Gravitation – JEE Mains Physics
1. Universal Law of Gravitation
- Every particle attracts every other particle with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them.
- Formula:
F = G * (m₁ * m₂) / r²
where G = gravitational constant.
2. Acceleration due to Gravity (g)
- Acceleration experienced by a body due to Earth's gravitational pull.
- On the surface:
g = GM / R²
where M = mass of Earth and R = radius of Earth.
3. Variation of g with Altitude and Depth
- At altitude h:
g' = g (1 - 2h/R)
for h ≪ R - At depth d:
g' = g (1 - d/R)
4. Kepler’s Laws of Planetary Motion
- First Law (Law of Orbits): Planets move in elliptical orbits with the Sun at one focus.
- Second Law (Law of Areas): The line joining a planet and the Sun sweeps out equal areas in equal time intervals.
- Third Law (Law of Periods): Square of the orbital period is proportional to the cube of the semi-major axis of the orbit:
T² ∝ r³
5. Gravitational Potential Energy
- Work done in bringing a mass from infinity to a point in a gravitational field.
- Formula:
U = -G * (M * m) / r
6. Gravitational Potential
- Potential energy per unit mass at a point in a gravitational field.
- Formula:
V = U / m = -G * M / r
7. Escape Velocity
- The minimum velocity needed to escape from the gravitational field of a planet without further propulsion.
- Formula:
vₑ = √(2GM / R)
8. Motion of a Satellite
- A satellite revolves around a planet due to the gravitational force providing the necessary centripetal force.
- For a circular orbit:
F_gravity = F_centripetal
9. Orbital Velocity
- Velocity needed to keep a satellite in orbit near a planet’s surface.
- Formula:
v₀ = √(GM / R)
10. Time Period and Energy of Satellite
- Time Period (T):
T = 2π√(r³ / GM)
- Energy of Satellite: Total energy (E) is negative and given by:
E = -GMm / (2r)