Important Formulas: Gravitation

 Page 1


GRAVITATION : Universal Law of Gravitation
F ?
2
2 1
r
m m
 or F =
 
2
2 1
r
m m
G
where G = 6.67 × 10
–11
 Nm
2
 kg
–2
 is the universal gravitational constant.
Newton's Law of Gravitation in vector form :
12
F
?
 = 
2
2 1
r
m Gm
 12
r ˆ
      &  
1 2
F
?
 = 
2
2 1
r
m Gm
    
Now
21 12
r ˆ r ˆ ? ?  , Thus 
12
2
2 1
21
r ˆ
r
m m G
F
?
?
?
.
Comparing above, we get 
21 12
F F
? ?
? ?
Gravitational Field E = 
m
F
 = 
2
r
GM
Gravitational potential : gravitational potential,
V = – 
r
GM
. E = – 
dr
dV
.
1. Ring. V = 
2 / 1 2 2
) r a ( or x
GM
?
?
   &   E = 
r
ˆ
) r a (
r GM
2 / 3 2 2
?
?
or  E = – 
2
x
cos GM ?
Page 2


GRAVITATION : Universal Law of Gravitation
F ?
2
2 1
r
m m
 or F =
 
2
2 1
r
m m
G
where G = 6.67 × 10
–11
 Nm
2
 kg
–2
 is the universal gravitational constant.
Newton's Law of Gravitation in vector form :
12
F
?
 = 
2
2 1
r
m Gm
 12
r ˆ
      &  
1 2
F
?
 = 
2
2 1
r
m Gm
    
Now
21 12
r ˆ r ˆ ? ?  , Thus 
12
2
2 1
21
r ˆ
r
m m G
F
?
?
?
.
Comparing above, we get 
21 12
F F
? ?
? ?
Gravitational Field E = 
m
F
 = 
2
r
GM
Gravitational potential : gravitational potential,
V = – 
r
GM
. E = – 
dr
dV
.
1. Ring. V = 
2 / 1 2 2
) r a ( or x
GM
?
?
   &   E = 
r
ˆ
) r a (
r GM
2 / 3 2 2
?
?
or  E = – 
2
x
cos GM ?
Gravitational field is maximum at a distance,
 r = ± 2 a and it is – 
2
a 3 3 GM 2
2. Thin Circular Disc.
 V = 
? ?
?
?
?
?
?
?
?
?
? ?
?
r r a
a
GM 2
2
1
2 2
2
   &   E = – 
? ?
?
?
?
?
?
?
?
?
?
?
?
?
2
1
2 2
2
a r
r
1
a
GM 2
 = – 
? ? ? ? cos 1
a
GM 2
2
3. Non conducting solid sphere
(a) Point P inside the sphere.  r < a, then
V = 
) r a 3 (
a 2
GM
2 2
3
? ?
 & E = – 
3
a
r GM
, and at the centre V = – 
a 2
GM 3
 and E = 0
(b) Point P outside the sphere .
r > a, then   V = 
r
GM
?     & E = – 
2
r
GM
4. Uniform Thin Spherical Shell / Conducting solid sphere
(a) Point P Inside the shell.
r < a , then V = 
a
GM ?
& E = 0
(b) Point P outside shell.
r > a, then  V = 
r
GM ?
& E = – 
2
r
GM
VARIATION OF ACCELERATION DUE TO GRAVITY :
1. Effect of Altitude
g
h
 = 
? ?
2
e
e
h R
GM
?
= g 
2
e
R
h
1
?
?
?
?
?
?
?
?
?
? ~    g 
?
?
?
?
?
?
?
?
?
e
R
h 2
1 when h << R.
2. Effect of depth g
d
 = g 
?
?
?
?
?
?
?
?
?
e
R
d
1
3. Effect of the surface of Earth
The equatorial radius is about 21 km longer than its polar radius.
We know, g = 
2
e
e
R
GM
 Hence g
pole
 > g
equator
.
SATELLITE VELOCITY (OR ORBITAL VELOCITY)
v
0
 = 
? ?
2
1
e
e
h R
GM
?
?
?
?
?
?
?
 =  
? ?
2
1
e
2
e
h R
R g
?
?
?
?
?
?
?
?
?
Page 3


GRAVITATION : Universal Law of Gravitation
F ?
2
2 1
r
m m
 or F =
 
2
2 1
r
m m
G
where G = 6.67 × 10
–11
 Nm
2
 kg
–2
 is the universal gravitational constant.
Newton's Law of Gravitation in vector form :
12
F
?
 = 
2
2 1
r
m Gm
 12
r ˆ
      &  
1 2
F
?
 = 
2
2 1
r
m Gm
    
Now
21 12
r ˆ r ˆ ? ?  , Thus 
12
2
2 1
21
r ˆ
r
m m G
F
?
?
?
.
Comparing above, we get 
21 12
F F
? ?
? ?
Gravitational Field E = 
m
F
 = 
2
r
GM
Gravitational potential : gravitational potential,
V = – 
r
GM
. E = – 
dr
dV
.
1. Ring. V = 
2 / 1 2 2
) r a ( or x
GM
?
?
   &   E = 
r
ˆ
) r a (
r GM
2 / 3 2 2
?
?
or  E = – 
2
x
cos GM ?
Gravitational field is maximum at a distance,
 r = ± 2 a and it is – 
2
a 3 3 GM 2
2. Thin Circular Disc.
 V = 
? ?
?
?
?
?
?
?
?
?
? ?
?
r r a
a
GM 2
2
1
2 2
2
   &   E = – 
? ?
?
?
?
?
?
?
?
?
?
?
?
?
2
1
2 2
2
a r
r
1
a
GM 2
 = – 
? ? ? ? cos 1
a
GM 2
2
3. Non conducting solid sphere
(a) Point P inside the sphere.  r < a, then
V = 
) r a 3 (
a 2
GM
2 2
3
? ?
 & E = – 
3
a
r GM
, and at the centre V = – 
a 2
GM 3
 and E = 0
(b) Point P outside the sphere .
r > a, then   V = 
r
GM
?     & E = – 
2
r
GM
4. Uniform Thin Spherical Shell / Conducting solid sphere
(a) Point P Inside the shell.
r < a , then V = 
a
GM ?
& E = 0
(b) Point P outside shell.
r > a, then  V = 
r
GM ?
& E = – 
2
r
GM
VARIATION OF ACCELERATION DUE TO GRAVITY :
1. Effect of Altitude
g
h
 = 
? ?
2
e
e
h R
GM
?
= g 
2
e
R
h
1
?
?
?
?
?
?
?
?
?
? ~    g 
?
?
?
?
?
?
?
?
?
e
R
h 2
1 when h << R.
2. Effect of depth g
d
 = g 
?
?
?
?
?
?
?
?
?
e
R
d
1
3. Effect of the surface of Earth
The equatorial radius is about 21 km longer than its polar radius.
We know, g = 
2
e
e
R
GM
 Hence g
pole
 > g
equator
.
SATELLITE VELOCITY (OR ORBITAL VELOCITY)
v
0
 = 
? ?
2
1
e
e
h R
GM
?
?
?
?
?
?
?
 =  
? ?
2
1
e
2
e
h R
R g
?
?
?
?
?
?
?
?
?
When h << R
e
  then v
0
 = 
e
R g
? v
0
 = 
6
10 4 . 6 8 . 9 ? ?
 = 7.92 × 10
3
 ms
–1
 = 7.92 km s
1
Time period of Satellite
T = 
? ?
? ?
2
1
e
2
e
e
h R
R g
h R 2
?
?
?
?
?
?
?
?
?
? ?
  = 
? ?
2
1
3
e
e
g
h R
R
2
?
?
?
?
?
?
?
?
? ?
Energy of a Satellite
U = 
r
m GM
e
?
 K.E. = 
r 2
m GM
e
 ; then total energy ? ? ? ?E = – 
e
e
R 2
m GM
Kepler's Laws
Law of area :
The line joining the sun and a planet sweeps out equal areas in equal
intervals of time.
Areal velocity   = 
time
swept area
= 
dt
) rd ( r
2
1
?
 =7 
2
1
 r
2
 
dt
d ?
 = constant .
Hence 
2
1
 r
2
 ? = constant.  Law of periods :
3
2
R
T
 = constant