Notes | EduRev

Gravitation:-

• Kepler’s first law (law of elliptical orbit):- A planet moves round the sun in an elliptical orbit with sun situated at one of its foci.
• Kepler’s second law (law of areal velocities):- A planet moves round the sun in such a way that its areal velocity is constant.
• Kepler’s third law (law of time period):- A planet moves round the sun in such a way that the square of its period is proportional to the cube of semi major axis of its elliptical orbit.
  T² ∝ R³
  Here R is the radius of orbit.
  T² = (4π²/GM)R ³
• Newton’s law of gravitation:-
  
  Every particle of matter in this universe attracts every other particle with a forcer which varies directly as the product of masses of two particles and inversely as the square of the distance between them.
  F= GMm/r²
  Here, G is universal gravitational constant. G = 6.67 ´10 ^-11 Nm² / kg²
• Dimensional formula of G: G = Fr²/Mm =[MLT^-2][L²]/[M²] = [M^-1L³T^-2]
• Acceleration due to gravity (g):- g = GM/R²
• Variation of g with altitude:- g' = g(1- 2h/R),  if h<<R. Here R is the radius of earth and h is the height of the body above the surface of earth.
• Variation of g with depth:- g' = g(1- d/R). Here g' be the value of acceleration due to gravity at the depth d.
• Variation with latitude:-
  At poles:- θ = 90°, g' = g
  At equator:- θ = 0°, g' = g (1-ω²R/g)
  Here ω is the angular velocity.
• As g = GM_e/R_e² , therefore g_pole > g_equator
• Gravitational Mass:- m = FR²/GM
• Gravitational field intensity:-
  E = F/m
  = GM/r²
• Weight:- W= mg
• Gravitational intensity on the surface of earth (E_s):-
  E_s = 4/3 (πRρG)
  Here R is the radius of earth, ρ is the density of earth and G is the gravitational constant.
• Gravitational potential energy (U):- U = -GMm/r
  (a) Two particles: U = -Gm₁m₂/r
  (b) hree particles: U = -Gm₁m₂/r₁₂ – Gm₁m₃/r₁₃ – Gm₂m₃/r₂₃
• Gravitational potential (V):- V(r) =  -GM/r
  At surface of earth,
  V_s=  -GM/R
  Here R is the radius of earth.
• Escape velocity (v_e):-
  
  It is defined as the least velocity with which a body must be projected vertically upward   in order that it may just escape the gravitational pull of earth.
  v_e = √2GM/R
  or, v_e = √2gR = √gD
  Here R is the radius of earth and D is the diameter of the earth.
• Escape velocity (v_e) in terms of earth’s density:- v_e = R√8πGρ/3
• Orbital velocity (v₀):-
  v₀ = √GM/r
  If a satellite  of mass m revolves in a circular orbit around the earth of radius R and h be the height of the satellite above the surface of the earth, then,
  r = R+h
  So, v₀ = √MG/R+h = R√g/R+h
  In the case of satellite, orbiting very close to the surface of earth, then orbital velocity will be,
  v₀ = √gR
• Relation between escape velocity v_e and orbital velocity v₀ :- v₀= v_e/√2  (if h<<R)
• Time period of Satellite:- Time period of a satellite is the time taken by the satellite to complete one revolution around the earth.
  T = 2π√(R+h)³/GM = (2π/R)√(R+h)³/g
  If h<<R, T = 2π√R/g
• Height of satellite:- h = [gR2T2/4π2]1/3 – R
• Energy of satellite:-
  Kinetic energy, K = ½ mv₀² = ½ (GMm/r)
  Potential energy, U = - GMm/r
  Total energy, E = K+U
  = ½ (GMm/r) + (- GMm/r)
  = -½ (GMm/r)
• Gravitational force in terms of potential energy:- F = – (dU/dR)
• Acceleration on moon:-
  g_m = GM_m/R_m² = 1/6 g_earth 
  Here Mm is the mass of moon and Rm is the radius of moon.
• Gravitational field:-
  (a) Inside:-
  
  (b) Outside:-
  
  
• GRAVITATIONAL POTENTIAL & FIELD DUE TO VARIOUS OBJECTS
  
  
  
  
  
  
  

Projectile:-

• Projectile fired at angle α  with the horizontal:- If a particle having initial speed u is projected at an angle α (angle of projection) with x-axis, then,
  
  Time of Ascent, t = (u sinα)/g
  Total time of Flight, T = (2u sinα)/g
  Horizontal Range, R = u²sin²α/g
  Maximum Height, H = u²sin²α/2g
  Equation of trajectory, y = xtanα-(gx²/2u²cos²α)
  Instantaneous velocity, V=√(u²+g²t²-2ugt sinα)
  and        
  β = tan-1(usinα-gt/ucosα)
• Projectile fired horizontally from a certain height:-
  Equation of trajectory: x² = (2u²/g)y
  Time of descent (timer taken by the projectile to come down to the surface of earth), T = √2h/g
  Horizontal Range, H = u√2h/g. Here u is the initial velocity of the body in horizontal direction.
  
  Instantaneous velocity:-
  V=√u²+g²t²
  If β be the angle which V makes with the horizontal, then,
  β = tan^-1(-gt/u)
• Projectile fired at angle α with the vertical:-
  Time of Ascent, t = (u cosα)/g
  Total time of Flight, T = (2u cosα)/g
  Horizontal Range, R = u²sin²α/g
  Maximum Height, H = u²cos²α/2g
  Equation of trajectory, y = x cotα-(gx²/2u²sin²α)
  Instantaneous velocity, V=√(u²+g²t²-2ugt cosα) and β = tan^-1(ucosα-gt/usinα)
• Projectile fired from the base of an inclined plane:-
  Horizontal Range, R = 2u² cos(α+β) sinβ/gcos²α
  Time of flight, T = 2u sinβ/ gcosα
  Here, α+β=θ