Chapter 6 – Gravitation (JEE Physics)
1. Introduction to Gravitation
Gravitation is a fundamental force of nature by which every object in the universe
attracts every other object.
It governs the motion of planets, satellites, stars, and galaxies.
2. Newton’s Law of Universal Gravitation
Every two particles of mass $m_1$ and $m_2$ attract each other with a force
directly proportional to the product of their masses and inversely proportional
to the square of the distance between them.
$$F = G \frac{m_1 m_2}{r^2}$$
$G$ is the universal gravitational constant:
$$G = 6.67 \times 10^{-11}\; \text{N m}^2\text{kg}^{-2}$$
3. Characteristics of Gravitational Force
- Always attractive
- Long-range force
- Central force
- Conservative force
4. Gravitational Field
Gravitational field at a point is defined as the gravitational force experienced
by a unit test mass placed at that point.
$$\vec{g} = \frac{\vec{F}}{m}$$
For a point mass:
$$g = G \frac{M}{r^2}$$
5. Gravitational Field Due to Earth
$$g = \frac{GM}{R^2}$$
At Earth’s surface:
$$g \approx 9.8\; \text{m/s}^2$$
6. Variation of $g$ with Height
$$g_h = g\left(\frac{R}{R+h}\right)^2$$
7. Variation of $g$ with Depth
$$g_d = g\left(1-\frac{d}{R}\right)$$
8. Gravitational Potential
Gravitational potential at a point is the work done per unit mass in bringing
a test mass from infinity to that point.
$$V = -G \frac{M}{r}$$
9. Gravitational Potential Energy
$$U = -G \frac{mM}{r}$$
Negative sign indicates bound system.
10. Relation Between $g$ and Potential
$$g = -\frac{dV}{dr}$$
11. Escape Velocity
Escape velocity is the minimum velocity required by an object to escape
the gravitational field of Earth without further propulsion.
$$v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$$
For Earth:
$$v_e \approx 11.2\; \text{km/s}$$
12. Orbital Velocity of Satellite
$$v_o = \sqrt{\frac{GM}{r}}$$
At Earth’s surface:
$$v_o \approx 7.9\; \text{km/s}$$
13. Time Period of Satellite
$$T = 2\pi \sqrt{\frac{r^3}{GM}}$$
14. Geostationary Satellite
- Time period = 24 hours
- Orbit in equatorial plane
- Appears stationary relative to Earth
15. Kepler’s Laws of Planetary Motion
First Law
Planets move in elliptical orbits with the Sun at one focus.
Second Law
The line joining a planet to the Sun sweeps equal areas in equal times.
Third Law
$$T^2 \propto r^3$$
16. Binding Energy of Satellite
$$E = -\frac{GMm}{2r}$$
17. Weightlessness
Weightlessness occurs when normal reaction becomes zero,
such as in free fall or orbiting satellite.
18. Important JEE Points
- Escape velocity independent of mass
- Orbital velocity < escape velocity
- Potential is scalar, field is vector
- Inside Earth, $g$ decreases linearly
19. Common Mistakes
- Forgetting negative sign in potential
- Confusing orbital and escape velocity
- Wrong radius used in satellite formulas
20. Final Revision Checklist
You are ready for full marks if you can:
- Apply Newton’s law correctly
- Derive $g$, $V$, $U$ expressions
- Solve satellite motion problems
- Use Kepler’s laws confidently