Gravitational Fields & Orbits

Newton's law of gravitation, field strength, potential, and orbital mechanics.

Spec Points Covered
  • Define a gravitational fieldA region of space in which a mass experiences a gravitational force. and distinguish it from electric and magnetic fields.
  • Apply $g = F/m$ and $g = GM/r^{2}$ to calculate gravitational field strengthThe gravitational force per unit mass at a point in a gravitational fieldA region of space in which a mass experiences a gravitational force.. Measured in N kg⁻¹..
  • State and apply Newton's law of gravitation: $F = GMm/r^{2}$.
  • Define gravitational potentialThe work doneEnergy transferred when a force moves an object. In electrical circuits, W = QV (chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). times potential difference). per unit mass in bringing a small test mass from infinity to that point. Always negative. Measured in J kg⁻¹. and explain why it is always negative.
  • Calculate work doneEnergy transferred when a force moves an object. In electrical circuits, W = QV (chargeA property of matter that causes it to experience a force in an electromagnetic field. Measured in coulombs (C). times potential difference). using $\Delta W = m\Delta V$ and the full GPE expression.
  • Relate g to the gradient of a V-r graph and ΔV to the area under a g-r graph.
  • Derive and apply $v^{2} = GM/r$ and $T^{2} = 4\pi^{2} r^{3}/GM$ for circular orbits.
  • Derive escape velocityThe rate of change of displacement. A vector quantity. Measured in m s⁻¹.The minimum speed at which an object must be launched from the surface of a body to escape its gravitational field entirely (reach infinity with zero kinetic energyThe capacity to do work. Measured in joules (J).The energyThe capacity to do work. Measured in joules (J). an object possesses due to its motion.). from energyThe capacity to do work. Measured in joules (J). conservation.
  • State the defining features of geostationary and low-Earth orbits.
Notes
01 A force field is a region where a body feels a non-contact force Force field 3.7.1.1 → 02 A gravitational field is always attractive and has infinite range Gravitational field 3.7.1.1 → 03 Gravitational field strength g is force per unit mass Gravitational field strength 3.7.1.1 → 04 Radial fields point inward; uniform fields have parallel lines 3.7.1.1 → 05 Newton's law of gravitation: F = GMm/r² Newton's law of gravitation 3.7.2.1 → 06 Gravitational field strength in a radial field: g = GM/r² $g = \frac{GM}{r^2}$ 3.7.2.2 → 07 Gravitational potential is always negative Gravitational potential 3.7.2.3 → 08 Calculating gravitational potential: V = -GM/r $V = -\frac{GM}{r}$ 3.7.2.3 → 09 The V-r and g-r graphs are linked by gradient and area $g = -\frac{\Delta V}{\Delta r}$ 3.7.2.3 → 10 Work done moving a mass in a gravitational field: ΔW = mΔV $\Delta W = m \Delta V$ 3.7.2.3 → 11 Equipotential surfaces are perpendicular to field lines Equipotential surface 3.7.2.3 → 12 Circular orbits: gravitational force provides centripetal force 3.7.2.4 → 13 Kepler's third law: T² is proportional to r³ Kepler's third law 3.7.2.4 → 14 Satellite energy: total energy = KE + GPE stays constant in orbit 3.7.2.4 → 15 Escape velocity: the minimum speed to leave a gravitational field Escape velocity 3.7.2.4 → 16 Geostationary orbits have a fixed position above the equator Synchronous orbit 3.7.2.4 →
Σ Key Equations Full Reference →
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.
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.
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.
Q Retrieval Practice All 15 Questions →
Q1. Define a gravitational fieldA region of space in which a mass experiences a gravitational force..
A region of space where a mass experiences a force due to the gravitational attraction of another mass.
Q2. State the equation for gravitational field strengthThe gravitational force per unit mass at a point in a gravitational field. Measured in N kg⁻¹. and define the unit.
  • g = F/m.
  • Gravitational field strengthThe gravitational force per unit mass at a point in a gravitational field. Measured in N kg⁻¹. is the force per unit mass.
  • Measured in N kg⁻¹ (equivalent to m s⁻²).
Q3. State Newton's law of gravitationThe gravitational force between two point masses is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. in words and as an equation.
  • The gravitational force between two point masses is proportional to the product of their masses and inversely proportional to the square of their separation.
  • F = GMm/r².
Q4. What happens to the gravitational force when the distance between two masses is doubled?
  • It reduces to (1/2)² = 1/4 of its original value.
  • This is the inverse square law.
Q5. Write the equation for gravitational field strength in a radial field and state how it is derived.
  • g = GM/r².
  • Derived by combining g = F/m with F = GMm/r².
  • The test mass m cancels.