# 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

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

#Calculation

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

W=W1_1-W1 #N

#Result

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)

#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
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

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

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

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

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

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

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

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

#Result

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

#Angular velocity after 3 seconds

#Result

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
N_o=400 #r.p.m #Initial Velocity
N=280 #r.p.m #Final speed
t=120 #seconds #Time

#Calculation

#Initial Angular velocity

#Final Angular velocity

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

#Angular acceleration

#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"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
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

#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

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

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

#Angular acceleration

#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

r_A=0.1 #m
r_B=0.15 #m

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

#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
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
W_o=735 #N #Weight of drum
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
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

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

#Angular acceleration

#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
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

#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"Linear acceleration of blocks A and B:alpha_A",round(alpha_A,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
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


acceleration of the weights is 0.98 m/s**2