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 = 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):- 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, v0 = √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:- (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
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