Newtons Gravitational Law Assignment Help

Classical Physics - Newtons Gravitational Law

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 

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 = ?Uand 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 / x+ R2]2 GM / R[ 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 / x                  (x > R)

E in  = G Mx / R                                              ( x > R)

Variation of 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 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 m
v02 = GMm / 2a, P= - GM(m / a) 

Net energy E = KE + PE = - GMm/2a note ve = √2 
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