Gravitation - Class 11 Physics

Gravitation

Overview: This chapter explores the fundamental force of gravity, planetary motion, and satellite mechanics.

1. Kepler's Laws of Planetary Motion

Law of Orbits (1st): All planets move in elliptical orbits with the Sun at one of the foci.
Law of Areas (2nd): The line joining the planet to the Sun sweeps out equal areas in equal intervals of time. (dA/dt = constant = L/2m). consequence of conservation of angular momentum.
Law of Periods (3rd): The square of time period is proportional to the cube of semi-major axis.

T² ∝ a³

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 (m₁m₂) / r²

G = 6.67 × 10^-11 N m²/kg²

3. Acceleration due to Gravity (g)

On surface of Earth: g = GM / R² ≈ 9.8 m/s².

Variation of g

At Altitude h: g_h = g (1 - 2h/R) [for h << R]
At Depth d: g_d = g (1 - d/R)

g is maximum at surface/poles, minimum at equator (due to rotation).

4. Gravitational Potential Energy (U)

Energy possessed by a body due to its position in gravitational field. Zero at infinity.

U = - (GMm) / r

Gravitational Potential (V)

Potential energy per unit mass.

V = - GM / r

5. Orbital & Escape Velocity

Escape Velocity (v_e)

Minimum speed required to escape Earth's gravity field.

v_e = √(2GM/R) = √(2gR) ≈ 11.2 km/s

Orbital Velocity (v_o)

Velocity required to keep a satellite in orbit close to Earth.

v_o = √(GM/R) = √(gR) ≈ 7.9 km/s

Relation: v_e = √2 v_o

Energy of Satellite

Kinetic K = GMm/2r. Potential U = -GMm/r. Total E = -GMm/2r.

Numericals - Gravitation

Numericals

Force Calc

Q1. Two spheres of mass 50kg and 10kg are 2m apart. Find grav-force. (G=6.67x10^-11)

F = G m1 m2 / r²

F = 6.67×10^-11 × 50 × 10 / 2²

F = 6.67×10^-11 × 500 / 4

F = 6.67 × 125 × 10^-11

F = 8.3 × 10^-9 N

Variation of g

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

g_h = g / 2

g_h = g R² / (R+h)²

1/2 = R² / (R+h)²

√2 = (R+h)/R

1.414 R = R + h

h = 0.414 R

h = 2650 km

Orbital Velocity

Q3. Calculate orbital velocity of a satellite near Earth's surface. (g=9.8, R=6.4x10⁶)

v_o = √(gR)

v_o = √(9.8 × 6.4 × 10⁶)

v_o = √(62.72 × 10⁶)

v_o ≈ 7.9 × 10³ m/s

v_o = 7.9 km/s

Gravitational Potential Energy

Q4. Calculate potential energy of a 100kg object at Earth's surface. (M=6x10²⁴kg, R=6.4x10⁶m)

U = - GMm / R

U = - (6.67×10^-11 × 6×10²⁴ × 100) / 6.4×10⁶

U = - (40 × 10¹⁵) / 6.4×10⁶

U = - 6.25 × 10⁹ J

Negative sign indicates binding.

Depth Calculation

Q5. Find depth d where g is 75% of surface value.

g_d = 0.75 g

g_d = g(1 - d/R)

0.75 = 1 - d/R

d/R = 0.25

d = R/4 = 1600 km

Escape Velocity

Q6. If a planet has mass 4 times Earth and same radius, find escape velocity. (v_earth = 11.2)

v_e = √(2GM/R)

v is proportional to √M

v' / v = √(M'/M)

v' / 11.2 = √(4M/M) = 2

v' = 22.4 km/s

Kepler's T-Squared Law

Q7. A satellite orbits at height R above surface (radius 2R). Geo-stationary orbits at 6R (radius 7R). T for first?

T₂ = 24 hrs, r₂ = 7R, r₁ = 2R

(T1/T2)² = (r1/r2)³

T1 = T2 × (2/7)^3/2

T1 = 24 × √(8/343)

T1 ≈ 24 × 0.15

T1 ≈ 3.6 hours

Weightlessness

Q8. What should be Earth's angular speed so value of g at equator becomes zero?

g' = g - Rω² = 0

ω = √(g/R)

ω = √(9.8 / 6.4×10⁶)

ω = 1.24 × 10^-3 rad/s

Energy to Orbit

Q9. Energy required to lift 500kg mass to orbit radius 4R (h=3R)?

Total Energy Change (Potential + Kinetic needed to orbit)

Actually Q asks work to lift or launch? Let's assume lift to height h.

ΔU = U_f - U_i

ΔU = -GMm/4R - (-GMm/R)

ΔU = (3/4) GMm/R = (3/4) mgR

Work = 0.75 × 500 × 10 × 6.4×10⁶

Work = 2.4 × 10¹⁰ J

Force Ratio

Q10. Two particles of equal mass move in a circle under mutual gravitation. Speed?

Grav force = Centripetal force

Gmm/(2R)² = mv²/R

Gm²/4R² = mv²/R

v² = Gm/4R

v = √(Gm/4R)

Formulas & Facts - Gravitation

Equations & Formulas

Concept	Formula
Grav Force	F = G m1 m2 / r²
Acc due to Gravity	g = GM / R²
g at Height h	g_h = g (1 - 2h/R)
g at Depth d	g_d = g (1 - d/R)
Grav Potential	V = - GM / r
Potential Energy	U = - GMm / r
Orbital Velocity	v_o = √(GM/r)
Escape Velocity	v_e = √(2GM/R)
Time Period	T² = (4π²/GM) r³
Total Satellite Energy	E = - GMm / 2r

50 NEET Facts

Key points for Gravitation.

1. Weakest Force Gravitational force is the weakest of the four fundamental forces in nature.

2. Central Force Acts along the line joining the centers of two masses. It is a conservative force.

3. Medium Independence Gravitational force does not depend on the intervening medium between masses.

4. Mass vs Weight Mass is constant everywhere. Weight (W=mg) varies with g. Weight is zero at Earth's center.

5. Inertial vs Gravitational Mass They are experimentally found to be equal (Equivalence Principle).

6. Value of G Universal Gravitational Constant G = 6.67 x 10^-11 Nm²/kg². First measured by Cavendish.

7. g Max/Min g is max at Poles (R is min) and min at Equator (R is max).

8. Effect of Rotation Rotation of Earth reduces g at equator by Rω². No effect at poles.

9. g at Infinity Value of g becomes zero at infinite distance.

10. Gravitational Shielding Impossible. You cannot shield a body from gravity.

11. Kepler's 1st Law Orbits are elliptical. Sun is at one focus, not the center.

12. Kepler's 2nd Law Areal velocity is constant. Implies conservation of angular momentum (L = constant).

13. Orbital Speed Variation Planets move faster when closer to Sun (Perihelion) and slower when far (Aphelion).

14. Escape Velocity Value Independent of mass of the body projected. Depends only on Mass and Radius of planet.

15. Escape Velocity Angle Does not depend on angle of projection.

16. Atmosphere Moon has no atmosphere because v_rms of gas molecules > v_escape.

17. Geostationary Satellite T = 24 hours. Height = 36,000 km. Orbits in equatorial plane. West to East.

18. Polar Satellite Low altitude (500-800 km). T approx 100 min. Scans entire Earth strip by strip.

19. Binding Energy Energy required to remove satellite to infinity. BE = +GMm/2r.

20. Negative Potential Potential is negative because work is done by the field (attractive forces) to bring mass from infinity.

21. Shell Theorem 1 Gravitational field inside a uniform spherical shell is Zero.

22. Shell Theorem 2 Outside spatial shell, it behaves as if entire mass is at center.

23. Weightlessness In a satellite, normal reaction is zero because grav force provides centripetal force entirely.

24. Black Hole Body so dense that Escape Velocity > Speed of Light (c).

25. Potential inside Earth Potential decreases from surface to center. Center is not zero potential; it is -3GM/2R.

26. g depth vs height Decrease in g at depth d is equal to decrease at height h=d/2 (for small h). g decreases faster going up.

27. Two Star System Both stars revolve around their common Center of Mass with same angular velocity ω.

28. Tidal Waves Caused by gravitational pull of Moon (and Sun) on Earth's oceans.

29. Intensity of Field E = F/m. Dimension is of acceleration [LT^-2].

30. Work around closed path Work done by gravity in a round trip is Zero (Conservative force).

31. Parking Orbit Another name for Geostationary orbit. Satellite appears fixed (parked).

32. Communication Satellites Usually geostationary to maintain constant link with ground antennas.

33. Remote Sensing Uses Polar satellites (low orbit, high resolution).

34. Gravitational Mass Determined by weighing. W = mg.

35. Inverse Square Law If force depended on 1/r (instead of r²), closed stable orbits would not exist.

36. Speed of Gravity In General Relativity, changes in gravity propagate at speed of light (Gravitational Waves).

37. Equipotential Surfaces For a point mass, they are spherical shells.

38. Air Resistance Satellites in low orbit eventually spiral down due to air drag, losing energy (actually speed increases as r decreases, paradox).

39. Relation ve vs vo Escape velocity is 41.4% higher than orbital velocity (close to surface).

40. Time Period at Surface A satellite skimming Earth surface would have T approx 84.6 minutes.

41. Zero Gravity Exists at the center of Earth. Or far in deep space (technically approaches zero).

42. Sphere of Influence Region around a planet where its gravity dominates over Sun.

43. Retrograde Orbit Orbit opposite to Earth's rotation (East to West). Harder to launch.

44. Launch Velocity Rockets are launched Eastward to gain from Earth's rotation speed.

45. Multi-stage Rocket Used to reduce mass of carrier as fuel burns, allowing higher final velocity.

46. Fall through Earth Tunnel If you drop a stone in a tunnel through Earth center, it executes SHM. T = 84.6 min.

47. Inertial Frame Earth is not strictly inertial due to rotation and revolution.

48. Sun's Mass Calculated using Earth's orbital period and radius (Kepler's 3rd Law).

49. Spring Balance Reading Varies with latitude. Max at poles, min at equator.

50. Beam Balance Measures mass. Reading is constant everywhere regardless of g.

📱 Practice MCQs for this topic inside our App

Gravitation

1. Kepler's Laws of Planetary Motion

2. Universal Law of Gravitation

3. Acceleration due to Gravity (g)

Variation of g

4. Gravitational Potential Energy (U)

Gravitational Potential (V)

5. Orbital & Escape Velocity

Escape Velocity (ve)

Orbital Velocity (vo)

Energy of Satellite

Numericals

Equations & Formulas

50 NEET Facts

Escape Velocity (v_e)

Orbital Velocity (v_o)