
Tension T1 63.64 N
Tension T2 54.55 N

Example 15.41,Page No.591

In [46]:
import math

#Initilization of Variables

W=2940 #N #Weight of cage
D=1.20 #m #Diameter of drum
R=0.6 #m #Radius of drum
W_o=735 #N #Weight of drum
k=0.55 #m #Radius of gyration
T1=4000 #N*m #Constant Torque Exerted by motor
g=9.8 #m/s**2 #Acceleration due to gravity

#Calculation


#Mass of the cage
M=W*g**-1 #Kg

#Net Force
#(P-W)=M*a    ........................1

#M.I of Drum
I=W_o*g**-1*k**2 

#Torque on Drum
#(T1-R*P)=I*alpha    ..................(2)

#Angular Acceleration
#alpha=(a*R**-1)

#Sub value of I and alpha in equation 2 we get
#(4000-0.6*P)=37.8125*a   ...................3

#After multipliying equation 1 by R we get
#(R*P-R*W)=R*M*a   .............................4

#Adding equations 3 and 4 we get equation as
#(4000-0.6*2940)=37.8125*a+0.6*300*a
#AFter further simplifying we get
a=2236*(217.8125)**-1 #m/s**2 #Acceleration

#Sub value of a in equation 1 
P=M*a+W #N

#PArt-2
#Time Required to raise the cage 20 m from ground

u=0 #m/s #Initial Velocity
a=10.265 #m/s**2 #acceleraation
s=20 #m #Height from ground

#time required
t=((2*s)*a**-1)**0.5 #s

#Result
print"Acceleration of the cage is",round(a,2),"m/s**2"
print"Tension in the cage is",round(P,2),"N"
print"Time required to raise the cage 20 m from ground",round(t,2),"s"

Acceleration of the cage is 10.27 m/s**2
Tension in the cage is 6019.71 N
Time required to raise the cage 20 m from ground 1.97 s

Example 15.42,Page No.593

In [47]:
import math

#Initilization of Variables

W_o=100 #N #Weight of cyclinder
W=10 #N #Weight of block
D=1 #m #diameter of cyclinder
R=0.5 #m #Radius of cyclinder
g=9.80 #m/s**2 #Acceleration due to gravity

#Calculation

#Mass of block
M=W*g**-1 #Kg

#Net Force
#(W-P)=W1*g**-1*alpha   .........................1

#M.I
I=M*R**2*2**-1 #kgm**2

#Torque equation
#T=I*alpha 

#Multiplying equation of net force with 2 we get
#P=25*g**-1*alpha      ..............................2

#Adding equations 1 and 2 we get
alpha=W*3.06**-1 #rad/s**2

#Initial velocity
u=0 #m/s
t=2 #sec #time

#Angular acceleration 
alpha=3.268 #rad/s**2

#Angular Velocity
u_o=u+alpha*t

#Result
print"Angular Velocity after 2 seconds is",round(u_o,2),"m/s**2"

Angular Velocity after 2 seconds is 6.54 m/s**2

Example 15.43,Page No.595

In [10]:
import math

#Initilization of Variables

W1=100 #N #Weight of Block
W2=300 #N #weight of pulley
k=0.25 #m #Radius of gyration
r_A=0.2 #m #Radius of pulley A
r_B=0.3 #m #Radius of pulley B
g=9.81 #m/s**2 #Acceleration due to gravity

#Calculation

#Motion of block B
#(W1-T_B)=W1*g**-1*r_B*alpha   ..................1

#Motion of block A
#(T_A-W1)=W1*g**-1*r_A*alpha    ..................2

#M.I of pulley
I=W2*g**-1*k**2

#Rotation of pulley
#T_B*r_B-T_A*r_A=I*alpha     .....................3

#After multiplying equation 1 by r_B we get
#30-r_B*T_B=W1*g**-1*r_B*alpha    ..................4               

#After multiplying equation 2 by r_A we get
#0.2*T_A-20=W1*g**-1*r_A*alpha    ..................5

#Adding equations 3,4,5 and further simplifying we get
alpha=10*g*31.75**-1 #Rad/s**2

#Linear acceleration of pulley
alpha_B=0.3*alpha #m/s**2
alpha_A=0.2*alpha #m/s**2 

#Tension in strings
T_A=W1+W1*g**-1*alpha_A #N
T_B=W1-W1*g**-1*alpha_B #N


#Result
print"Angular Acceleration of pulley",round(alpha,2),"rad/s**2"
print"Linear acceleration of blocks A and B:alpha_A",round(alpha_A,2),"rad/s**2"
print"                                     :alpha_B",round(alpha_B,2),"rad/s**2"
print"Tension in the strings",round(T_A,2),"N"
print"                      ",round(T_B,2),"N"

Angular Acceleration of pulley 3.09 rad/s**2
Linear acceleration of blocks A and B:alpha_A 0.62 rad/s**2
                                     :alpha_B 0.93 rad/s**2
Tension in the strings 106.3 N
                       90.55 N

Example 15.44,Page No.598

In [48]:
import math

#Initilization of Variables

#Weights
W1=200 #N
W2=800 #N

F=400 #N #Force applied
mu=0.3 #Coefficient of friction
g=9.80 #m/s**2 #Acceleration due to gravity

#Calculation

#Total Weights
W=W1+W2 #N

#Total Mass 
M=W*g**-1 #Kg

#Force of friction
F2=mu*W #N

#acceleration
a=-((-F+F2)*g)*W**-1 #m/s**2 

#PArt-2

#Force of Friction
F3=mu*W1 #N

#Reverse Effective Force on it
F4=W1*g**-1*a #N

#Tension in thread
T=F3+F4 #N

#Result
print"acceleration of the weights is",round(a,2),"m/s**2"
print"Tension in the string is",round(T,2),"N"

acceleration of the weights is 0.98 m/s**2
Tension in the string is 80.0 N