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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.
     Notes | EduRevT∝ R3
    Here R is the radius of orbit.
    T2 = (4π2/GM)R 3
  • Newton’s law of gravitation:-
     Notes | EduRev
    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 = GMe/Re2 , therefore gpole > gequator
  • Gravitational Mass:- m = FR2/GM
  • Gravitational field intensity:-
    E = F/m
    = GM/r2
  • Weight:- W= mg
  • Gravitational intensity on the surface of earth (Es):-
    Es = 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,
    Vs=  -GM/R
    Here R is the radius of earth.
  • Escape velocity (ve):-
     Notes | EduRev
    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.
    ve = √2GM/R
    or, ve = √2gR = √gD
    Here R is the radius of earth and D is the diameter of the earth.
  • Escape velocity (ve) in terms of earth’s density:- ve = 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,
    v= √gR
  • Relation between escape velocity ve and orbital velocity v0 :- v0= ve/√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:-
    gm = GMm/Rm2 = 1/6 gearth 
    Here Mm is the mass of moon and Rm is the radius of moon.
  • Gravitational field:-
    (a) Inside:-
     Notes | EduRev
    (b) Outside:-
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  • GRAVITATIONAL POTENTIAL & FIELD DUE TO VARIOUS OBJECTS
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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,
     Notes | EduRev
    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
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