# Chapter 4 :Circular Motion and Gravitation¶

## Example 4.1 , Page no:24¶

In [1]:
import math
from __future__ import division

#initialisation of variables
t=2; #time in sec

#CALCULATIONS
s=2*3.14*r; #calculating s using circumference of circle
#=2*3.14*r in ft
v=s/t; #calculating velocity using v=s/t in ft/sec
ac=(v*v)/r; #calculating centripetal accelaration in #ft/sec square.

#RESULTS
print"Centripetal Accelaration =",round(ac,3);

Centripetal Accelaration = 14.789


## Example 4.2 , Page no:24¶

In [2]:
import math
from __future__ import division

#initialisation of variables
m=0.5; #weight in kg
v=4; #velocity in metre/sec

#CALCULATIONS
F=(m*v*v)/r; #calculating centripetal force in Newton

#RESULTS
print"Centripetal Force =",round(F);

Centripetal Force = 8.0


## Example 4.3 , Page no:24¶

In [3]:
import math
from __future__ import division

#initialisation of variables
F=1; #force in Newton
m=0.1; #m in kg

#CALCULATIONS
v=math.sqrt((F*r)/m); #calculating v in m/sec

#RESULTS
print"Velocity in metre/sec =",round(v,3);

Velocity in metre/sec = 2.646


## Example 4.4 , Page no:24¶

In [4]:
import math
from __future__ import division

#initialisation of variables
g=32; #gravitational constant in ft/sec square.
w=160; #weight in lb
v=10; #velocity in ft/sec

#CALCULATIONS
m=w/g;  #calculating mass in slugs
F=(m*v*v)/r; #calculating centripetal force in lb

#RESULTS
print"Centripetal Force in lb =",round(F);

Centripetal Force in lb = 25.0


## Example 4.5 , Page no:24¶

In [5]:
import math
from __future__ import division

#initialisation of variables
m=1000; #mass in kg
v=9; #velocity in metre/sec

#CALCULATIONS
F=(m*v*v)/r; #calculating centripetal force in Newton.

#RESULTS
print"Centripetal Force in Newton =",round(F);

Centripetal Force in Newton = 2700.0


## Example 4.6 , Page no:25¶

In [6]:
import math
from __future__ import division

#initialisation of variables
g=32; #gravitational constant in ft/sec square.
w=3200; #weight in lb
F=2000; #Maximum Force in lb

#CALCULATIONS
m=w/g; #calculating mass in slugs
v=math.sqrt((F*r)/m); #calculating velocity in ft/sec
v1=v*0.682;

#RESULTS
print"Velocity in min/hr =",round(v1);

Velocity in min/hr = 55.0


## Example 4.7 , Page no:25¶

In [7]:
import math
from __future__ import division

#initialisation of variables
g=9.8; #gravitaional constant in metre/sec
m=1; #mass in kg
v=5; #velocity in metre/sec

#CALCULATIONS
F=(m*v*v)/r; #calculationg centripetal force in Newton
w=m*g; #calculating weight in Newton
T=F-w; #calculating Tension in string at top position in Newton
T1=F+w; #calculating Tension at bottom of string in Newton.

#RESULTS
print"Tension in the string at the top position in Newton =",round(T,2);
print"Tension in the string at the bottom position in Newton =",round(T1,2);

Tension in the string at the top position in Newton = 40.2
Tension in the string at the bottom position in Newton = 59.8


## Example 4.8 , Page no:25¶

In [8]:
import math
from __future__ import division

#initialisation of variables
G=3.44*10**-8; #universal gravitational constant in lb.ft square/slug square
w=2000; #weight in lb
g=32; #gravitational constant in ft/sec square

#CALCULATIONS
m=w/g; #calculating mass in slugs
F=(G*m*m)/(r*r); #calculating gravitational force in lb

#RESULTS
print"Gravitational force in lb =",round(F,7);

Gravitational force in lb = 1.3e-06


## Example 4.9 , Page no:26¶

In [9]:
import math
from __future__ import division

#initialisation of variables
G=6.67*10**-11; #universal gravitational constant in Nm square/kg square.
m1=5.98*10**24; #mass of earth in kg
m2=7.36*10**22; #mass of moon in kg

#CALCULATIONS
F=(G*m1*m2)/(r*r); #calculating gravitationalforce in Newton
v=math.sqrt((G*m1)/r); #calculating velocity of moon in m/sec
s=2*3.14*r; #calculating circumference of moon's orbit in metre
t=s/v; #calculating time in sec
t1=t/86400;

#RESULTS
print"Gravitational Force in Newton =",'%.2E'%F;
print"Velocity in metre/sec =",round(v,3);
print"Time in sec =",round(t);
print"Time in days =",round(t1,3);

Gravitational Force in Newton = 1.99E+20
Velocity in metre/sec = 1019.173
Time in sec = 2366154.0
Time in days = 27.386


## Example 4.10 , Page no:26¶

In [10]:
import math
from __future__ import division

#initialisation of variables
r=6.4*10**6; #radius of earth in m
g=9.8; #gravitational constant in m/sec square

#CALCULATIONS
v=math.sqrt(r*g); #calculating velocity in m/sec

#RESULTS
print"Velocity in metre/sec =",round(v,3);

Velocity in metre/sec = 7919.596


## Example 4.11 , Page no:26¶

In [11]:
import math
from __future__ import division

#initialisation of variables

#CALCULATIONS
g=(6400/7400)*(6400/7400)*9.8; #calculating g at 1000km using g=(r earth/r)*g

#RESULTS
print"Accelaration due to gravity at 1000km =",round(g,3);

Accelaration due to gravity at 1000km = 7.33


## Example 4.12 , Page no:27¶

In [12]:
import math
from __future__ import division

#initialisation of variables
g=32; #gravitational constant in ft/sec square
w=128; #mass in lb

#CALCULATIONS
F=(1/2)*(1/2)*128; #calculating F in lb
m=w/g; #calculating m in slugs

#RESULTS
print"Weight at height above the earths surface of one earth radius =",round(F);
print"Mass of the girl in slugs=",round(m);

Weight at height above the earths surface of one earth radius = 32.0
Mass of the girl in slugs= 4.0


## Example 4.13 , Page no:27¶

In [13]:
import math
from __future__ import division

#initialisation of variables
T=24*60*60; #time in sec
re=6.4*10**6; #radius of earth in m
g=9.8; #gravitational constant in m/sec square

#CALCULATIONS
r=(((6.4*10**6*6.4*10**6)*9.8*(8.64*10**4*8.64*10**4))/(4*3.14*3.14))**(1/3); #calculating r in metre
h=r-re; #h =altitude above earth's surface
h1=h/1000;

#RESULTS
print"Height above the earths Surface in metre =",round(h);
print"Height above the earths Surface in kilometre =",round(h1,3);

Radius in metre = 42354355.0
Height above the earths Surface in metre = 35954355.0
Height above the earths Surface in kilometre = 35954.355


# Example 4.14 , Page no:27¶

In [14]:
import math
from __future__ import division

#initialisation of variables
re=6.4*10**6; #radius of earth in m
g=9.8; #gravitational constant in m/sec square

#CALCULATIONS
G=6.67*10**-11; #Universal gravitational constant in Nm square/kg square
m=(g*re*re)/G; #calculating mass of earth in kg

#RESULTS
print"Mass of Earth in kg =",'%.2E'%m;

Mass of Earth in kg = 6.02E+24


## Example 4.15 , Page no:28¶

In [15]:
import math
from __future__ import division

#initialisation of variables
G=6.67*10**-11; #Universal gravitational constant in Nmsquare/kg square
mm=7.36*10**22; #mass of moon in kg
r=1.74*10**6; #radius of moon in m
m=75; #weight of man in kg

#CALCULATIONS
g=(G*mm)/(r*r); #calculating g in m/sec square
w=m*g; #calculating weight in Newton

#RESULTS
print"Accelaration due to gravity at its surface in m/sec square=",round(g,3);
print"Mans weight on moon in Newton =",round(w,3);

Accelaration due to gravity at its surface in m/sec square= 1.621
Mans weight on moon in Newton = 121.609


## Example 4.16 , Page no:28¶

In [16]:
import math
from __future__ import division

#initialisation of variables

Velocity in m/sec = 1668.532