Gravitation
Important Points(Part - II)
The gravitational potential energy of a mass m at a point above the surface of the earth at a height h is given by 
. The negative sign implies that as R increases, the gravitational potential energy decreases and becomes zero at infinity.
The body is moved from the surface of earth to a point at a height h above the surface of earth then change in potential energy will be mgh.
Gravitational potential at a point above the surface of the earth at a height h is _ GM/(R + h).
Its unit is joule/kilogram.
Gravitational mass, 
is defined by Newton's law of gravitation.
= 
= 
= 
= 
Let 
= angular speed of the satellite, 
= orbital speed of the satellite, then 
= 
, where R = radius of the earth and h = height of the satellite above the surface of the earth. Let g = acceleration due to gravity on the surface of the earth, T = time period of the satellite, M = mass of the earth. Then different quantities connected with satellite at height h are as follows:
(a) 
= 
= R 
.
(b) T = 
and frequency of revolution, v = 
.
(c) v0 = 
R = 
Very near the surface of the earth, we get the values by putting h = 0. That is:
(i) 
= 
= 
(ii)
= 
= 
(iii)T 
= 5078 sec = 1 hour 24.6 minute.
Altitude or height of satellite above the earth's surface, h = 
.
Angular momentum, L = mv
= 
.
Above the surface of the earth, the acceleration due to gravity varies inversely as the square of the distance from the
centre of the earth.
= 
.
The gravitational potential energy of a satellite of mass m is U = 
, where r is the radius of the orbit.
It is negative.
Kinetic energy of the satellite is K = 
= 
.
Total energy of the satellite E = U + K = _
Negative sign indicates that it is the binding energy of the statellite.
Total energy of a satellite at a height equal to the radius of the earth is given by
= 
= 
.
where g = 
is the acceleration due to gravity on the surface of the earth.
When the total energy of the satellite becomes zero or greater than zero, the satellite escapes from the gravitational pull of the earth.
If the radius of planet decreases by
n%, keeping the mass unchanged, the acceleration due to gravity on its surface increases by 2n% i.e. 
= 
.
If the mass of the planet increases by m% keeping the radius constant, the acceleration due to gravity on its surface increases by m% 
where R = constant.
If the density of planet decreases by p% keeping the radius constant, the acceleration due to gravity decreases by p%.
If the radius of the planet decreases by q% keeping the density constant, the acceleration due to gravity decreases by q% .