Physics · Gravitation · NEET
Core set: (1) Force between two masses F = Gm1m2/r squared. (2) Surface gravity g = GM/R squared, and g = (4/3) pi G rho R. (3) g at height h: g_h = g(1 - 2h/R) for small h, or g_h = g R squared/(R+h) squared for any h. (4) g at depth d: g_d = g(1 - d/R). (5) Gravitational PE: U = -GMm/r. (6) Gravitational potential: V = -GM/r. (7) Escape velocity ve = sqrt(2GM/R) = sqrt(2gR) = 11.2 km/s for Earth. (8) Orbital velocity vo = sqrt(GM/r). (9) Satellite time period T = 2 pi sqrt(r cubed/GM). (10) Satellite energies: KE = +GMm/2r, PE = -GMm/r, Total E = -GMm/2r. (11) Kepler's third law T squared proportional to r cubed. Learn these 11 lines and most Gravitation MCQs become plug-and-play.
They differ only by a factor of root 2. Orbital velocity vo = sqrt(GM/r) is the speed to STAY in a circle. Escape velocity ve = sqrt(2GM/r) is the speed to LEAVE forever. So ve = sqrt(2) x vo at the same radius (about 1.41 times faster). At the Earth's surface vo is about 7.9 km/s and ve is about 11.2 km/s. Hook: 'escape needs root-2 more than orbit.'
At small height h: g_h = g(1 - 2h/R), the factor is 2h/R. At depth d: g_d = g(1 - d/R), the factor is just d/R (no 2). So g falls TWICE as fast per km going up as going down near the surface. That is why 'height h = 1 km equals depth d = 2 km' gives the same g. Both give g = 0 far away or at the centre respectively.
By question count: escape velocity scaling ve proportional to R sqrt(rho), satellite energy E = -GMm/2r, Kepler's third law T squared proportional to r cubed, variation of g with height and depth, and gravitational potential/PE change. If you are short on time, master these five families first, they cover the majority of Gravitation PYQs.
Replace M with (4/3) pi R cubed rho (mass = density x volume). Then g = (4/3) pi G rho R and ve = R sqrt(8 pi G rho/3), so ve is proportional to R sqrt(rho). This 'density form' is the fast path for scaling questions like 'a planet with twice the radius and twice the density' because you compare R and rho directly instead of recomputing M.
A body weighs 200 N on the surface of the earth. How much will it weigh half way down to the centre of the earth?
The escape velocity from the earth's surface is v. The escape velocity from the surface of another planet of radius four times that of earth and the same mean density is:
A satellite has period 24 h at height 6 R_E from the earth's surface. The period of another satellite at height 2.5 R_E from the surface is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
The sheet gets you most MCQs because Gravitation is heavily formula-driven, but you must also know two concept ideas: total satellite energy is negative (bound orbit), and a planet moves fastest at perihelion (Kepler's second law). Pair the 11 formulas with these two facts and you cover nearly the whole chapter.
G is the universal gravitational constant, G = 6.67 x 10 to the -11 N m squared/kg squared, the same everywhere in the universe. Small g is the acceleration due to gravity, about 9.8 m/s squared on Earth's surface, and it changes with height, depth and planet. They are linked by g = GM/R squared.
For a bound orbit, KE = +GMm/2r but PE = -GMm/r, so total E = KE + PE = -GMm/2r, which is negative. The negative sign means the satellite is bound to Earth and needs positive energy (the binding energy = +GMm/2r) to escape to infinity.
They share the factor root 2: ve = sqrt(2) x vo at the same radius. So orbital velocity vo = ve/sqrt(2). At Earth's surface ve = 11.2 km/s, so vo is about 11.2/1.41 = 7.9 km/s. Learn one and derive the other in the exam.
