Gravitational Potential (Copy)

Definition of Gravitational Potential

• Gravitational potential (ϕ) at a point in a gravitational field is defined as:

  The work done per unit mass in bringing a small test mass from infinity to that point.

• It is a scalar quantity, unlike gravitational field strength (which is a vector).
• Mathematically:

  ϕ = W / m

  where:
  ϕ = gravitational potential (J/kg)
  W = work done in bringing mass m from infinity (J)
  m = test mass (kg)

• Since gravity is attractive, the work done is negative — energy is released as the mass is brought in.
• Therefore, gravitational potential is always negative in a gravitational field.

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Gravitational Potential in a Radial Field

• For a point mass M, the gravitational potential at a distance r from the centre is:

  ϕ = –GM / r

  where:
  G = gravitational constant = 6.674 × 10⁻¹¹ Nm²/kg²
  M = mass producing the field (kg)
  r = distance from the centre (m)

• This equation is derived by integrating the gravitational field strength:

  ϕ = –∫(g dr) = –∫(GM / r²) dr = –GM / r

• ϕ is negative and becomes more negative as r decreases.
• At infinity, ϕ = 0 (this is the reference point).

Units of Gravitational Potential

• From the definition ϕ = W / m:
  • Work done = joules (J)
  • Mass = kg
  • So gravitational potential has units of J/kg

Graph of ϕ vs r

• The graph of ϕ = –GM / r is:
  • Negative
  • Asymptotic to 0 as r → ∞
  • Becomes more negative as r decreases
  • Non-linear curve (hyperbolic shape)

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Gravitational Potential Energy (Eₚ)

• The gravitational potential energy of a mass m at a distance r from a point mass M is given by:

  Eₚ = mϕ = –GMm / r

  where:
  Eₚ = gravitational potential energy (J)
  G = gravitational constant
  M = source mass (kg)
  m = test mass (kg)
  r = distance between centres (m)

• Like potential, gravitational potential energy is also negative.
• Interpretation:
  • At infinity: Eₚ = 0
  • As the test mass approaches the source mass: Eₚ becomes more negative
  • Energy must be supplied (positive work done) to move the mass away from the gravitational source.

Gravitational Potential Difference and Work Done

• The work done in moving a mass m between two points in a gravitational field is:

  W = mΔϕ = m(ϕ₂ – ϕ₁)

• If a mass moves from a point of potential –20 J/kg to –10 J/kg:

  W = m(–10 – (–20)) = m × 10 → Positive work done = energy supplied to move mass outward

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Summary of Key Equations

Important Notes

• Both ϕ and Eₚ are negative because:
  • Work is released when moving from infinity towards a mass.
• The zero reference level is at infinity.
• For escape velocity calculations, we use:
  • Total energy at surface = Eₖ + Eₚ = 0
  • Hence, ½mv² – GMm / r = 0
  • Solve to get: v = √(2GM / r)

Example Problems

Example 1:
Calculate the gravitational potential 10,000 km from Earth’s centre (mass = 6.0 × 10²⁴ kg)

ϕ = –GM / r = –(6.674 × 10⁻¹¹ × 6.0 × 10²⁴) / (1.0 × 10⁷)
ϕ ≈ –4.0 × 10⁷ J/kg

Example 2:
A 500 kg satellite is in orbit at 7000 km from Earth’s centre. Find its gravitational potential energy.

Eₚ = –GMm / r = –(6.674 × 10⁻¹¹ × 6.0 × 10²⁴ × 500) / (7.0 × 10⁶)
Eₚ ≈ –2.86 × 10¹⁰ J

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course