Gravitation

Universal Law of Gravitation
F = G m1m2 / r2

Gravitational Force is independent of the medium. The gravitational force inside a body is zero.

Gravitational Constant
G = 6.67 x 10-11 Nm2/kg2

Variation with Height
g = GM / (R+h)2gh = g (R/(R+h)2)        gh = g (1 - 2h/R)

Variation with Depth
g = gh(1 - d/R)

for every percentage decrease in radius, with mass constant ; there is 2 % increase in acceleration due to gravity

Variation with Latitude
g' = g - Rω2cos2Φ

Relation between Variation of height and depth
If acceleration due to gravity is same at a certain height and depth ;

then d = 2h

Gravitational Field Intensity
Gravitational field is the field of attraction around a mass. Gravitational Field Intensity is the amount of Force experienced by an object of unit mass in a gravitational Field.

... Intensity = Force / mass

Gravitational Potential Energy
Gravitational Potential Energy is the work done in bringing a charge of unit mass from infinity to a point against the gravitational field.

Potential Energy (U) = Force. Displacement

= - GMm /r

... U = Vm

Gravitational Potential
Gravitational potential is the amount of work done per unit mass in a gravitational field.

Potential(V) = - GM / r

... V = U / m

Gravitational Potential is always directed opposite to Gravitational Field.

Cases of Potential :
 * 1) Due to point mass :  V = - GM/r  ... at r = 0, Z is negative infinite . The graph will be negative , curved and parallel to V and r axes.
 * 2) Inside a Solid Sphere : V = - GM (1.5R2 - 0.5r2) / R3. At r=R ,V = - GM/R . At r = 0 ,V = -1.5GM/R  The graph is negative, parabolic.
 * 3) Outside a Solid Sphere : V = -GM/r, at r = infinity , V = 0 ; On surface , V = -GM/R
 * 4) Due to a Spherical Shell : Outside  V = -GM/r . Inside V = - GM/r
 * 5) On the Axis of a Ring : V = - GM / sqrt(R2+x2)

Law of Orbits
Kepler's Law of Orbits states that all planets revolve around the sun in elliptical orbits with sun as the focus.

Law of Areas
Kepler's Law of Areas states that a planet sweeps equal Areas in equal intervals of time around the sun.

Law of Periods
Kepler's Law of Periods states that the square of Time Period is directly proportional to the cube of Distance.

... T2 α r3

Critical Velocity
vc = sqrt (GM/r)

vc = sqrt (ghr)

Time Period
T = 2π sqrt (r3 / GM)

Binding Energy
B.E. = - T.E.

B.E. = GMm / r   {At rest}

B.E. = (GMm) / 2r

Escape Velocity
ve = sqrt (2ghr)

ve= sqrt (2GM/r)

The escape velocity remains independent of angle of projection.

Escape Velocity for an object on earth is 11.2 km/s