Definition: Gravitation is the universal force of attraction that acts between all matter possessing mass. It is a fundamental interaction in physics described by Newton’s Law of Universal Gravitation and refined by Einstein’s General Theory of Relativity, governing the motion of celestial bodies and the structure of the universe.
Newton’s Law of Universal Gravitation
At the heart of classical mechanics lies the Law of Universal Gravitation, proposed by Sir Isaac Newton in 1687. Newton hypothesized that every particle in the universe attracts every other particle with a force that is directly proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
The force of attraction between two point masses $m_1$ and $m_2$ separated by a distance $r$ is given by $F = G \frac{m_1 m_2}{r^2}$, where $G$ is the Universal Gravitational Constant.
The value of $G$ is approximately $6.67 \times 10^{-11} \, \text{N m}^2/\text{kg}^2$. It is crucial to remember that this force is always attractive and acts along the line joining the centers of the two masses. Because $G$ is extremely small, the gravitational force between everyday objects is negligible; however, it becomes the dominant force when dealing with massive bodies like planets and stars.
Gravitational Field and Potential
To understand how an object “feels” the presence of another mass without direct contact, we use the concept of the Gravitational Field. The field at a point is defined as the gravitational force experienced by a unit test mass placed at that point. Mathematically, the intensity $E$ is given by $E = \frac{GM}{r^2}$.
Gravitational Potential ($V$) is defined as the amount of work done by an external agent in bringing a unit mass from infinity to a specific point in the gravitational field without acceleration. The potential at a distance $r$ from a mass $M$ is $V = -\frac{GM}{r}$. The negative sign indicates that the potential is zero at infinity and decreases (becomes more negative) as we move closer to the mass, signifying that the field is attractive.
Kepler’s Laws of Planetary Motion
Before Newton, Johannes Kepler formulated three empirical laws based on the observational data of Tycho Brahe. These laws are essential for understanding orbital mechanics in competitive exams:
- Law of Orbits: All planets move in elliptical orbits with the Sun at one of the two foci.
- Law of Areas: A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time. This is a direct consequence of the Conservation of Angular Momentum.
- Law of Periods: The square of the orbital period ($T$) of a planet is directly proportional to the cube of the semi-major axis ($a$) of its orbit ($T^2 \propto a^3$).
Satellite Motion and Escape Velocity
A satellite in a circular orbit around a planet is held in place by the gravitational pull, which provides the necessary centripetal force. By equating the gravitational force to the centripetal force ($mv^2/r$), we find the orbital velocity $v_o = \sqrt{\frac{GM}{r}}$.
Escape Velocity is the minimum speed required for an object to break free from a planet’s gravitational influence and never return. It is calculated by setting the total mechanical energy (kinetic + potential) equal to zero at infinity. The formula for escape velocity from the surface of a planet of mass $M$ and radius $R$ is $v_e = \sqrt{\frac{2GM}{R}} = \sqrt{2gR}$.
Important Facts / Formulas
| Concept | Formula |
|---|---|
| Gravitational Force | $F = G \frac{m_1 m_2}{r^2}$ |
| Acceleration due to Gravity ($g$) | $g = \frac{GM}{R^2}$ |
| Gravitational Potential | $V = -GM/r$ |
| Orbital Velocity | $v_o = \sqrt{GM/r}$ |
| Escape Velocity | $v_e = \sqrt{2GM/R}$ |
Previous Year Question Hints
- Variation of $g$: Expect questions on how $g$ changes with altitude ($h$) and depth ($d$). Remember $g_h = g(1 – 2h/R)$ and $g_d = g(1 – d/R)$.
- Energy of Satellites: Often, you will be asked to calculate the total energy of a satellite, which is always negative ($E = -GMm/2r$).
Quick Revision Summary
- Gravitational force is a conservative force and follows the inverse square law.
- The gravitational field intensity is the gradient of the gravitational potential.
- Kepler’s Second Law is equivalent to the conservation of angular momentum.
- The orbital period of a satellite is independent of its own mass.
- Escape velocity is independent of the mass of the escaping object.
- The gravitational potential energy of a system of two masses is $U = -GMm/r$.
- At the center of a solid sphere, the gravitational field is zero, but the potential is not.
- Weightlessness in a satellite is due to the state of free-fall, not the absence of gravity.
