1. Introduction and Universal Law of Gravitation
Gravitation is the fundamental force of attraction that exists between any two objects with mass. Sir Isaac Newton's Universal Law of Gravitation states that every particle of matter 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. This can be expressed as $F = G \frac{m_1 m_2}{r^2}$, where $G$ is the gravitational constant. This law explains phenomena ranging from the falling of an apple to the orbits of planets around the Sun, playing a critical role in celestial mechanics and the structure of the universe.
2. Free Fall and Acceleration due to Gravity ('g')
When an object falls under the sole influence of gravity, it is said to be in free fall. In the absence of air resistance, all objects fall with the same acceleration, regardless of their mass. This acceleration is known as the acceleration due to gravity, denoted by '$g$'. On the Earth's surface, the average value of $g$ is approximately $9.8 \, \text{m/s}^2$. This value decreases slightly with altitude. Understanding free fall and the constant acceleration due to gravity is essential for calculating the motion of falling objects, such as raindrops or objects thrown upwards, and is a direct application of Newton's second law.
3. Mass and Weight
Mass is an intrinsic property of matter that measures its inertia and the amount of substance it contains. It is a scalar quantity and remains constant regardless of location. Weight, on the other hand, is the force of gravity acting on an object's mass. It is calculated as $W = mg$, where $m$ is mass and $g$ is the acceleration due to gravity at that location. Therefore, weight is a force and a vector quantity. An object's weight will change if it moves to a location with a different gravitational acceleration, such as the Moon, where $g$ is much lower than on Earth, meaning an object will weigh less there but its mass remains the same.
4. Kepler's Laws of Planetary Motion
Johannes Kepler, using observational data, formulated three laws that describe the motion of planets around the Sun. Kepler's First Law states that planets orbit the Sun in elliptical paths, with the Sun at one focus. Kepler's Second Law posits that a line joining a planet and the Sun sweeps out equal areas during equal intervals of time, meaning planets move faster when closer to the Sun and slower when farther away. Kepler's Third Law relates the orbital period ($T$) of a planet to the semi-major axis ($a$) of its orbit by $T^2 \propto a^3$. These laws were a crucial precursor to Newton's law of gravitation.
5. Gravitational Potential Energy and Escape Speed
Gravitational potential energy ($U$) is the energy an object possesses due to its position in a gravitational field. Near the Earth's surface, it is given by $U = mgh$. At larger distances, the formula $U = -G \frac{m_1 m_2}{r}$ is used, where the potential energy is defined as zero at infinite separation. Escape speed is the minimum speed an object needs to be launched from the surface of a celestial body to escape its gravitational pull completely and not return. For Earth, this speed is approximately $11.2 \, \text{km/s}$. This concept is vital for space exploration and understanding the limits of celestial bodies.
6. Earth Satellites
Earth satellites, whether natural like the Moon or artificial like communication satellites, are celestial bodies or spacecraft that orbit the Earth. Their motion is governed by Earth's gravitational pull, which acts as the centripetal force keeping them in orbit. Satellites in circular orbits travel at a constant speed, and the gravitational force provides the necessary acceleration. The orbital period depends on the altitude of the satellite. For example, geostationary satellites orbit at an altitude where their orbital period matches the Earth's rotation period, making them appear stationary in the sky, which is crucial for broadcasting and weather monitoring.
7. Additional: Gravitational Field and Potential
A gravitational field is a region of space around a massive object where another massive object will experience a gravitational force. It is a vector quantity, and its strength at any point is equal to the force per unit mass experienced by a test mass placed at that point. Gravitational potential is the work done per unit mass to move an object from infinity to a specific point in the gravitational field. It is a scalar quantity, and the gravitational field is the negative gradient of the gravitational potential ($ \vec{g} = -\nabla V $). These concepts provide a more comprehensive understanding of how gravity affects objects in space.