Chapter 21 Kinematics of rigid body

Example 21.1 Linear and angular velocity linear and angular acceleration in rotation

In [4]:
import math
#Initilization of variables
N=1800 # r.p.m # Speed of the shaft
t=5 # seconds # time taken to attain the rated speed # case (a)
T=90 # seconds # time taken by the unit to come to rest # case (b)
# Calculations
omega=(2*math.pi*N)/(60)
# (a)
# we take alpha_1,theta_1 & n_1 for case (a)
alpha_1=omega/t # rad/s**2 #
theta_1=(omega**2)/(2*alpha_1) # radian
# Let n_1 be the number of revolutions turned,
n_1=theta_1*(1/(2*math.pi))
# (b)
# similarly we take alpha_1,theta_1 & n_1 for case (b)
alpha_2=(omega/T) # rad/s**2 # However here alpha_2 is -ve
theta_2=(omega**2)/(2*alpha_2) # radians
# Let n_2 be the number of revolutions turned,
n_2=theta_2*(1/(2*math.pi))
# Results
print('(a) The no of revolutions the unit turns to attain the rated speed is %f'%n_1)
print('(b) The no of revolutions the unit turns to come to rest is %f'%n_2)

(a) The no of revolutions the unit turns to attain the rated speed is 75.000000
(b) The no of revolutions the unit turns to come to rest is 1350.000000

Example 21.2 Absolute and relative velocity in plane motion

In [5]:
import math
# Initilization of variables
r=1 # m # radius of the cylinder
v_c=20 # m/s # velocity of the cylinder at its centre
# Calculations
# The velocity of point E is given by using the triangle law as,
v_e=math.sqrt(2)*v_c # m/s 
# Similarly the velocity at point F is given as,
v_f=2*v_c # m/s 
# Results
print('The velocity of point E is %f m/s'%v_e)
print('The velocity of point F is %f m/s'%v_f)

The velocity of point E is 28.284271 m/s
The velocity of point F is 40.000000 m/s

Example 21.3 Absolute and relative velocity in plane motion

In [6]:
import numpy
# Initilization of Variables
v_1=3 # m/s # uniform speed of the belt at top
v_2=2 # m/s # uniform speed of the belt at the bottom
r=0.4 # m # radius of the roller
# Calculations
# equating eq'ns 2 & 4 and solving for v_c & theta' (angular velocity). We use matrix to solve the eqn's
A=numpy.matrix([[1,r],[1,-r]])
B=numpy.matrix([[v_1],[v_2]])
C=numpy.linalg.inv(A)*B
# Results
print('The linear velocity (v_c) at point C is %f m/s'%C[0])
print('The angular velocity at point C is %f radian/seconds'%C[1])
# The answer of angular velocity is incorrect in the book

The linear velocity (v_c) at point C is 2.500000 m/s
The angular velocity at point C is 1.250000 radian/seconds

Example 21.4 Absolute and relative velocity in plane motion

In [7]:
import math
# Initilization of Variables
l=1 # m # length of bar AB
v_a=5 # m/s # velocity of A
theta=30 # degree # angle made by the bar with the horizontal
# Calculations
# From the vector diagram linear velocity of end B is given as,
v_b=v_a/math.tan(theta*math.pi/180) # m/s 
# Now let the relative velocity be v_ba which is given as,
v_ba=v_a/math.sin(theta*math.pi/180) # m/s
# Now let the angular velocity of the bar be theta_a which is given as,
theta_a=(v_ba)/l # radian/second
# Velocity of point A
v_a=(l/2)*theta_a # m/s
# Magnitude of velocity at point C is,
v_c=v_a # m/s # from the vector diagram
# Results
print('(a) The angular velocity of the bar is %f radian/second'%theta_a)
print('(b) The velocity of end B is %f m/s'%v_b)
print('(c) The velocity of mid point C is %f m/s'%v_c)

(a) The angular velocity of the bar is 10.000000 radian/second
(b) The velocity of end B is 8.660254 m/s
(c) The velocity of mid point C is 5.000000 m/s

Example 21.5 Absolute and relative velocity in plane motion

In [9]:
import math
# Initilization of Variables
r=0.12 # m # length of the crank
l=0.6 # m # length of the connecting rod
N=300 # r.p.m # angular velocity of the crank
theta=30 # degree # angle made by the crank with the horizontal
# Calculations
# Now let the angle between the connecting rod and the horizontal rod be phi
phi=math.asin(((r*math.sin(theta*math.pi/180))/(l))*math.pi/180) # degree
# Now let the angular velocity of crank OA be omega_oa, which is given by eq'n
omega_oa=(2*math.pi*N)/(60) # radian/second
# Linear velocity at A is given as,
v_a=r*omega_oa # m/s
# Now using the sine rule linear velocity at B can be given as,
v_b=(v_a*math.sin(35.7*math.pi/180))/(math.sin(84.3*math.pi/180)) # m/s
# Similarly the relative velocity (assume v_ba) is given as,
v_ba=(v_a*math.sin(60*math.pi/180))/(math.sin(84.3*math.pi/180))
# Angular velocity (omega_ab) is given as,
omega_ab=v_ba/l # radian/second
# Results
print('(a) The angular velocity of the connecting rod is %f radian/second'%omega_ab)
print('(b) The velocity of the piston when the crank makes an angle of 30 degree is %f m/s'%v_b)

(a) The angular velocity of the connecting rod is 5.468436 radian/second
(b) The velocity of the piston when the crank makes an angle of 30 degree is 2.210830 m/s

Example 21.6 Instantaneous Centre of rotation in plane motion

In [10]:
import math
# Initiization of variables
r=1 # m # radius of the cylinder
v_c=20 # m/s # velocity at the centre
# Calculations
# Angular velocity is given as,
omega=v_c/r # radian/second
# Velocity at point D is
v_d=omega*math.sqrt(2)*r # m/s # from eq'n 1
# Now, the velocity at point E is,
v_e=omega*2*r # m/s 
# Results
print('The velocity at point D is %f m/s'%v_d)
print('The velocity at point E is %f m/s'%v_e)

The velocity at point D is 28.284271 m/s
The velocity at point E is 40.000000 m/s

Example 21.7 Instantaneous Centre of rotation in plane motion

In [12]:
import numpy
# Initilization of Variables
r=5 # cm # radius of the roller
AB=0.1 # m
v_a=3 # m/s # velocity at A
v_b=2 # m/s # velocity at B
# Calculations
# Solving eqn's 1 & 2 using matrix for IA & IB we get,
A=([[-2,3],[1,1]])
B=numpy.matrix([[0],[AB]])
C=numpy.linalg.inv(A)*B
d1=C[1]*10**2 # cm # assume d1 for case 1
# Similary solving eqn's 3 & 4 again for IA & IB we get,
P=numpy.matrix([[-v_b,v_a],[1,-1]])
Q=numpy.matrix([[0],[AB]])
R=numpy.linalg.inv(P)*Q
d2=R[1]*10**2 # cm # assume d2 for case 2
# Results
print('The distance d when the bars move in the opposite directions are %f cm'%d1)
print('The distance d when the bars move in the same directions are %f cm'%d2)

The distance d when the bars move in the opposite directions are 4.000000 cm
The distance d when the bars move in the same directions are 20.000000 cm

Example 21.8 Instantaneous Centre of rotation in plane motion

In [13]:
import math
# Initilization of Variables
v_c=1 # m/s # velocity t the centre
r1=0.1 # m 
r2=0.20 # m
EB=0.1 # m
EA=0.3 # m
ED=math.sqrt(r1**2+r2**2) # m
# Calculations
# angular velocity is given as,
omega=v_c/r1 # radian/seconds
# Velocit at point B
v_b=omega*EB # m/s 
# Velocity at point A
v_a=omega*EA # m/s
# Velocity at point D
v_d=omega*ED # m/s
# Results
print('The velocity at point A is %f m/s'%v_a)
print('The velocity at point B is %f m/s'%v_b)
print('The velocity at point D is %f m/s'%v_d)

The velocity at point A is 3.000000 m/s
The velocity at point B is 1.000000 m/s
The velocity at point D is 2.236068 m/s

Example 21.9 Instantaneous Centre of rotation in plane motion

In [14]:
import math
# Initilization of variables
l=1 # m # length of bar AB
v_a=5 # m/s # velocity at A
theta=30 # degree # angle made by the bar with the horizontal
# Calculations
IA=l*math.sin(theta*math.pi/180) # m
IB=l*math.cos(theta*math.pi/180) # m
IC=0.5 # m # from triangle IAC
# Angular veocity is given as,
omega=v_a/(IA) # radian/second
v_b=omega*IB # m/s
v_c=omega*IC # m/s
# Results
print('The velocity at point B is %f m/s'%v_b)
print('The velocity at point C is %f m/s'%v_c)

The velocity at point B is 8.660254 m/s
The velocity at point C is 5.000000 m/s

Example 21.11 Instantaneous Centre of rotation in plane motion

In [15]:
import math
# Initilization of variables
v_a=2 # m/s # velocity at end A
r=0.05 # m # radius of the disc
alpha=30 # degree # angle made by the bar with the horizontal
# Calculations 
# Soving eqn's 1 & 2 and substuting eqn 1 in it we get eq'n for omega as,
omega=(v_a*(math.sin(alpha*math.pi/180))**2)/(r*math.cos(alpha*math.pi/180)) # radian/second
# Results
print('The anguar veocity of the bar is %f radian/second'%omega)

The anguar veocity of the bar is 11.547005 radian/second

Example 21.12 Instantaneous Centre of rotation in plane motion

In [16]:
import math
# Initilization of variables
l=0.6 # m 
r=0.12 # m 
theta=30 # degree # angle made by OA with the horizontal
phi=5.7 # degree # from EX 21.5
N=300
# Calculations
# Let the angular velocity of the connecting rod be (omega_ab) which is given from eqn's 1 & 4 as,
omega_oa=(2*math.pi*N)/(60) # radian/ second
# Now,in triangle IBO.
IB=(l*math.cos(phi*math.pi/180)*math.tan(theta*math.pi/180))+(r*math.sin(theta*math.pi/180)) # m
IA=(l*math.cos(phi*math.pi/180))/(math.cos(theta*math.pi/180)) # m
# from eq'n 5
v_b=(r*omega_oa*IB)/(IA) # m/s
# From eq'n 6
omega_ab=(r*omega_oa)/(IA) # radian/second
# Results
print('The velocity at B is %f m/s'%v_b)
print('The angular velocity of the connecting rod is %f radian/second'%omega_ab)

The velocity at B is 2.213062 m/s
The angular velocity of the connecting rod is 5.468436 radian/second

Example 21.13 Instantaneous Centre of rotation in plane motion

In [17]:
import math
# Initilization of variables
omega_ab=5 # rad/s # angular veocity of the bar
AB=0.20 # m
BC=0.15 # m
CD=0.3 # m
theta=30 # degree # where theta= angle made by AB with the horizontal
alpha=60 # degree # where alpha=angle made by CD with the horizontal
# Calculations
# Consider triangle BIC
IB=math.sin(alpha*math.pi/180)*BC*1 # m
IC=math.sin(theta*math.pi/180)*BC*1 # m
v_b=omega_ab*AB # m/s
# let the angular velocity of the bar BC be omega_bc
omega_bc=v_b/IB # radian/second
v_c=omega_bc*IC # m/s
# let the angular velocity of bar DC be omega_dc
omega_dc=v_c/CD # radian/second
# Results
print('The angular velocity of bar BC is %f rad/s'%omega_bc)
print('The angular velocity of bar CD is %f rad/s'%omega_dc)

The angular velocity of bar BC is 7.698004 rad/s
The angular velocity of bar CD is 1.924501 rad/s