Chapter 7: Gravitation

Chapter 7: Gravitation

1. Kepler’s Laws of Planetary Motion

• First Law: Planets move in elliptical orbits with Sun at one focus
• Second Law: Equal areas are swept in equal time
• Third Law: T² ∝ R³

2. Universal Law of Gravitation

F = G (m₁m₂ / r²)

• Force between two masses
• G = gravitational constant

3. Acceleration Due to Gravity

g = GM / R²

Variation with Height

gₕ = g (1 - 2h/R)

Variation with Depth

g_d = g (1 - d/R)

4. Gravitational Potential Energy

PE = -GMm / r

• Negative because force is attractive

5. Gravitational Potential

V = -GM / r

6. Escape Velocity

v = √(2GM / R)

• Minimum velocity to escape Earth
• Independent of mass

7. Orbital Velocity

v = √(GM / R)

• Velocity required to stay in orbit

8. Energy of Satellite

Total Energy = -GMm / 2R

• Sum of KE and PE

9. MCQ One-Liners

• Gravitational force is always attractive.
• Force ∝ 1/r².
• Kepler’s first law → elliptical orbit.
• Kepler’s second law → equal areas.
• g depends on mass and radius.
• g decreases with height.
• g decreases with depth.
• Escape velocity is √2 times orbital velocity.
• Orbital velocity is independent of satellite mass.
• Potential energy is negative.
• Gravitational potential is negative.
• Energy of satellite is negative.
• Unit of G is N·m²/kg².
• Weight = mg.
• Free fall acceleration = g.
• Gravitational field is vector.
• Potential is scalar.
• g is maximum at surface.
• At center of Earth → g = 0.
• Satellite motion is circular motion.

10. Numerical Problems with Solutions

1. Gravitational Force

m₁=2kg, m₂=3kg, r=2m, G=6.67×10⁻¹¹

F = Gm₁m₂/r² = 6.67×10⁻¹¹×6/4 = 1×10⁻¹⁰ N

2. Acceleration Due to Gravity

G=6.67×10⁻¹¹, M=6×10²⁴, R=6.4×10⁶

g = GM/R² ≈ 9.8 m/s²

3. Escape Velocity

g=10, R=6400 km

v = √(2gR) = √(2×10×6.4×10⁶) ≈ 11.3 km/s

4. Orbital Velocity

g=10, R=6400 km

v = √(gR) ≈ 7.9 km/s

5. Potential Energy

M=Earth, m=1kg, R=6.4×10⁶

PE = -GMm/R ≈ -6.25×10⁷ J

6. Satellite Energy

PE = -100 J

Total Energy = PE/2 = -50 J

7. Variation with Height

h = R

gₕ = g(1 - 2h/R) = g(1 - 2) = -g (approx concept)

8. Ratio of Velocities

v_escape : v_orbit = √2 : 1