Key Equations

Gravitational Fields & Orbits — AQA A-Level Physics

On Data Sheet
Not on Data Sheet
Gravitational field strength (uniform field)
$$g = \frac{F}{m}$$
  • Where:
    • $g$ = gravitational field strength (N kg⁻¹)
    • $F$ = gravitational force / weight (N)
    • $m$ = mass (kg)
Definition of g. Also gives the acceleration due to gravity (m s⁻²).
Change in GPE (large distances)
$$\Delta \text{GPE} = GMm\left(\frac{1}{r_1} - \frac{1}{r_2}\right)$$
  • Where:
    • $M$ = mass producing the field (kg)
    • $m$ = mass being moved (kg)
    • $r₁$ = initial distance from centre (m)
    • $r₂$ = final distance from centre (m)
For large distances where g varies. Do not use GPE = mgΔh here.
Newton's law of gravitation
$$F = \frac{GMm}{r^2}$$
  • Where:
    • $F$ = gravitational force (N)
    • $G$ = Newton's gravitational constant (6.67 × 10⁻¹¹ N m² kg⁻²)
    • $M$ = mass of first body (kg)
    • $m$ = mass of second body (kg)
    • $r$ = distance between centres of mass (m)
Inverse square law. r is centre-to-centre, not surface-to-surface.
Orbital speed
$$v^2 = \frac{GM}{r}$$
  • Where:
    • $v$ = orbital speed (m s⁻¹)
    • $G$ = Newton's gravitational constant
    • $M$ = mass of central body (kg)
    • $r$ = orbital radius (m)
Derived by equating gravitational force to centripetal force. Speed is independent of satellite mass.
Gravitational field strength (radial field)
$$g = \frac{GM}{r^2}$$
  • Where:
    • $g$ = gravitational field strength (N kg⁻¹)
    • $G$ = Newton's gravitational constant
    • $M$ = mass producing the field (kg)
    • $r$ = distance from centre of mass (m)
Derived from g = F/m and F = GMm/r². Inverse square law.
Escape velocity
$$v = \sqrt{\frac{2GM}{r}}$$
  • Where:
    • $v$ = escape velocity (m s⁻¹)
    • $G$ = Newton's gravitational constant
    • $M$ = mass of body being escaped (kg)
    • $r$ = distance from centre of mass (m)
Derived from ½mv² = GMm/r. Independent of escaping object's mass. Not on data sheet.
Gravitational potential
$$V = -\frac{GM}{r}$$
  • Where:
    • $V$ = gravitational potential (J kg⁻¹)
    • $G$ = Newton's gravitational constant
    • $M$ = mass producing the field (kg)
    • $r$ = distance from centre of mass (m)
Always negative. Scalar quantity. Zero at infinity.
Kepler's third law (orbital period)
$$T^2 = \frac{4\pi^2 r^3}{GM}$$
  • Where:
    • $T$ = orbital period (s)
    • $r$ = orbital radius (m)
    • $G$ = Newton's gravitational constant
    • $M$ = mass of central body (kg)
T² ∝ r³. Derived from v² = GM/r and v = 2πr/T. Must be able to derive.
Field strength from potential gradient
$$g = -\frac{\Delta V}{\Delta r}$$
  • Where:
    • $g$ = gravitational field strength (N kg⁻¹)
    • $ΔV$ = change in gravitational potential (J kg⁻¹)
    • $Δr$ = change in distance (m)
g equals the negative gradient of a V-r graph. Area under g-r graph gives ΔV.
Work done in a gravitational field
$$\Delta W = m \Delta V$$
  • Where:
    • $ΔW$ = work done / change in GPE (J)
    • $m$ = mass being moved (kg)
    • $ΔV$ = change in gravitational potential (J kg⁻¹)
Equals the change in gravitational potential energy.
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