# Chapter 14: Kinematics of a Rigid Body in Plane Motion¶

## Example 14.14-2, Page no 265¶

In [9]:
import math

#Initilization of variables
d=500 #mm
wo=0 #rpm
w=300 #rpm
t=20 #s
t1=2 #s

#Calculations
v=(d*(2*1000)**-1)*w1 #m/s
a_n=(d*(2*1000)**-1)*w1**2 #m/s**2
a_t=(d*(2*1000)**-1)*alpha #m/s**2
a=(a_n**2+a_t**2)**0.5 #m/s**2
theta=arccos(a_n/a)*(180/pi) #degrees

#Result
print'The computed values are:'
print'v=',round(v,3),"m/s"
print'a=',round(a,2),"m/s**2"
print'theta=',round(theta,1),"degrees"

# The answers may wary in decimal points.

The computed values are:
v= 0.785 m/s
a= 2.5 m/s**2
theta= 9.0 degrees


## Example 14.14-3, Page no 266¶

In [10]:
import math

#Initilization of variables
s_BC=2 #m
s_C=2.5 #m

#Calculations
s_B=(s_BC**2+s_C**2)**0.5 #m
theta=arctan(s_BC*s_C**-1)*(180/pi) #degrees

#Result
print'The absolute displacement is',round(s_B,1),"m",'and the angle made by the vector is',round(theta,1),"degrees."

The absolute displacement is 3.2 m and the angle made by the vector is 38.7 degrees.


## Example 14.14-4, Page no 266¶

In [11]:
import math
import numpy as np

#Initilization of variables
V_A=20 #mi/h
V_B=70 #mi/h
# as theta1=60 degrees,
sintheta1=sqrt(3)*2**-1
costheta1=2**-1
# also phi=45 degrees, thus
sinphi=sqrt(2)**-1
cosphi=sqrt(2)**-1

#Result
#Vector's in matrix form
v_A=np.array([-V_A*cosphi,V_A*sinphi]) #mi/h
v_B=np.array([V_B*costheta1,V_B*sintheta1]) #mi/h
a=v_A[0]+v_B[0] #mi/h
b=v_A[1]+v_B[1] #mi/h
v_ab=(a**2+b**2)**0.5 #mi/h
theta=arctan(b/a)*(180/pi) #degrees
#The relative velocity v_ba is just different in sign while the magnitude stays the same

#Resul
print'The relative velocity is',round(v_ab,1),"mi/h",'making an angle',round(theta,1),"degrees"

# The ans may wary due to decimal point descrepancy.

The relative velocity is 77.6 mi/h making an angle 74.4 degrees


## Example 14.14-9, Page no 271¶

In [14]:
import math

#Initilization of variables
l=2.5 #m
v_A=4 #m/s
a_A=5 #m/s**2
theta=30 #degrees

#Calculations
#Vector triangle yields v_a.b=2.93 m/s
v_ab=2.93 #m/s
#Ploygon yields alpha_a/b=2.75 m/s**2
alpha_ab=2.75 #m/s**2

#Result
print'The value of angular velocity is',round(w,2),"rad/s"
print'The value of angular acceleration is',round(alpha,1),"rad/s**2"

The value of angular velocity is 1.17 rad/s
The value of angular acceleration is 1.1 rad/s**2


## Example 14.14-10, Page no 272¶

In [29]:
import math

#Initilization of variables
l=24 #in
l_c=4 #in
# as th=30 degrees,
sinth=2**-1

#Calculations
v=(l_c*12**-1)*w #ft/s
betaa=arcsin((l_c*sinth)/l)*(180/pi) #degrees
# betaa yeilds 4.8 degrees, thus value of cosbetaa is,
cosbetaa=0.996
theta=60-betaa #degrees
# here theta yeilds 55.2 degrees, thus value of costheta is,
costheta=0.57
#Component of velocity along connecting rod is
v1=v*costheta #ft/s
v_p=v1/cosbetaa #ft/s

#Result
print'The absoulte velocity is',round(v_p,2),"ft/s"

The absoulte velocity is 2.4 ft/s


## Example 14.14-13, Page no 274¶

In [30]:
import math

#Initilization of variables
v_pc=3.68 #ft/s
l=2 #ft

#Calculations

#Result

The angular velocity is 1.84 rad/s


## Example 14.14-14, Page no 274¶

In [31]:
import math

#This problem is a combination of numerical and graphical solution
#The program only deals with the numerical solution parts the rest can be verified by graphical solution
#Initilization of variables
r=4*12**-1 #ft
l=2 #ft

#Calculations
ac_n=r*w**2 #ft/s**2
a_pc_n=l*w2**2 #ft/s**2

#Result
print'The  value of ac_n is',round(ac_n,1),"ft/s**2"
print'The value of a_pc_n is',round(a_pc_n,2),"ft/s**2"

The  value of ac_n is 52.6 ft/s**2
The value of a_pc_n is 6.77 ft/s**2


## Example 14.14-15, Page no 275¶

In [47]:
import math

#Initilization of variables
AB=250 #mm
BC=150 #mm
AC=179 #mm
# as theta1=45 degrees,
sintheta1=(2**0.5)**-1
costheta1=(2**0.5)**-1

#Calculations
v_c=(BC*1000**-1)*w_bc #m/s
AC=((AB**2+BC**2)-(2*AB*BC*costheta1))**0.5 #m
betaa=arcsin((BC*sintheta1)/AC)*(180/pi) #degrees
gammaa=arcsin((AB*sintheta1)/AC)*(180/pi)#degrees answer in the textbook is incorrect
ang=60-betaa #degrees
# ang yeilds 23.7 degrees, thus
sinang=0.40056
cosang=0.916
D=arcsin((AC*sinang)/CD)*(180/pi) #degrees
# D yeilds 63.2 degrees,thus
sinD=0.8925
n=360-(theta+gammaa+90) #degrees
# n yeilds 101.8 degrees, thus
cosn=-0.2045
v_cd=v_c*cosn #m/s
delta=180-(90+D) #degrees
# Delts yeilds 26.8 degrees, thus
cosdelta=0.8925
v_D=v_cd/cosdelta #m/s

#Result
#Answer in the textbook is incorrect

The angular Velocity of AD is -1.72 rad/s clockwise.


## Example 14.14-18, Page No 277¶

In [78]:
import math

#Initilization of variables
# as theta1=73.9 degrees,theta2=60 degrees and theta3=46.1 degrees
sintheta1=0.96
sintheta2=sqrt(3)*2**-1
sintheta3=0.72
V=900 #mm/s

#Calculations
BC=((350*350)+(86.6*86.6))**0.5 #mm
CD=400 #mm
v_cb=(V*sintheta2)/(sintheta1) #mm/s
v_c=((V*sintheta3))/(sintheta1) #mm/s

#Result

The angular velocities are: w_dc= 1.69 rad/s and w_bc= 2.25 rad/s


## Example 14.14-19, Page No 278¶

In [80]:
import math
import numpy as np

#Calculations
#After equating the i and j terms we obtain simplified equations
#Solving by matrix method
A=np.array([[346,86.7],[200,-350]])
B=np.array([[-3700],[-1790]])
C=np.linalg.solve(A,B)

#Result
#The signs only indicate that the originally assumed orientations are incorrect and are opposite to those assumed

The angular accelerations are alpha_DC= -10.475 rad/s**2 and alpha_BC= -0.87 rad/s**2


## Example 14.14-20, Page No 279¶

In [96]:
import math

#Initilization of variables
d=3 #m
r=d*2**-1 #m

#Calculations
vo=r*w #m/s
ao=r*alpha #m/s**2
#Here OB is r
OB=r #m
v_bo=OB*w #m/s
v_B=v_bo+vo #m/s
#Also
a_bo=r*alpha #m/s**2 (directed left)
a_bo_n=r*w**2 #m/s**2
a_h=ao+a_bo #m/s**2
a_v=a_bo_n #m/s**2
a_B=((a_h**2)+(a_v**2))**0.5 #m/s**2
phi=arctan(a_h/a_v)*(180/pi) #degrees

#Result
print'The linear velocity at B is',round(v_B),"m/s",'and the acceleration is',round(a_B,1),"m/s**2",'making an angle of',round(phi,2),"degrees with horizontal"

The linear velocity at B is 24.0 m/s and the acceleration is 96.7 m/s**2 making an angle of 7.13 degrees with horizontal


## Example 14.14-21, Page No 281¶

In [100]:
import math

#Initilization of variables
OA=0.6 #m
# as theta=30 degrees,
sintheta=2**-1
costheta=(3**0.5)*2**-1
v_O=12 #m/s
a_O=6 #m/s**2

#Calculations
#Velocity Calculations
v_AO=OA*w #m/s
v_Ah=v_AO*sintheta+v_O #m/s horizontal component
v_Av=v_AO*costheta #m/s
v_A=((v_Ah**2)+(v_Av**2))**0.5 #m/s
phi=arctan(v_Av/v_Ah)*(180/pi) #degrees
#Acceleration Calculations
a_AOt=OA*alpha #m/s**2
a_AOn=OA*w**2 #m/s**2
a_Ah=-a_O-a_AOn*costheta-a_AOt*sintheta #m/s**2
a_Av=-a_AOn*sintheta+a_AOt*costheta #m/s**2
a_A=((a_Ah**2)+(a_Av**2))**0.5 #m/s**2
phi2=arctan(a_Av*a_Ah**-1)*(180/pi) #degrees

#Result
print'The velocity is',round(v_A),"m/s",'making an angle of',round(phi,1),"degrees with horizontal."
print'The acceleration is',round(a_A),"m/s**2",'making an angle of',round(phi2,1),"degrees with horizontal"

The velocity is 15.0 m/s making an angle of 16.1 degrees with horizontal.
The acceleration is 44.0 m/s**2 making an angle of 22.9 degrees with horizontal


## Example 14.14-22, Page No 282¶

In [101]:
import math

#Initilization of variables
AL=5 #ft
d=10 #ft displacement

#Calculations
s_o=3*theta#ft

#Result
print'The displacement So is',round(s_o),"ft"

The displacement So is 6.0 ft


## Example 14.14-23, Page No 283¶

In [103]:
import math

#Initilization of variables
#Speed and acceleration at the center
v=12 #in/s
a=18 #in/s**2

#Calculations
v_D=((a+v*0.5)*a**-1)*v #in/s
#Speed at point F
v_F=((v/2)*v**-1)*v_D #in/s
#Acceleration at D
a_D=(24/a)*a #in/s**2
#Acceleration at F
a_F=((v/2)*v**-1)*24 #in/s**2

#Result
print'The velocity and acceleration of weight A are',round(v_F),"in/s",'and',round(a_F),"in/s**2 respectively."

The velocity and acceleration of weight A are 8.0 in/s and 12.0 in/s**2 respectively.


## Example 14.14-24, Page No 283¶

In [104]:
import math

#Calculations
#Speed and acceleration of D
sD=((18-6)*18**-1)*12 #in/s
#Speed and acceleration of F
sF=(6*12**-1)*8 #in/s
aF=(6*12**-1)*12 #in/s^2

#Result
print'The velocity and acceleration of weight A are',round(sF),"in/s",'and',round(aF),"in/s**2 respectively."

The velocity and acceleration of weight A are 4.0 in/s and 6.0 in/s**2 respectively.


## Example 14.14-26, Page No 284¶

In [106]:
import math

#Initilization of variables
v_BG=300 #mm/s
v_G=300 #mm/s
a_BGt=500 #mm/s**2
a_BGn=3600#mm/s**2
a_Gh=500 #mm/s**2
a_Bv=1800 #mm/s**2

#Calculations

The velocity and acceleration of point B are 424.0 mm/s and 3362.0 mm/s**2 respectively.