Gravitational Fields (Copy)

A2 Level Physics – Section 13: Gravitational Fields (Detailed Notes)

13.1 Gravitational Field

1. Definition of Gravitational Field

• A gravitational field is a region in which a mass experiences a force due to another mass.
• It is a field of force and acts at a distance.
• Gravitational field strength (g) = force per unit mass:

  g = F / m

  • g: N/kg
  • F: gravitational force (N)
  • m: mass of the object experiencing the force (kg)

2. Gravitational Field Lines

• Field lines show the direction of the force on a small test mass.
• Lines are:
  • Radially inward for point masses or spherical masses.
  • Parallel and equally spaced in uniform gravitational fields (e.g. near Earth’s surface).
  • Denser lines = stronger field.

13.2 Gravitational Force Between Point Masses

1. Point Mass Assumption for Spheres

• A uniform spherical mass acts as if all its mass were concentrated at its center, for external points.
• Enables us to apply Newton’s law of gravitation to planets, stars, satellites, etc.

2. Newton’s Law of Universal Gravitation

F = G·(m₁·m₂) / r²

• F = gravitational force between two point masses (N)
• G = universal gravitational constant = 6.67 × 10⁻¹¹ N·m²/kg²
• m₁, m₂ = interacting masses (kg)
• r = separation between centers of mass (m)

Key Properties:

• Always attractive
• Inverse square law: doubling distance → force becomes 1/4

3. Circular Orbits and Gravitational Force

In a stable circular orbit, gravitational force provides the centripetal force:

G·Mm / r² = mv² / r
⇒ v² = GM / r

Also:
T = 2πr / v, so
⇒ T² = (4π² / GM)·r³

(This is Kepler’s third law: T² ∝ r³ for orbital motion around a central mass M)

4. Geostationary Orbit

A geostationary satellite:

• Remains fixed over a point on Earth’s equator
• Has an orbital period of 24 hours
• Orbits in the same direction as Earth’s rotation (west to east)
• Must be directly above the equator
• Altitude ≈ 35,800 km from Earth’s surface

Applications: telecommunications, weather monitoring

13.3 Gravitational Field of a Point Mass

1. Deriving g = GM / r²

From:

• Newton’s Law: F = GMm / r²
• Gravitational field strength: g = F / m

Substitute F:

g = (GMm / r²) / m = GM / r²

2. Using g = GM / r²

• g decreases as 1 / r² from center of mass.
• Unit: N/kg (same as m/s²)

3. g is Constant Near Earth’s Surface

• For small changes in height, r ≈ R (Earth’s radius)
• g ≈ constant (≈ 9.81 m/s²)
• Justified because change in r is negligible compared to Earth’s radius (≈ 6.37 × 10⁶ m)

13.4 Gravitational Potential

1. Gravitational Potential (ϕ)

• ϕ is the work done per unit mass in bringing a small test mass from infinity to a point in the field.
• Scalar quantity
• Defined as zero at infinity

ϕ = W / m

2. Gravitational Potential from Point Mass

ϕ = –GM / r

• Negative: work must be done against gravity to escape the field
• Unit: J/kg

3. Gravitational Potential Energy

• Eₚ = m·ϕ = –GMm / r
• Represents the energy required to move mass m from r to infinity
• Eₚ is zero at infinity, negative at all finite distances

Key Concept:

• Closer to the mass → more negative potential energy
• Work done to move mass out of the field = gain in potential energy (less negative)