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 g 

E₈ = F / m = GM / r²

Gravitational potential (V_g)  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. 

V_g = 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. 

U_g = - GMm / r note that W = ?U_g and U_g = 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

E_g = G M .x / (R² + x²)^3/2

Gravitational field intercity inside the shell – 0

E_g = 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 K_E = 1/2 mv₀² = GMm / 2a, P_E = - GM(m / a) 

Net energy E = K_E + P_E = - GMm/2a note v_e = √2 v₀. 

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