Gravitation

1. Kepler's Laws

Three laws governing planetary motion around the Sun.

1. Law of Orbits: Planets move in elliptical orbits with Sun at one focus.
2. Law of Areas: Areal velocity is constant (dA/dt = L/2m). (Conservation of Angular Momentum).
3. Law of Periods: T² ∝ R³ (Square of time period is proportional to cube of semi-major axis).

(Asked in NEET 2018, 2021)

2. Universal Law of Gravitation

Every particle attracts every other particle with a force proportional to product of masses and inversely proportional to square of distance.

F = - (G m₁ m₂ / r²) r̂
G = 6.67 × 10^-11 N m²/kg².

Acceleration due to Gravity (g)

Surface: g = GM / R²
Height h: g_h = g (1 - 2h/R) for h << R.
Depth d: g_d = g (1 - d/R).
Center of Earth: g = 0.

(Asked in NEET 2016, 2019, 2020, 2022)

3. Potential & Energy

Potential (V): -GM/r
Potential Energy (U): -GMm/r (Zero at infinity).
Work done W = ΔU = m(V₂ - V₁).

Escape Velocity

Minimum speed to escape gravitational pull.

v_e = √(2GM/R) = √(2gR)
For Earth, v_e = 11.2 km/s.

4. Satellite Motion

Body revolving around a planet.

Orbital Speed (v_o): √(GM/r) = √(GM/(R+h)).
Near surface: v_o = √(gR) ≈ 8 km/s.
Relation: v_e = √2 v_o.

Energy of Satellite

KE = ½ m v_o² = GMm / 2r
PE = -GMm / r
Total Energy E = -GMm / 2r (Negative means bound system).

Geostationary Satellite

Time Period T = 24 Hours.
Height h ≈ 36,000 km.
Orbits in Equatorial Plane.
Used for Communication.

(Asked in NEET 2017, 2019)

Numericals: Gravitation

Q1. Two spheres of mass 50 kg and 10 kg are 0.5 m apart. Calculate gravitational force between them.

Solution:
F = G m₁ m₂ / r²
F = (6.67 × 10^-11 × 50 × 10) / (0.5)²
F = (6.67 × 10^-11 × 500) / 0.25
F = 6.67 × 2000 × 10^-11
F = 13340 × 10^-11 = 1.33 × 10^-7 N.

Q2. At what height above Earth's surface does the value of g become half of that on surface? (R = 6400 km)

Solution:
g_h = g [R / (R+h)]²
g/2 = g [R / (R+h)]²
1/2 = [R / (R+h)]² → 1/√2 = R / (R+h)
R+h = √2 R = 1.414 R
h = 0.414 R = 0.414 × 6400 ≈ 2650 km.

Q3. Find percentage decrease in weight of a body when taken to a depth of 32 km below surface.

Solution:
g_d = g (1 - d/R).
Decrease = g - g_d = g(d/R).
% Decrease = (Change/Original) × 100 = (d/R) × 100.
% Decrease = (32 / 6400) × 100 = 3200 / 6400 = 0.5 %.

Q4. A planet has mass 4 times Earth and radius 2 times Earth. Find escape velocity if v_e (Earth) = 11.2 km/s.

Solution:
v = √(2GM/R).
v' = √(2G(4M)/(2R)) = √2 × √(2GM/R).
v' = √2 v = 1.414 × 11.2 ≈ 15.8 km/s.

Q5. Calculate work done in lifting a body of mass 10 kg to a height equal to radius of Earth (R).

Solution:
W = ΔU = -GMm/(2R) - (-GMm/R)
W = GMm/2R = (gR²)m / 2R = mgR/2.
(Can use formula W = mgh/(1+h/R) with h=R → mgR/2).
W = 10 × 9.8 × 6.4×10⁶ / 2
W = 98 × 3.2 × 10⁶ = 3.13 × 10⁸ J.

Q6. A satellite revolves close to earth surface. Its KE is E. How much energy required to make it escape?

Solution:
Orbital Velocity v_o = √(GM/R). KE = ½mv_o² = GMm/2R = E.
Escape Velocity v_e = √(2GM/R) = √2 v_o.
Required KE for escape = ½mv_e² = ½m(2v_o²) = 2(½mv_o²) = 2E.
Additional Energy Needed = 2E - E = E.

Q7. Distance between Earth and Moon is d. Mass of Earth is 81 times Moon. Where is gravitational field zero?

Solution:
Field zero at distance x from moon.
G M_m / x² = G M_E / (d-x)²
1 / x² = 81 / (d-x)²
Take square root: 1/x = 9 / (d-x)
d - x = 9x → d = 10x → x = d/10 from Moon.

Q8. Geostationary satellite orbits at 6R from center. Find time period of another satellite at 3.5R.

Solution:
T² ∝ R³.
(T₂/T₁)² = (R₂/R₁)³
(T₂/24)² = (3.5R / 7R)³ (Note: h=6R means distance 7R usually, assuming questions means distance r). Let's assume distance given.
If r₁=7R (approx geo), r₂=3.5R = r₁/2.
T₂² = 24² (1/2)³ = 576 / 8 = 72.
T₂ = √72 ≈ 8.5 hours.

Q9. Three particles of mass m are at vertices of equilateral triangle of side a. Find PE of system.

Solution:
Pairs: (1,2), (2,3), (3,1). All distances same 'a'.
U = U₁₂ + U₂₃ + U₃₁
U = -3 (Gmm / a) = -3 Gm²/a.

Q10. To what radius must Earth be compressed to become a Black hole? (M = 6×10²⁴ kg)

Solution:
Schwarzschild Radius R_s = 2GM/c²
R = 2 × 6.67 × 10^-11 × 6 × 10²⁴ / (3 × 10⁸)²
R = 80 × 10¹³ / 9 × 10¹⁶
R = 8.8 × 10^-3 m ≈ 9 mm.

Important Formulae

1. Newton's Law

F = G m₁ m₂ / r²

g = GM / R²
Density ρ = 3g / 4πGR

2. Variation of g

Altitude h (h<

g_h = g (1 - 2h/R)

Depth d:

g_d = g (1 - d/R)

Rotation (Latitude λ):

g' = g - Rω² cos²λ

3. Potential & Velocity

Gravitational Potential:

V = -GM/r

Escape Velocity:

v_e = √(2GM/R)

Orbital Velocity:

v_o = √(GM/r)

4. Satellite Energy

Total Energy:

E = -GMm / 2r

KE = |E| = GMm / 2r
PE = 2E = -GMm / r
PE = -2 KE

20 NEET Golden Facts

1. Conservation of L: Kepler's 2nd Law (Law of Areas) is based on Conservation of Angular Momentum.
2. Center Force: Gravitational force is a Central Conservative force. Work done in closed path is zero.
3. Weightlessness: Inside a hollow spherical shell, field is zero, potential is constant (same as surface). Weight is zero.
4. Escape Speed: Independent of mass of projection body and angle of projection. Depends on Planet's mass and radius.
5. Maximum g: Value of g is maximum at Poles (Radius min) and minimum at Equator (Radius max + Rotation effect).
6. Binding Energy: Energy required to remove satellite from orbit to infinity. BE = +GMm/2r.
7. Geo-stationary: Must orbit in equatorial plane, West to East, T=24h, h=35800km.
8. Double Star: Two stars rotate about common center of mass with same angular speed ω.
9. No Atmosphere: Moon has no atmosphere because RMS speed of gas molecules > Escape velocity on Moon (2.3 km/s).
10. 1% Rule: For small % change in radius (h<
11. Inertial Mass: Measured by F/a. Gravitational Mass: Measured by Weight/g. They are equivalent.
12. Black Hole: Body whose escape velocity exceeds speed of light c. R = 2GM/c² (Schwarzschild radius).
13. Tidal Waves: Caused by gravitational pull of Moon (and Sun) on Ocean water.
14. Pendulum: Time period of simple pendulum is infinite at center of earth (g=0).
15. Decrease in g: g decreases faster with height (2h) than with depth (d) near surface.
16. Speed at Infinity: If body thrown with v > v_e, speed at infinity is √(v² - v_e²).
17. Work Done: To move mass m from surface to height h: W = mgh / (1 + h/R).
18. Orbital Speed: Decreases as radius increases (v ∝ 1/√r). Distant planets move slower.
19. Astronauts: Feel weightless in satellite because "Falling" freely towards earth (Normal reaction is zero).
20. Universal Constant: G is scalar and constant everywhere in universe. Independent of medium.

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