# CHAPTER11 : GRAVITATION¶

## Example E1 : Pg 204¶

In :
import math
#example 11.1
#calculation of the initial acceleration of the particles

#given data
m1=1.#masss(in kg) of particle1
m2=2.#masss(in kg) of particle2
r=50.*10.**-2.#separation(in m) between the two particles
G=6.67*10.**-11.#universal constant of gravitation(in N-m**2/kg**2)

#calculation
F=G*m1*m2/(r*r)#force of gravitation
a1=F/m1#initial acceleration of the particle1
a2=F/m2#initial acceleration of the particle2

print '%s %.2f %s' %("the initial acceleration of the particle1 towards particle2 is",a1,"m/s**2")
print '%s %.2f %s' %("the initial acceleration of the particle2 towards particle1 is",a2,"m/s**2")

the initial acceleration of the particle1 towards particle2 is 0.00 m/s**2
the initial acceleration of the particle2 towards particle1 is 0.00 m/s**2


## Example E2 : Pg 207¶

In :
import math
#example 11.2
#calculation of the work done in bringing three particles together

#given data
m1=100.*10.**-3.#masss(in kg) of particle1
r=20.*10.**-2.#separation(in m) between the two particles
G=6.67*10.**-11.#universal constant of gravitation(in N-m**2/kg**2)

#calculation
#since the work done by the gravitational force is equal to change in the potential energy
U=3.*(-G*m1*m1/r)

print '%s %.2f %s' %("the work done in bringing three particles is",U,"J")

the work done in bringing three particles is -0.00 J


## Example E4 : Pg 210¶

In :
import math
#example 11.4
#calculation of the gravitational field

#given data
F=2.#gravitational force(in N)
m=50.*10.**-3.#mass(in kg) of the particle

#calculation
E=F/m#gravitational field

print '%s %.2f %s' %("the gravitational field along the direction of force is",E,"N/kg\n")

the gravitational field along the direction of force is 40.00 N/kg



## Example E7 : Pg 214¶

In :
import math
#example 11.7
#calculation of the gravitational field due to the moon at its surface

#given data
M=7.36*10.**22.#mass(in kg) of the moon
G=6.67*10.**-11.#universal constant of gravitation(in N-m**2/kg**2)

#calculation
E=G*M/(a*a)#formula of gravitational field

print '%s %.2f %s' %("the gravitational field due to the moon at its surface is",E,"N/kg\n")

the gravitational field due to the moon at its surface is 1.62 N/kg



## Example E8 : Pg 215¶

In :
import math
#example 11.8
#calculation of the value of acceleration due to gavity

#given data
h=5.*10.**3.#height(in m) above the earth's surface
g0=9.8#gravitational acceleration(in m/s**2) of the earth
d=5.*10.**3.#depth(in m) below the earth's surface

#calculation
gh=g0*(1.-(2.*h/R))#formula of gravitational acceleration at height h above the earth's surface
gd=g0*(1.-(d/R))#formula of gravitational acceleration at depth d below the earth's surface

print '%s %.2f %s' %("the value of gravitational acceleration at height 5 km above the earth surface is",gh,"m/s**2\n")
print '%s %.2f %s' %("the value of gravitational acceleration at depth 5 km below the earth surface is",gd,"m/s**2\n")

the value of gravitational acceleration at height 5 km above the earth surface is 9.78 m/s**2

the value of gravitational acceleration at depth 5 km below the earth surface is 9.79 m/s**2



## Example E9 : Pg 216¶

In :
import math
#example 11.9
#calculation of the speed and time period of the satellite

#given data
h=600.*10.**3.#height(in m) of the satellite
M=6.*10.**24.#mass(in kg) of the earth
G=6.67*10.**-11.#universal constant of gravitation(in N-m**2/kg**2)

#calculation
a=h+R#distance of satellite from centre of the earth
v=math.sqrt(G*M/a)#speed of satellite
T=(2.*math.pi*a)/v#time period of satellite

print '%s %.2f %s %.2f %s' %("the speed of the satellite is",v,"m/s\n or",v*10**-3,"km/s\n")
print '%s %.2f %s' %("the time period of the satellite is",T,"s\n")

the speed of the satellite is 7561.18 m/s
or 7.56 km/s

the time period of the satellite is 5816.86 s



## Example E10 : Pg 218¶

In :
import math
#example 11.10
#calculation of the escape velocity from the moon

#given data
M=7.4*10.**22.#mass(in kg) of the moon
G=6.67*10.**-11.#universal constant of gravitation(in N-m**2/kg**2)

#calculation
v=math.sqrt(2.*G*M/R)#formula of the escape velocity

print '%s %.2f %s' %("the escape velocity from the moon is",v*10**-3,"km/s")

the escape velocity from the moon is 2.38 km/s


# WORKED EXAMPLES¶

## Example E2w : Pg 220¶

In :
import math
#example 11.2w
#calculation of the distance from the earth's surface where resultant gravitational field due to the earth and the moon is zero

#given data
Me=6.*10.**24.#mass(in kg) of the earth
Mm=7.4*10.**22.#mass(in kg) of the moon
d=4.*10.**5.*10.**3.#distance(in m) between the earth and the moon

#calculation
#gravitational field due to the earth at that point
#E1 = G*Me/x**2.........................(1)
#gravitational field due to the moon at that point
#E2 = G*Mm/(d-x)**2.....................(2)
#E1 = E2.....given
x=(d*math.sqrt(Me/Mm))/(1.+math.sqrt(Me/Mm))

print '%s %.2f %s' %("the distance from the earth surface where resultant gravitational field due to the earth and the moon is zero is",x*10**-3,"km\n")

the distance from the earth surface where resultant gravitational field due to the earth and the moon is zero is 360018.01 km



## Example E4w : Pg 220¶

In :
import math
#example 11.4w
#calculation of the separation between the particles under mutual attraction

#given data
mA=1.#mass(in kg) of particle A
mB=2.#mass(in kg) of particle B
R=1.#initial distance(in m) between the two particles
vB=3.6*10.**-2./(60.*60.)#speed(in m/s) of the particle B
G=6.67*10.**-11.#universal constant of gravitation(in N-m**2/kg**2)

#calculation
v=(mB*vB)/mA#principle of conservation of linear momentum
U1=-G*mA*mB/R#initial potential energy of the pair
d=U1/(U1-(mB*vB*vB/2.)-(mA*v*v/2.))#principle of conservation of energy

print '%s %.2f %s' %("the speed of particle A is",v,"m/s\n")
print '%s %.2f %s' %("the separation between the particles under mutual attraction is",d,"m\n")

the speed of particle A is 0.00 m/s

the separation between the particles under mutual attraction is 0.31 m



## Example E5w : Pg 220¶

In :
import math
#example 11.5w
#calculation of the work done by an external agent

#given data
#E = (10 N/kg)(i + j).....given gravitational field
Ex=10.#value of X-component of gravitational field(in N/kg)
Ey=10.#value of Y-component of gravitational field(in N/kg)
m=2.#mass(in kg) of the gravitational field
x0=0#value of X component of initial location(in m)
x1=5.#value of X component of final location(in m)
y0=0#value of Y component of initial location(in m)
y1=4.#value of Y component of final location(in m)

#calculation
def fx(x) :
Fx=m*Ex#value of X component of force
return Fx

def fy(x):
Fy=m*Ey
return Fy

#calculation
W1=100.;#integrate('fx','x',x0,x1)#work done by X component of external force
W2=80.;#integrate('fy','x',y0,y1)#work done by Y component of external force

W=W1+W2

print '%s %.2f %s' %("the work done by the external agent is",-W,"J\n")

the work done by the external agent is -180.00 J



## Example E9w : Pg 222¶

In :
import math
#example 11.9w
#calculation of the maximum height attained by the particle

#given data
v0=9.8*10.**3.#speed(in m/s) the particle is fired
g=9.8#gravitational acceleration(in m/s**2) of the earth

#calculation
#by the principle of conservation of energy
#(-G*M*m/R) + (m*v0*v0/2) = -(G*M*m/(R+H))
H=(R*R/(R-(v0*v0/(2.*g))))-R

print '%s %.2f %s' %("the maximum height attained by the particle is",H*10**-3,"km\n")

the maximum height attained by the particle is 20906.67 km



## Example E10w : Pg 222¶

In :
import math
#example 11.10w
#calculation of the stretch produced in the spring
#given data
d=1.*10.**-2.#stretch(in m) of the spring
h=800.*10.**3.#height(in m) above the earths surface

#calculation
#The extension in the spring on the surface is
#1*10**-2 = (G*M*m)/(k*R**2)...........(1)
#The extension in the spring at height h above the surface
#x = (G*M*m)/(k*(R+h)**2).............(2)
#from above equations,we get
x=d*((R**2.)/(R+h)**2.)

print '%s %.2f %s' %("the stretch produced in the spring is",x*10**2,"cm\n")

the stretch produced in the spring is 0.79 cm



## Example E11w : Pg 222¶

In :
import math
#example 11.11w
#calculation of time period of the pendulum if used at the equator

#given data
t=2.#time period (in s) of the pendulum at North pole
g=9.8#gravitational acceleration(in m/s**2) of the earth
G=6.67*10.**-11.#universal constant of gravitation(in N-m**2/kg**2)
w=(2.*math.pi)/(24.*60.*60.)#angular velocity(in rad/s) of the earth

#calculation
#By equilibrium conditions,we get
#t = 2*%pi*sqrt(l/g)..............................(1)
#tdash = 2*%pi*sqrt(l/(g-(w*w*R)).................(2)
#from equations (1) and (2),we get
tdash=t*(1.+(w*w*R/(2.*g)))

print '%s %.2f %s' %("the value of time period of the pendulum if used at the equator is",tdash,"s\n")

the value of time period of the pendulum if used at the equator is 2.00 s



## Example E12w : Pg 222¶

In :
import math
#example 11.12w
#calculation of the speed of projection of the satellite into an orbit

#given data
r=8000.*10.**3.#radius(in m) of the orbit of the satellite
g=9.8#gravitational acceleration(in m/s**2) of the earth

#calculation
#using Newtons second law
#(G*M*m/(r*r)) = m*v*v/r
v=math.sqrt(g*R*R/r)
t=(2.*math.pi*r/v)#time period of the satellite

print '%s %.2f %s' %("the speed of projection of the satellite into the orbit is",v*10**-3,"km/s\n")
print '%s %.2f %s' %("the time period of the satellite in the orbit is",t*(1/(60)),"minutes\n")

the speed of projection of the satellite into the orbit is 7.08 km/s

the time period of the satellite in the orbit is 0.00 minutes



## Example E13w : Pg 223¶

In :
import math
#example 11.13w
#calculation of the speed and the angular speed of the satellite S2 relative to the satellite S1

#given data
T1=1.#period of revolution(in h) of satellite S1
T2=8.#period of revolution(in h) of satellite S2
R1=10.**4.#radius(in km) of the orbit of satellite S1

#calculation
#by Kelplers third law
#(R2/R1)**3 = (T2/T1)**2
R2=R1*(((T2/T1)**2.)**(1./3.))
v1=(2.*math.pi*R1/T1)#speed(in km/h) of satellite S1
v2=(2.*math.pi*R2/T2)#speed(in km/h) of satellite S2
v=abs(v2-v1)#speed of satellite S2 with respect to satellite S1
w=v/(R2-R1)#angular speed of satellite S2 as observed by an astronaut in satellite S1

print '%s %.2f %s' %("the speed of the satellite S2 with respect to the satellite S1 is",v,"km/h\n")
print '%s %.2f %s' %("the angular speed of the satellite S2 as observed by an astronaut in the satellite S1 is",w,"rad/h\n")

the speed of the satellite S2 with respect to the satellite S1 is 31415.93 km/h

the angular speed of the satellite S2 as observed by an astronaut in the satellite S1 is 1.05 rad/h