Chapter 2 : Statics of particle

Example 2.1 Page No : 22

In [1]:
import math 

# Given Data
P = 40.; 			# N   Magnitude of vector P
Q = 60. 			# N Magnitude of vector Q
# imagine triangle for triangle law of vectors 
B = 180.-25;			# degree , Angle between vector P and vector Q

# Calculations and Results
#R- resultant vector
B = B*math.pi/180;			# conversion into radian
#R**2 = P**2+Q**2-2*P*Q*math.cos(B);    Comath.sine Law
R = math.sqrt(P**2+Q**2-2*P*Q*math.cos(B));			# N

print "Maginitude of resultant is R =  %.2f N"%(R);

#A- Angle between resultant and P vector, Unknown

# math.sin(A)/Q  =  =  math.sin(B)/R  math.sine law

A = math.asin(Q*math.sin(B)/R);			# radian


A = A*180/math.pi;			#			# Conversion into degree

alpha = A+20;			# degree
print "Angle of resultant  vector R with x axis is %.2f Degrees"%(alpha);

Maginitude of resultant is R =  97.73 N
Angle of resultant  vector R with x axis is 35.04 Degrees

Example 2.2 Page No : 29

In [2]:
import math 

# Given Data
R = 25.; 			# kN   Magnitude of resultant vector
alpha = 45.;			#degree
# T1 and T2 are tensions in rope 1 and rope 2 respectively 
A = 30.;			# degree , Angle between vector T1 and resultant
B = alpha;			# degree , Angle between vector T2 and resultant
C = 180.-(A+B);			# degree , Angle between vector T1 and T2


# Calculations and Results
# conversion of angles into radian
A = A*math.pi/180;
B = B*math.pi/180;
C = C*math.pi/180;


# math.sin(A)/T2  =  =  math.sin(B)/T1  =  =  math.sin(C)/R .............. math.sine law

T1 = (R*math.sin(B))/math.sin(C);			#kN
T2 = (R*math.sin(A))/math.sin(C);			#kN

print "Tension in rope 1 is T1 = %.2f kN and in rope 2 is T2 = %.2f kN "%(T1,T2);

# Minimum value of T2 o# Resultsurs when T1 and T2 are perpendicular to each other i.e C = 90 degree
C = 90.;			#degree
A = 30.;			# degree
B = 180-(A+C);			#degrees
alpha = B;			#degrees
B = B*math.pi/180;			# radian
T2 = R*math.sin(B);			# kN
T1 = R*math.cos(B);			#kN
print "Minimum tension in rope 2 is T2 = %.2f kN "%(T2);
print "corrosponding T1 = %.2f kN  "%(T1);
print "alpha = %.2f degrees"%(alpha);

Tension in rope 1 is T1 = 18.30 kN and in rope 2 is T2 = 12.94 kN 
Minimum tension in rope 2 is T2 = 21.65 kN 
corrosponding T1 = 12.50 kN  
alpha = 60.00 degrees

Example 2.3 Page No : 31

In [3]:
import math 

# Given Data
F1 = 150.;			# N 
F2 = 80.;			# N
F3 = 110.;			#N
F4 = 100.			# in N

F1x = 129.			#in N
F2x = -27.4
F3x = 0.
F4x = 96.6
F1y = 75.
F2y = 75.2
F3y = -110.
F4y = -25.9

# Calculations and Results
Rx = F1x+F2x+F3x+F4x;			#N Horizontal component of R- resultant
Ry = F1y+F2y+F3y+F4y;			#N Vertical component of R- resultant

#R = Rx i +Ry j

print "R =  %.2f i + %.2f j "%( Rx,Ry);

alpha = math.atan(Ry/Rx);			#Radian, Angle made by resultant with +ve x axis
alpha = alpha*180/math.pi;			#Conversion into degrees

R = math.sqrt(Rx**2+Ry**2);			# N , Magnitude of resultant
print "alpha =  %.2f degrees and R =  %.2f N"%(alpha,R);

R =  198.20 i + 14.30 j 
alpha =  4.13 degrees and R =  198.72 N

Example 2.4 Page No : 38

In [4]:
import math 

# Given Data
W = 3500.; 			# lb   weight of automobile
alpha = 2.;			#degree
# TAB and TAC are tensions in cable AB and cable AC respectively 
A = 90+30.;			# degree , Angle between vector T1 and resultant
B = alpha;			# degree , Angle between vector T2 and resultant
C = 180-(A+B);			# degree , Angle between vector T1 and T2

# Calculations
# conversion of angles into radian
A = A*math.pi/180;
B = B*math.pi/180;
C = C*math.pi/180;

# math.sin(A)/TAB  =  =  math.sin(B)/TAC  =  =  math.sin(C)/W .............. math.sine law
TAB = (W*math.sin(A))/math.sin(C);			#N
TAC = (W*math.sin(B))/math.sin(C);			#N

# Results
print "Tension in cable AB is TAB = %.2f lb and in Cable AC  is TAC = %.2f lb "%(TAB,TAC);

Tension in cable AB is TAB = 3574.19 lb and in Cable AC  is TAC = 144.03 lb

Example 2.5 Page No : 39

In [5]:
import math 

# Given Data
mass = 30.;			# kg
W = mass*9.81;			# N, Weight of package
alpha = 15.;			#degree

# Calculations
alpha = alpha*math.pi/180;			# Conversion into radian
F = W*math.sin(alpha);			#N

# Results
print "F =  %.2f N"%(F);

F =  76.17 N

Example 2.6 Page No : 39

In [6]:
import math 
alpha = math.atan(7./4);			#rad
beta = math.atan(1.5/4);			#rad
T_AB = 200.;			#N tension in cable AB
T_AE = -300.;			#N, tension in cable AE
# R =  T_AB+T_AC+T_AE+F_D = 0        ...Equillibrium Condition...........1


# Calculations
T_ABx = -T_AB*math.sin(alpha);			# Xcomponent of T_AB
T_ABy = T_AB*math.cos(alpha);			#Y component of T_AB

# T_ACx = T_AC*math.sin(beta); Xcomponent of T_AC
# T_ACy = T_AC*math.cos(beta); Y component of T_AC

# Sum Fx  = 0 gives -T_AB*math.sin(alpha) N + T_AC*math.sin(beta) +F_D = 0..........2
#Sum Fy = 0 gives T_AB*math.cos(alpha) N +T_AC*math.cos(beta) +T_AE  = 0................3

T_AC = (-T_AB*math.cos(alpha)-T_AE)/math.cos(beta);			#N, From 3

F_D = T_AB*math.sin(alpha)-T_AC*math.sin(beta);			#N, From 2

# Results
print "Value of drag force is F_D = %.2f N and tension in cable AC is T_AC =  %.2f N"%(F_D,T_AC);

Value of drag force is F_D = 98.36 N and tension in cable AC is T_AC =  214.42 N

Example 2.7 Page No : 50

In [7]:
import math 

# Given Data
dx = -40.;			#m
dy = 80.;			#m
dz = 30.;			#m
f = 2500.;			#N, Mafnitude of force F

# Calculations and Results
d = math.sqrt(dx**2+dy**2+dz**2);			#m, total dismath.tance of vector AB
#F = f*lambda, lambda - unit vector =  AB/d. So we can calculate each component by multiplying this unit vector
Fx = f*dx/d;			#N , X component of F
Fy = f*dy/d;			#N , Y component of F
Fz = f*dz/d;			#N , Z component of F

print "Component of F along X axis is %.2f N"%(Fx);
print "Component of F along Y axis is %.2f N"%(Fy);
print "Component of F along Z axis is %.2f N"%(Fz);
print "We may write F as  F  =  %.2f i + %.2f j + %.2f k"%(Fx,Fy,Fz);

thetax = math.acos(Fx/f);			# radian , angle with +ve x axis
thetay = math.acos(Fy/f);			# radian , angle with +ve y axis
thetaz = math.acos(Fz/f);			# radian , angle with +ve z axis

#Conversion of angles into degree
thetax = thetax*180/math.pi;			#degree
thetay = thetay*180/math.pi;			#degree
thetaz = thetaz*180/math.pi;			#degree

print "Angle made by F with +ve X axis %.2f degree"%(thetax);

print "Angle made by F with +ve Y axis %.2f degree"%(thetay);
print "Angle made by F with +ve Z axis %.2f degree"%(thetaz);

F = 800. 			# N , given force
theta = 145. 			# Degrees , angle with posiyive X axis 

theta = theta*math.pi/180;			# Conversion into radian

Fx = F*math.sin(theta);			#N, Horizontal component
Fy = F*math.cos(theta);			# N, Vertical Component
print "Horizontal component of F is %.2f N"%(Fx);
print "Vertial component of F is %.2f N"%(Fy);
print "We may write F as  F  =  %.2f i + %.2f j"%(Fx,Fy);

F = 300. 			# N , given force
AB = math.sqrt(8**2+6**2);			# m Length of AB
math.cos_alpha = 8/AB;
math.sin_alpha = -6/AB;
Fx = F*math.cos_alpha;			#N, Horizontal component
Fy = F*math.sin_alpha;			# N, Vertical Component
print "Fx = %.2f"%Fx
print "Fy = %.2f"%Fy

Component of F along X axis is -1060.00 N
Component of F along Y axis is 2120.00 N
Component of F along Z axis is 795.00 N
We may write F as  F  =  -1060.00 i + 2120.00 j + 795.00 k
Angle made by F with +ve X axis 115.09 degree
Angle made by F with +ve Y axis 32.01 degree
Angle made by F with +ve Z axis 71.46 degree
Horizontal component of F is 458.86 N
Vertial component of F is -655.32 N
We may write F as  F  =  458.86 i + -655.32 j
Fx = 240.00
Fy = -180.00

Example 2.8 Page No : 51

In [8]:
import math 

# Given Data
T_AB = 4200.;			#N , Tension in cable AB
T_AC = 6000.;			#N , Tension in cable AC
# Vector AB = -(5m)i+(3m)j+(4m)k
#Vector Ac =  -(5m)i+(3m)j+(5m)k
ABx = -5.;			#m
ABy = 3.;			#m
ABz = 4.;			#m
ACx = -5.;			#m
ACy = 3.;			#m
ACz = -5.;			#m

# Calculations and Results
AB = math.sqrt((-5)**2+3**2+4**2);			#m, Magnitude of vector AB
AC = math.sqrt((-5)**2+3**2+5**2);			#m, Magnitude of vector AC
#vT_AB = T_AB*lambdaAB, lambdaAB - unit vector =  vAB/AB. So we can calculate each component by multiplying this unit vector
T_ABx = T_AB*ABx/AB;			#N , X component of T_AB
T_ABy = T_AB*ABy/AB;			#N , Y component of T_AB
T_ABz = T_AB*ABz/AB;			#N , Z component of T_AB

print "Component of T_AB along X axis is %.2f N"%(T_ABx);
print "Component of T_AB along Y axis is %.2f N"%(T_ABy);
print "Component of T_AB along Z axis is %.2f N"%(T_ABz);
print "We may write T_AB as  T_AB  =  %.2f i + %.2f j + %.2f k"%(T_ABx,T_ABy,T_ABz);

#vT_AC = T_AC*lambdaAC, lambdaAC - unit vector =  vAC/AC. So we can calculate each component by multiplying this unit vector
T_ACx = T_AC*ACx/AC;			#N , X component of T_AC
T_ACy = T_AC*ACy/AC;			#N , Y component of T_AC
T_ACz = T_AC*ACz/AC;			#N , Z component of T_AC

print "Component of T_AC along X axis is %.2f N"%(T_ACx);
print "Component of T_AC along Y axis is %.2f N"%(T_ACy);
print "Component of T_AC along Z axis is %.2f N"%(T_ACz);
print "We may write T_AC as  T_AC  =  %.2f i + %.2f j + %.2f k"%(T_ACx,T_ACy,T_ACz);

Rx = T_ABx+T_ACx;			#N ,X component of R
Ry = T_ABy+T_ACy;			#N ,Y component of R
Rz = T_ABz+T_ACz;			#N ,Z component of R

print "Component of R along X axis is %.2f N"%(Rx);
print "Component of R along Y axis is %.2f N"%(Ry);
print "Component of R along Z axis is %.2f N"%(Rz);
print "We may write R as  R  =  %.2f i + %.2f j + %.2f k"%(Rx,Ry,Rz);

R = math.sqrt(Rx**2+Ry**2+Rz**2);			#N, Magnitude of resultant

thetax = math.acos(Rx/R);			# radian , angle with +ve x axis
thetay = math.acos(Ry/R);			# radian , angle with +ve y axis
thetaz = math.acos(Rz/R);			# radian , angle with +ve z axis

#Conversion of angles into degree
thetax = thetax*180/math.pi;			#degree
thetay = thetay*180/math.pi;			#degree
thetaz = thetaz*180/math.pi;			#degree

print "Angle made by R with +ve X axis %.2f degree"%(thetax);

print "Angle made by R with +ve Y axis %.2f degree"%(thetay);
print "Angle made by F with +ve Z axis %.2f degree"%(thetaz);

Component of T_AB along X axis is -2969.85 N
Component of T_AB along Y axis is 1781.91 N
Component of T_AB along Z axis is 2375.88 N
We may write T_AB as  T_AB  =  -2969.85 i + 1781.91 j + 2375.88 k
Component of T_AC along X axis is -3905.67 N
Component of T_AC along Y axis is 2343.40 N
Component of T_AC along Z axis is -3905.67 N
We may write T_AC as  T_AC  =  -3905.67 i + 2343.40 j + -3905.67 k
Component of R along X axis is -6875.52 N
Component of R along Y axis is 4125.31 N
Component of R along Z axis is -1529.79 N
We may write R as  R  =  -6875.52 i + 4125.31 j + -1529.79 k
Angle made by R with +ve X axis 147.38 degree
Angle made by R with +ve Y axis 59.64 degree
Angle made by F with +ve Z axis 100.80 degree