Chapter 13:Kinematics of Curvilinear Motion,Circular Motion,,Rotation And Translation¶

Example 13.1,Page No.471¶

In [1]:
import math

#Initilization of Variables

t=4 #s #time

#Calculation

#Angular Acceleration of the body

#Result
print"Angular Acceleration of the body is",round(alpha,2),"Rad/s**2"

Angular Acceleration of the body is 2.0 Rad/s**2


Example 13.2,Page No.471¶

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

#Initilization of Variables

N_o=20 #Initial r.p.m of wheel
n=50 #No. of revolution
n2=100
t=70 #s #time

#Calculation

#Angular Dispalcement
theta=2*pi*n

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

#Angular Velocity at the end of 70 s

#Time Required for the speed to reach 100 r.p.m

#Final Angular Velocity

#Time required for speed to reach 100 revolutions
t=(w2-w_o)*alpha**-1 #s

#Result
print"Angular Velocity at the end of 70 s is",round(w,2),"rad/s**2"
print"Time required for speed to reach 100 revolutions is",round(t,2),"s"

Angular Velocity at the end of 70 s is 6.88 rad/s**2
Time required for speed to reach 100 revolutions is 122.5 s


Example 13.3,Page No.472¶

In [3]:
import math

#Initilization of Variables

#Calculation

#Total angle turned

#Result

Total angle turned is 41.34 rad


Example 13.4,Page No.472¶

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

#Initilization of Variables

#Part-1
w_o=0 #Initial angular velocity
t=90 #s #time

#Part-2

w2=0 #final angular velocity

#Calculation

#Part-1

#Angular Velocity

#Speed in r.p.m
N=60*w*(2*pi)**-1 #r.p.m

#Part-2

#Time taken by flywheel in seconds to come to rest
t1=-w_o2*alpha2**-1 #s

#Result
print"Speed in r.p.m is",round(N,2)
print"Time taken by flywheel in seconds to come to rest is",round(t1,2),"s"

Speed in r.p.m is 859.44
Time taken by flywheel in seconds to come to rest is 180.0 s


Example 13.5,Page No.473¶

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

#Initilization of Variables

N=200 #r.p.m #initial speed
N2=160 #r.p.m
t=10 #s #time
f3=0 #Final angular velocity

#Calculation

#Uniform retardation

#total angular displacement
n=theta*(2*pi)**-1 #revolutions

#Time taken by wheel before it comes to rest
t=-f1*alpha**-1

#Result
print"Number of revolutions is",round(n,2),"revolution"
print"Time taken by wheel before it comes to rest is",round(t,2),"s"

Number of revolutions is 83.28 revolution
Time taken by wheel before it comes to rest is 49.98 s


Example 13.6,Page No.474¶

In [6]:
import math

#Initilization of Variables

#Angle of rotation
#theta=2*t**3-5*t**2+8*t+6
t=0 #s
t2=4 #s

#Calculation

#After deriving above equation we get

#Again differentiating above equation we get

#for t2=4

#Result

Angular Velocity at t=0 is 8.0 rad/s
Angular acceleration at t=4 is -10.0 rad/s**2
Angular Velocity at t=0 is 64.0 rad/s
Angular acceleration at t=4 is 38.0 rad/s**2


Example 13.6(A),Page No.474¶

In [7]:
import math

#Initilization of Variables

#angular rotation
#theta=9*32**-1*t**3
t=1.6 #s

#Calculation

#after differentiating above equation twice we get

#Result

Angular accelerations is 2.7 rad/s**2


Example 13.7,Page No.475¶

In [8]:
import math

#Initilization of Variables

alpha1=0 #Initial angular acceleration

#Calculation

#Integrating law of rotation we get
#f2=t**3-3*t+C   .......1
#put t=0 weg et
C=2

#now at t=5 #s
t=5 #s
f2=t**3-3*t+C

#Integrting equation 1 we get
#theta=t**4*4**-1-3*t**2*2**-1+2*t
#Sub values and further simplifying we get
theta=t**4*4**-1-3*t**2*2**-1+2*t

#Result

Angular velocity when t=5s is 112.0 rad/s
Angular displacement when t=5s is 128.75 radians


Example 13.8,Page No.476¶

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

#Initilization of Variables

#Angle of rotation of body
#theta=theta1+a*t+b*t**2
f=3*pi
f2=8*pi
t=0 #s
t2=2 #s

#Calculations

#differentiating above equation and further sub values and simplifuing we getweget
a=f
b=(f2-a)*4**-1

#Result
print"Constants a is",round(a,2)
print"Constants b is",round(b,2)

Constants a is 9.42
Constants b is 3.93


Example 13.9,Page No.480¶

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

#Initilization of Variables

#Velocity
V_A=4 #m/s
theta=30 #Degrees

#Calculation

#Velocity
V_B=V_A*(tan(theta*pi*180**-1))**-1 #m/s

#Result
print"Velocity of point B is",round(V_B,2),"m/s"

Velocity of point B is 6.93 m/s


Example 13.9(A),Page No.480¶

In [11]:
import math

#Initilization of Variables

V_C=20 #m/s #Velocity

#Calculation

#Length
L_DE=(r**2+r**2)**0.5 #m
L_DF=2 #m #Diameter

#Velocity
V_E=L_DE*f #m/s
V_F=f*L_DF #m/s

#Result
print"Velocity of point E:V_E",round(V_E,2),"m/s"
print"Velocity of point E:V_F",round(V_F,2),"m/s"

Velocity of point E:V_E 28.28 m/s
Velocity of point E:V_F 40.0 m/s


Example 13.9(B),Page No.481¶

In [12]:
import math

#Initilization of Variables

D=50 #cm
V_A=L_AL=5 #m/s
V_B=L_BM=3 #m/s

#Calculation

#V_A=f*L_AO
#V_B=f*L_BO

#After further simplifying and resolving we get
x=L_BO=3*f**-1

#Linear Velocity
V_C=f*(r+x) #m/s

#Result
print"Linear Velocity of roller is",round(V_C,2),"m/s"

Linear Velocity of roller is 4.0 m/s


Example 13.9(C),Page No.482¶

In [13]:
import math
from math import sin, cos, tan, radians, pi

#Initilization of Variables

r=1 #m
u=20 #m/s

#Calculation

#Velocity component of point E
#u_E=u+u*sin(u*t)
#at t=0
t=0
u_E=u+u*sin(u*t*pi*180**-1)
v_E=u*cos(u*t)
V_E=(u_E**2+v_E**2)**0.5 #m/s
u_F=u+u*cos(u*t*pi*180**-1) #m/s

#Result
print"Velocity of point E is",round(V_E,2),"m/s"
print"Velocity of point F is",round(u_F,2),"m/s"

Velocity of point E is 28.28 m/s
Velocity of point F is 40.0 m/s


Example 13.9(D),Page No.485¶

In [14]:
import math

#Initilization of Variables

g=9.81 #Acceleration due to gravity
W1=W2=80*1000*g
D1=0.75*10**3 #mm
R1=0.75*500 #mm
a1=0.025 #m/s**2
D2=1.2*10**3 #mm
R2=1.2*500 #mm
a2=0.0625 #m/s**2

#Calculation

#Horizontal Forces
P1=W1*a1*R1**-1 #N
P2=W2*a2*R2**-1 #N

#Result
print"Horizontal Force required to maintain uniform speed is",round(P1,2),"N"
print"Horizontal Force for truck and trailer is",round(P2,2),"N"

Horizontal Force required to maintain uniform speed is 52.32 N
Horizontal Force for truck and trailer is 81.75 N