Gravitational Field

A gravitational field is a region of space surrounding a body that has mass, where another body experiences a force of gravitational attraction. It is a vector field, meaning it has both magnitude and direction.

1. Newton’s Law of Universal Gravitation

Newton’s law of universal gravitation states that the force of attraction between any two point masses (m1 and m2) is directly proportional to the product of their masses and inversely proportional to the square of the distance (r) between them.

Mathematically: F = G(m1m2) / r²

Where G is the Universal Gravitational Constant (6.67 x 10⁻¹¹ Nm²/kg²). Note that G is a scalar and its value is constant throughout the universe.

2. Gravitational Potential

The gravitational potential (V) at a point in a gravitational field is the work done per unit mass in bringing a mass from infinity to that point.

Expression: V = -GM / r

The negative sign indicates that the work is done by the field (attraction). Gravitational potential is a scalar quantity and its unit is J/kg.

3. Conservative and Non-Conservative Fields

• Conservative Field: A field in which the work done in moving a body between two points is independent of the path taken. Examples include gravitational and electrostatic fields.
• Non-Conservative Field: A field in which the work done depends on the path taken. Examples include frictional forces and air resistance.

4. Acceleration due to Gravity (g)

The acceleration experienced by a body falling freely under the influence of gravity is denoted by 'g'. On Earth's surface, g = GM / R², where M is the mass of the Earth and R is its radius.

5. Variation of g on Earth’s Surface

The value of g is not constant everywhere on Earth due to:

• The Shape of the Earth: The Earth is an oblate spheroid (flattened at the poles and bulging at the equator). Since R is smaller at the poles, g is higher at the poles than at the equator.
• Altitude (h): g decreases as we move higher above the Earth's surface.
• Depth (d): g decreases as we move towards the center of the Earth.
• Rotation: The Earth's rotation slightly reduces the effective value of g at the equator.

6. Mass vs. Weight

• Mass: The quantity of matter in a body. It is a scalar, measured in kg, and constant everywhere.
• Weight: The force of gravity acting on a body (W = mg). It is a vector, measured in Newtons, and varies with location.

7. Escape Velocity and Satellites

Escape Velocity (Ve): The minimum velocity required for a body to escape the gravitational pull of a planet permanently. Expression: Ve = √(2GM/R) = √(2gR). For Earth, this is approximately 11.2 km/s.

Parking (Geostationary) Orbit: An orbit where a satellite moves with the same angular velocity as the Earth, appearing stationary to an observer on the ground. Its period is 24 hours.

Weightlessness: A state where the apparent weight of a body is zero, such as in a freely falling elevator or an orbiting spacecraft.