COMPLETE FORMULA SHEET

PRERNA FOR IAS

COMPLETE FORMULA SHEET

1. Acceleration Due to Gravity (g)

Acceleration due to gravity, denoted by g, is the acceleration experienced by an object due to Earth's gravitational pull. It is given by the formula g = GM/R², where G is the gravitational constant, M is Earth’s mass, and R is Earth’s radius. Near Earth’s surface, its value is approximately 9.8 m/s². Gravity acts equally on all objects regardless of their mass in a vacuum. This is why a feather and an iron ball fall at the same rate when air resistance is absent. The concept of g is fundamental in understanding falling bodies and gravitational forces.

2. Variation of g

The value of gravitational acceleration g is not constant everywhere. It decreases with height above Earth because the distance from Earth’s center increases. It also decreases with depth below Earth’s surface and becomes zero at the center of Earth. Due to Earth’s rotation and slightly flattened shape, g is greater at the poles and smaller at the equator. The variation of g explains differences in weight at different locations. Understanding these changes is important in geophysics, satellite motion, and space science. It shows that gravity depends on distance from Earth’s center and local conditions.

3. Gravitational Field, Potential and Potential Energy

A gravitational field is the region around a mass where another mass experiences gravitational force. Gravitational potential is the work done per unit mass in bringing an object from infinity to a point in the field. Gravitational potential energy is the energy possessed by a body due to its position in a gravitational field. It is given by U = -GMm/r, which is always negative because gravity is an attractive force. As an object moves closer to Earth, its potential energy decreases and kinetic energy increases. These concepts are essential for understanding planetary motion and satellite dynamics.

4. Kepler’s Laws of Planetary Motion

Kepler’s Laws describe how planets move around the Sun. The First Law states that planets move in elliptical orbits with the Sun at one focus. The Second Law states that a line joining a planet and the Sun sweeps out equal areas in equal intervals of time, meaning planets move faster when closer to the Sun. The Third Law states that the square of a planet’s orbital period is proportional to the cube of its average distance from the Sun. These laws accurately explain planetary motion and formed the basis for Newton’s theory of gravitation and modern astronomy.

5. Escape Velocity

Escape velocity is the minimum speed required for an object to escape a planet’s gravitational pull without further propulsion. For Earth, it is approximately 11.2 km/s. The formula is Ve = √(2GM/R). If an object is launched with a speed lower than escape velocity, it will eventually return to Earth. If launched at or above escape velocity, it can escape Earth’s gravity permanently. Escape velocity depends on the planet’s mass and radius but not on the object’s mass. This concept is important in space missions, rocket launches, and interplanetary exploration.

6. Orbital Velocity and Satellites

Orbital velocity is the speed required for a satellite to remain in a stable orbit around a planet. For a satellite near Earth’s surface, it is about 7.9 km/s. It is given by Vo = √(GM/r). Satellites are classified into Low Earth Orbit (LEO), Medium Earth Orbit (MEO), and Geostationary Orbit (GEO). Geostationary satellites revolve around Earth in 24 hours and appear stationary from the ground. Satellites are used for communication, navigation, weather forecasting, and scientific research. Orbital motion results from the balance between gravitational attraction and the satellite’s forward velocity.

7. Weightlessness – The Big Misconception

Weightlessness does not mean the absence of gravity. Astronauts in space feel weightless because they and their spacecraft are in continuous free fall around Earth. At the altitude of the International Space Station, Earth’s gravity is still very strong. Apparent weightlessness occurs because there is no supporting force acting on the astronauts. The apparent weight is given by W = m(g – a). When the acceleration equals g, the apparent weight becomes zero. This phenomenon also occurs briefly in freely falling elevators. Understanding weightlessness is important for space travel, astronaut training, and orbital mechanics.

8. One-Line Revision of Gravitation

Gravitation is the universal force of attraction between masses. Newton’s Law of Gravitation states that gravitational force is proportional to the product of masses and inversely proportional to the square of the distance between them. The gravitational constant G is universal, while g varies from place to place. Orbital velocity keeps satellites in orbit, while escape velocity allows objects to leave Earth permanently. Gravitational potential energy is always negative, and weightlessness occurs due to free fall, not the absence of gravity. These concepts form the foundation of celestial mechanics, satellite technology, and modern space exploration.

9. Complete Formula Sheet of Gravitation

The gravitation chapter contains several important formulas. Gravitational force is F = GM₁M₂/r². Acceleration due to gravity is g = GM/R². At height h, gh = gR²/(R+h)². At depth d, gd = g(1−d/R). Gravitational potential is V = -GM/r, and potential energy is U = -GMm/r. Orbital velocity is Vo = √(GM/r), while escape velocity is Ve = √(2GM/R). Orbital period is T = 2π√(r³/GM). These formulas are frequently used in JEE, NEET, and other competitive examinations and help solve numerical problems efficiently.

10. Topper Tricks and Mnemonics

Mnemonic techniques help students remember important gravitation concepts quickly. A useful relation is Ve = √2 Vo, meaning escape velocity is √2 times orbital velocity. The value of g is maximum at Earth’s surface and decreases both above and below it. Gravitational potential energy is always negative because work is required to move an object away from Earth’s gravitational field. Kepler’s Second Law is linked to conservation of angular momentum. The mnemonic “FLAT” helps remember gravity variations: Faster at poles, Less at altitude, At center zero, and Tidal effects exist. Such tricks improve recall during examinations.