Topic 12: Gravitational Fields - Edexcel Physics A-level

Gravitational Fields

• Force Fields: Areas where an object experiences a non-contact force.
  • Represented as vectors (direction of force) and field lines (strength of force).
• Gravitational Field: A force field where objects with mass experience a force.

Gravitational Field Strength

• Types:
  • Uniform Field: Same gravitational force throughout.
    • Represented by parallel, equally spaced field lines.
  • Radial Field: Force depends on object position.
    • Force decreases as the object moves away from the center.
  • Near Earth's surface, the field is almost uniform.
• Gravitational Field Strength (g): Force per unit mass.
  • Constant in uniform fields, varies in radial fields.
  • Formula: (g = \frac{F}{m})

Newton's Law of Universal Gravitation

• Gravity acts on any objects with mass and is always attractive.
• Magnitude: Proportional to the product of masses, inversely proportional to the square of the distance between their centers.
  • Formula: (F = \frac{Gm_1m_2}{r^2})
  • G: Gravitational constant

Gravitational Field Strength in a Radial Field

• Derived using Newton's law and general formula:
  • Formula: (g = \frac{Gm_1}{r^2})
  • (m_1): Mass of the object creating the field

Gravitational Potential

• Gravitational Potential (V): Work done per unit mass from infinity to a point.
  • Potential at infinity is zero; energy released as potential energy is reduced.
  • Formula for radial field: (V = -\frac{GM}{r})
• Gravitational Potential Difference: Energy needed to move a unit mass between two points.
  • Work done: (W = m\Delta V)

Comparing Electric and Gravitational Fields

• Similarities:
  • Both follow inverse-square law.
  • Represented with field lines, can be uniform or radial.
  • Use similar equations for force and field strength.
• Differences:
  • Gravitational force is always attractive; electric can be repulsive.
  • Gravitational acts on mass; electric on charge.

Orbital Motion

• Kepler's Third Law: (T^2 \propto r^3)
  • (T): Orbital period
  • (r): Radius of orbit
• Derivation:
  1. Gravitational force acts as centripetal force for orbiting objects.
  2. Equate centripetal: (C = \frac{mv^2}{r}) and gravitational force: (G = \frac{GMm}{r^2}).
  3. Rearrange to find velocity: (v^2 = \frac{GM}{r}).
  4. Express velocity in terms of (r) and (T): (v = \frac{2\pi r}{T}).
  5. Substitute into the equation to derive Kepler's law.
• Apply Newton's laws to orbital motion; circular motion equations apply if orbit is circular.