# 5: Central Forces¶

## Example number 4, Page number 200¶

In [5]:
#importing modules
import math
from __future__ import division

#Variable declaration
m1=m2=0.5;           #mass(kg)
r2=0.1;              #distance(m)
r1=float("inf");     #distance(m)
G=6.67*10**-11;      #gravitational constant

#Calculation
delta_U=G*m1*m2*((1/r2)-(1/r1));        #gravitational energy(J)

#Result
print "gravitational energy is",delta_U*10**10,"*10**-10 J"

gravitational energy is 1.6675 *10**-10 J


## Example number 5, Page number 201¶

In [8]:
#importing modules
import math
from __future__ import division

#Variable declaration
m1=1;           #mass(kg)
m2=2;           #mass(kg)
m3=3;           #mass(kg)
a=0.5;              #side(m)
G=6.67*10**-11;      #gravitational constant

#Calculation
delta_U=-G*((m1*m2)+(m2*m3)+(m3*m1))/a;        #net energy(J)

#Result
print "net energy is",round(delta_U*10**11,2),"*10**-11 J"
print "answer given in the book is wrong"

net energy is -146.74 *10**-11 J
answer given in the book is wrong


## Example number 8, Page number 203¶

In [12]:
#importing modules
import math
from __future__ import division

#Variable declaration
r=4*10**8;        #distance(m)
M1=6*10**24;      #mass of earth(kg)
M2=7.5*10**22;    #mass of moon(kg)

#Calculation
x=r/(1+math.sqrt(M2/M1));      #distance where potential becomes zero(m)

#Result
print "distance where potential becomes zero is",round(x/10**8,1),"*10**8 m"

distance where potential becomes zero is 3.6 *10**8 m


## Example number 10, Page number 204¶

In [18]:
#importing modules
import math
from __future__ import division

#Variable declaration
g=9.8;            #acceleration due to gravity(m/s**2)

#Calculation
v=math.sqrt(3*g*R/2);    #initial velocity(km/s)

#Result
print "initial velocity is",round(v,1),"km/s"
print "answer given in the book is wrong"

initial velocity is 306.7 km/s
answer given in the book is wrong


## Example number 11, Page number 205¶

In [21]:
#importing modules
import math
from __future__ import division

#Variable declaration
m=10**-26;           #mass(kg)
G=6.67*10**-11;      #gravitational constant

#Calculation
V=math.sqrt(G*m/(4*R));     #velocity of particle(m/s)

#Result
print "velocity of particle is",round(V*10**14,2),"*10**-14 m/s"

velocity of particle is 5.77 *10**-14 m/s


## Example number 13, Page number 206¶

In [25]:
#importing modules
import math
from __future__ import division

#Variable declaration
m=1;           #mass(kg)
G=6.67*10**-11;      #gravitational constant

#Calculation
F=G*m/r**2;     #force(N/kg)
U=F*r;          #gravitational potential(J/kg)

#Result
print "gravitational potential is",U*10**10,"*10**-10 J/kg"

gravitational potential is 6.67 *10**-10 J/kg


## Example number 15, Page number 207¶

In [27]:
#importing modules
import math
from __future__ import division

#Variable declaration
rmax=1.2*10**12;      #semi minor axis(m)
rmin=0.06*10**12;     #semi major axis(m)

#Calculation
e=(rmax-rmin)/(rmax+rmin);      #eccentricity of orbit

#Result
print "eccentricity of orbit is",round(e,1)

eccentricity of orbit is 0.9


## Example number 16, Page number 207¶

In [30]:
#importing modules
import math
from __future__ import division

#Variable declaration
Vmin=21;        #minimum velocity(km/sec)
rmax=4*10**10;      #apogee position(m)
rmin=1.6*10**9;     #perigee position(m)

#Calculation
Vmax=Vmin*rmax/rmin;    #maximum velocity(km/sec)

#Result
print "maximum velocity is",int(Vmax),"km/sec"

maximum velocity is 525 km/sec


## Example number 17, Page number 208¶

In [36]:
#importing modules
import math
from __future__ import division

#Variable declaration
e=0.05;      #eccentricity of orbit
Vmin=36;        #minimum velocity(km/sec)
rmin=2.5*10**10;     #perigee position(m)

#Calculation
rmax=rmin*((1+e)/(1-e));      #apogee position(m)
Vmax=Vmin*rmin/rmax;          #velocity at apogee point(km/s)

#Result
print "apogee position is",round(rmax/10**10,2),"*10**10 m"
print "velocity at apogee point is",round(Vmax,2),"km/s"

apogee position is 2.76 *10**10 m
velocity at apogee point is 32.57 km/s


## Example number 18, Page number 208¶

In [38]:
#importing modules
import math
from __future__ import division

#Variable declaration
Vmin=23;        #minimum velocity(km/sec)
Vmax=25;          #velocity at apogee point(km/s)

#Calculation
e=(Vmax-Vmin)/(Vmax+Vmin);      #eccentricity of orbit

#Result
print "eccentricity of orbit is",round(e,4)

eccentricity of orbit is 0.0417


## Example number 20, Page number 208¶

In [40]:
#importing modules
import math
from __future__ import division

#Variable declaration
T=27*24*3600;      #time period(sec)
G=6.67*10**-11;    #gravitational constant

#Calculation
M=4*math.pi**2*r**3/(G*T**2);     #mass of earth(kg)

#Result
print "mass of earth is",round(M/10**24,3),"*10**24 kg"

mass of earth is 5.968 *10**24 kg


## Example number 21, Page number 209¶

In [43]:
#importing modules
import math
from __future__ import division

#Variable declaration
T1=225;          #time period of venus(days)
T2=365;          #time period of earth(days)

#Calculation
a1bya2=(T1/T2)**(2/3);     #ratio of semi major axis

#Result
print "ratio of semi major axis is",round(a1bya2,3)

ratio of semi major axis is 0.724


## Example number 22, Page number 209¶

In [45]:
#importing modules
import math
from __future__ import division

#Variable declaration
a2=1;            #assume
a1=1.25*a2;      #axis of planet
T2=365;          #time period of earth(days)

#Calculation
T1=T2*math.sqrt((a1/a2)**3);          #time period of planet(days)

#Result
print "time period of planet is",int(T1),"days"

time period of planet is 510 days


## Example number 23, Page number 209¶

In [47]:
#importing modules
import math
from __future__ import division

#Variable declaration
r1=1;               #assume
T1=24;          #time period of earth(hours)

#Calculation
T2=T1-(T1*((r2/r1)**2));          #change in time period(hours)

#Result
print "change in time period is",T2,"hours"

change in time period is 15.36 hours


## Example number 24, Page number 210¶

In [50]:
#importing modules
import math
from __future__ import division

#Variable declaration
R=1;       #assume
a1=R/2;
a=R;
T=365;     #time period of earth(days)

#Calculation
T1=T*math.sqrt((a1/a)**3)/2;     #time interval to reach sun(days)

#Result
print "time interval to reach sun is",round(T1,1),"days"

time interval to reach sun is 64.5 days


## Example number 25, Page number 210¶

In [56]:
#importing modules
import math
from __future__ import division

#Variable declaration

mass of sun is 2 *10**30 kg