Understanding Gravitational Fields and Potential Energy
Gravitational Field Strength
The purpose of a gravitational field manifests at a point where a test mass is placed. The gravitational field strength (or field intensity vector) at a point is equal to the force exerted on a unit mass placed at that point. The vector equation is:
g = -G (M / r³) r
Where G is the gravitational constant, M is the source mass, r is the distance from the source mass, and r is the position vector.
The field strength at a point is characterized by:
- Magnitude: g = GM / r²
- Direction: Along the line connecting the source mass (creator of the field) with the point.
- Sense: Towards the mass creating the field (e.g., towards Earth).
- Point of Application: The point where the field is being evaluated.
The force with which the Earth attracts bodies in close proximity is called weight (P). The term g is known as the acceleration due to gravity; it represents the force with which a unit mass is attracted by the Earth. The relationship is:
P = mg
Variations in Acceleration Due to Gravity (g)
Variations on the Earth’s Surface
Although often assumed spherical, the Earth’s radius is smaller at the poles than at the equator. Since g is inversely proportional to the square of the distance from the Earth’s center, its value varies across the globe.
Furthermore, the measured weight on Earth results from two forces: the gravitational force and the centrifugal force due to Earth’s rotation acting on the body.
- At the equator, the centrifugal force is maximum, reducing the effective gravitational field strength (g is lower).
- At the poles, there is no rotation; thus, the centrifugal force is zero, and the gravitational field strength is maximum (g is higher).
Variations with Altitude
The Earth’s gravitational field strength is inversely proportional to the square of the distance (r) from the center of the Earth. Therefore, as altitude (h) above the surface increases, the distance r = R + h (where R is Earth’s radius) increases, and the value of g decreases.
Variations with Depth
Consider a point inside the Earth at a depth h below the surface, which is at a distance r from the center (r = R – h). Assuming uniform density (ρ), only the mass (M’) within the sphere of radius r contributes to the gravitational field at that point. As depth increases (r decreases), the effective mass M’ decreases significantly, leading to a decrease in g towards the center (where g = 0).
Potential Energy of the Gravitational Field
As previously established, a scalar quantity called potential energy (Ep) can be defined for any conservative force field. The gravitational field is an example, being both a central and conservative force field.
Consider a source mass M creating a gravitational field. Imagine moving a test mass m from point A to point B within this field. Potential energy measures the capacity of the field forces to do work on a particle due to its position within the field. Its value equals the work done by the field forces when moving the particle from its current position to a reference point where potential energy is defined as zero (often taken at infinity).
When the field does positive work, the potential energy (Ep) decreases; work is done at the expense of the particle’s potential energy. As the test mass m moves closer to the source mass M, its potential energy decreases, and the gravitational force performs positive work.