Newtons Gravitational Law
Newton in 1665 formulated F α m1 m2
∝ 1 / r2 F = Gm1m2 / r2 where G = 6.67 x 10-11. Its unit is Nm-2 and is called universal gravitational constant. The value of G was first experimentally determined by Cavendish in 1736. The value of G measured for small distances is about 1% less than the value of G measured for large distances.
Gravitational field intercity gravitational force per unit mass is called gravitational field intensity. Gravitational field intensity of earth is g
E8 = F / m = GM / r2
Gravitational potential (Vg) the amount of work done to bring a unit mass form infinity to that point under the influence of gravitational field of a given mass M, without changing the velocity.
Vg = GM /r
Gravitational potential energy the amount of work done to bring a mass m from infinity to that point under the influence of gravitational field of a given mass M without changing the velocity.
Ug = - GMm / r note that W = ?Ug and Ug = mVg.
Gravitational field intensity due to a ring of radius R, mass M, at any point on the axial line at a distance x from the centre of the ring is
Eg = G M .x / (R2 + x2)3/2
Gravitational field intercity inside the shell – 0
Eg = 2 GM / R2 [1 – x / x2 + R2]2 GM / R2 [ 1 – cos θ]
In terms of angleθ.
Gravitational field intensity inside the shell = 0,
E surface = GM / R2
Gravitational potential due to a shell
Vin = V sur = - GM / r (x ≤ r)
V out = - GM / x (x > r)
Gravitational potential due to a solid sphere
V in = - GM / 2 R3 (3 R2 – r2) V out = -- GM / x (x > R)
V centre = - 3 GM / 2R
Gravitational field due to a solid sphere
E sur = GM /R2, E out = GM / x2 (x > R)
E in = G Mx / R3 ( x > R)
Variation of g due to height g’ = g (1 – h /R) if h <<R
G’ g / (1 + h / R)2 if h is comparable to R
Variation of g due to depth
g’ = g (1 – x/ R) where x is the depth.
= O at the centre of the earth
Variation of g with rotation of earth/ latitude
G’ = g (1 – Rω2 / g cos2 λ)
That is g is maximum at the poles and minimum at the equator
Orbital velocity v0 √(G )M / r
Where v0 is speed with which a planet or a satellite moves in its orbit and r is the radius of the orbit.
Escape velocity v0 = √2GM / r
Escape velocity is the minimum velocity required to escape from the surface of the earth/planet from its gravitational field.
Time period T = 2 πr / v0 or T2 = 4π2 r3 / GM
Kinetic energy KE = 1/2 mv02 = GMm / 2a, PE = - GM(m / a)
Net energy E = KE + PE = - GMm/2a note ve = √2 v0.
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